------------------------------ One Mixed Nash Equiilbrium for von Neumann Poker with number of cards from 2 to, 40, and bet sizes from, 1, to, 10 By Shalosh B. Ekhad Player 1 and Player 2 each get (different) cards from a deck of n cards numb\ ered 1,..., n. They each see their own card, but not the other's They each put 1 dollar in the pot Player 1 can either check, in which case the cards are compared, and whoever\ has the larger card gets the pot (and hence wins 1 dollar), or bet an a\ dditional amount of b dollars Now Player 2 can decide to cut his losses, and fold, and the pot goes immedi\ ately to player 1, who is going to make a dollar, and player 2 is going \ to lose a dollar or else to also put b dollars in the pot Now the cards are compared, and whoever has the larger card gets the pot, wi\ nning b+1 dollars (and the other player loses b+1 dollars Here is ONE Mixed one, for card sizes from 2 to, 40, and bet sizes from, 1, to, 10 For each number of cards n, and bet size, we list: the Player 1 strategy, as a list of length n, let's call it, P1, telling th\ e player that if he got card number x, he should bet with prob. P1[x] an\ d check otherwise. the Player 2 strategy, as a list of length n, let's call it, P2, telling th\ e player that if he got card number y, he should call with prob. P2[y] a\ nd fold otherwise. We also list the value, that we call V -------------------------------------------- The deck has, 2, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 2, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2}, . They can \ see their own cards but, of course, not their opponent's. They each put \ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {2} If Player 2's card is in the following set then he should DEFINITELY fold {1} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 2, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2}, . They can \ see their own cards but, of course, not their opponent's. They each put \ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {2} If Player 2's card is in the following set then he should DEFINITELY fold {1} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 2, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2}, . They can \ see their own cards but, of course, not their opponent's. They each put \ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {2} If Player 2's card is in the following set then he should DEFINITELY fold {1} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 2, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2}, . They can \ see their own cards but, of course, not their opponent's. They each put \ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {2} If Player 2's card is in the following set then he should DEFINITELY fold {1} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 2, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2}, . They can \ see their own cards but, of course, not their opponent's. They each put \ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {2} If Player 2's card is in the following set then he should DEFINITELY fold {1} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 2, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2}, . They can \ see their own cards but, of course, not their opponent's. They each put \ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {2} If Player 2's card is in the following set then he should DEFINITELY fold {1} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 2, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2}, . They can \ see their own cards but, of course, not their opponent's. They each put \ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {2} If Player 2's card is in the following set then he should DEFINITELY fold {1} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 2, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2}, . They can \ see their own cards but, of course, not their opponent's. They each put \ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {2} If Player 2's card is in the following set then he should DEFINITELY fold {1} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 2, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2}, . They can \ see their own cards but, of course, not their opponent's. They each put \ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {2} If Player 2's card is in the following set then he should DEFINITELY fold {1} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 2, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2}, . They can \ see their own cards but, of course, not their opponent's. They each put \ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {2} If Player 2's card is in the following set then he should DEFINITELY fold {1} -------------------------------------------- The deck has, 3, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 3, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/18, and in decimal, 0.05555555556 If Player 1's card is in the following set then he should DEFINITELY check {2} If Player 1's card is in the following set then he should DEFINITELY bet {3} If his card is, 1, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {3} If Player 2's card is in the following set then he should DEFINITELY fold {1} If his card is, 2, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 3, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {3} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 3, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {3} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 3, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {3} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 3, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {3} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 3, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {3} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 3, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {3} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 3, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {3} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 3, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {3} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 3, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {3} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} -------------------------------------------- The deck has, 4, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 4, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4}, . The\ y can see their own cards but, of course, not their opponent's. They eac\ h put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/12, and in decimal, 0.08333333333 If Player 1's card is in the following set then he should DEFINITELY check {2, 3} If Player 1's card is in the following set then he should DEFINITELY bet {4} If his card is, 1, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {3, 4} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 4, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4}, . The\ y can see their own cards but, of course, not their opponent's. They eac\ h put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/12, and in decimal, 0.08333333333 If Player 1's card is in the following set then he should DEFINITELY check {2, 3} If Player 1's card is in the following set then he should DEFINITELY bet {4} If his card is, 1, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {4} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} If his card is, 3, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 4, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4}, . The\ y can see their own cards but, of course, not their opponent's. They eac\ h put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/20, and in decimal, 0.05000000000 If Player 1's card is in the following set then he should DEFINITELY check {2, 3} If Player 1's card is in the following set then he should DEFINITELY bet {4} If his card is, 1, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {4} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} If his card is, 3, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 4, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4}, . The\ y can see their own cards but, of course, not their opponent's. They eac\ h put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {4} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 4, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4}, . The\ y can see their own cards but, of course, not their opponent's. They eac\ h put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {4} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 4, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4}, . The\ y can see their own cards but, of course, not their opponent's. They eac\ h put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {4} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 4, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4}, . The\ y can see their own cards but, of course, not their opponent's. They eac\ h put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {4} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 4, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4}, . The\ y can see their own cards but, of course, not their opponent's. They eac\ h put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {4} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 4, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4}, . The\ y can see their own cards but, of course, not their opponent's. They eac\ h put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {4} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 4, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4}, . The\ y can see their own cards but, of course, not their opponent's. They eac\ h put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {4} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} -------------------------------------------- The deck has, 5, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 5, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/12, and in decimal, 0.08333333333 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {5} If his card is, 1, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {4, 5} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} If his card is, 3, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 5, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {5} If his card is, 1, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {4, 5} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 5, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 9/100, and in decimal, 0.09000000000 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {5} If his card is, 1, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {5} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} If his card is, 4, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 5, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/15, and in decimal, 0.06666666667 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {5} If his card is, 1, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {5} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} If his card is, 4, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 5, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/28, and in decimal, 0.03571428571 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {5} If his card is, 1, then he should bet with probability, 5/7, and check with probability, 2/7 If Player 2's card is in the following set then he should DEFINITELY call {5} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} If his card is, 4, then he should call with probability, 1/7, and fold with probability, 6/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 5, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4, 5} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {5} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 5, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4, 5} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {5} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 5, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4, 5} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {5} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 5, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4, 5} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {5} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 5, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4, 5} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {5} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} -------------------------------------------- The deck has, 6, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 6, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 4/45, and in decimal, 0.08888888889 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4} If Player 1's card is in the following set then he should DEFINITELY bet {5, 6} If his card is, 1, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {4, 5, 6} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2} If his card is, 3, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 6, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5} If Player 1's card is in the following set then he should DEFINITELY bet {6} If his card is, 1, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {5, 6} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} If his card is, 4, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 6, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5} If Player 1's card is in the following set then he should DEFINITELY bet {6} If his card is, 1, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {5, 6} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 6, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 4/45, and in decimal, 0.08888888889 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5} If Player 1's card is in the following set then he should DEFINITELY bet {6} If his card is, 1, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {6} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} If his card is, 5, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 6, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/14, and in decimal, 0.07142857143 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5} If Player 1's card is in the following set then he should DEFINITELY bet {6} If his card is, 1, then he should bet with probability, 5/7, and check with probability, 2/7 If Player 2's card is in the following set then he should DEFINITELY call {6} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} If his card is, 5, then he should call with probability, 3/7, and fold with probability, 4/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 6, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/20, and in decimal, 0.05000000000 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5} If Player 1's card is in the following set then he should DEFINITELY bet {6} If his card is, 1, then he should bet with probability, 3/4, and check with probability, 1/4 If Player 2's card is in the following set then he should DEFINITELY call {6} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} If his card is, 5, then he should call with probability, 1/4, and fold with probability, 3/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 6, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 7/270, and in decimal, 0.02592592593 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5} If Player 1's card is in the following set then he should DEFINITELY bet {6} If his card is, 1, then he should bet with probability, 7/9, and check with probability, 2/9 If Player 2's card is in the following set then he should DEFINITELY call {6} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} If his card is, 5, then he should call with probability, 1/9, and fold with probability, 8/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 6, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {6} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 6, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {6} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 6, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {6} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} -------------------------------------------- The deck has, 7, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 7, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 2/21, and in decimal, 0.09523809524 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5} If Player 1's card is in the following set then he should DEFINITELY bet {6, 7} If his card is, 1, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {4, 5, 6, 7} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 7, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 2/21, and in decimal, 0.09523809524 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {7} If his card is, 1, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {5, 6, 7} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 7, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {7} If his card is, 1, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {6, 7} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} If his card is, 5, then he should call with probability, 2/5, and fold with probability, 3/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 7, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 2/21, and in decimal, 0.09523809524 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {7} If his card is, 1, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {6, 7} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 7, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 25 The value (to Player1) of the game is, ---, and in decimal, 0.08503401361 294 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {7} If his card is, 1, then he should bet with probability, 5/7, and check with probability, 2/7 If Player 2's card is in the following set then he should DEFINITELY call {7} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} If his card is, 6, then he should call with probability, 5/7, and fold with probability, 2/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 7, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/14, and in decimal, 0.07142857143 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {7} If his card is, 1, then he should bet with probability, 3/4, and check with probability, 1/4 If Player 2's card is in the following set then he should DEFINITELY call {7} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} If his card is, 6, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 7, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/18, and in decimal, 0.05555555556 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {7} If his card is, 1, then he should bet with probability, 7/9, and check with probability, 2/9 If Player 2's card is in the following set then he should DEFINITELY call {7} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} If his card is, 6, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 7, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 4/105, and in decimal, 0.03809523810 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {7} If his card is, 1, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {7} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} If his card is, 6, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 7, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 3/154, and in decimal, 0.01948051948 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {7} If his card is, 1, then he should bet with probability, 9/11, and check with probability, 2/11 If Player 2's card is in the following set then he should DEFINITELY call {7} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} If his card is, 6, then he should call with probability, 1/11, 10 and fold with probability, -- 11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 7, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 0, and in decimal, 0. If Player 1's card is in the following set then he should DEFINITELY check {1, 2, 3, 4, 5, 6, 7} If Player 1's card is in the following set then he should DEFINITELY bet {} If Player 2's card is in the following set then he should DEFINITELY call {7} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} -------------------------------------------- The deck has, 8, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 8, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8}, . They can see their own cards but, of course, \ not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 2/21, and in decimal, 0.09523809524 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {7, 8} If his card is, 1, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {5, 6, 7, 8} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} If his card is, 4, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 8, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8}, . They can see their own cards but, of course, \ not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 3/28, and in decimal, 0.1071428571 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {1, 7, 8} If Player 2's card is in the following set then he should DEFINITELY call {6, 7, 8} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} If his card is, 5, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 8, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8}, . They can see their own cards but, of course, \ not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 27 The value (to Player1) of the game is, ---, and in decimal, 0.09642857143 280 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7} If Player 1's card is in the following set then he should DEFINITELY bet {8} If his card is, 1, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {7, 8} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} If his card is, 6, then he should call with probability, 4/5, and fold with probability, 1/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 8, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8}, . They can see their own cards but, of course, \ not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 2/21, and in decimal, 0.09523809524 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7} If Player 1's card is in the following set then he should DEFINITELY bet {8} If his card is, 1, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {7, 8} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} If his card is, 6, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 8, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8}, . They can see their own cards but, of course, \ not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 5/56, and in decimal, 0.08928571429 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7} If Player 1's card is in the following set then he should DEFINITELY bet {8} If his card is, 1, then he should bet with probability, 5/7, and check with probability, 2/7 If Player 2's card is in the following set then he should DEFINITELY call {7, 8} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 8, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8}, . They can see their own cards but, of course, \ not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 9/112, and in decimal, 0.08035714286 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7} If Player 1's card is in the following set then he should DEFINITELY bet {8} If his card is, 1, then he should bet with probability, 3/4, and check with probability, 1/4 If Player 2's card is in the following set then he should DEFINITELY call {8} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} If his card is, 7, then he should call with probability, 3/4, and fold with probability, 1/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 8, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8}, . They can see their own cards but, of course, \ not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 5/72, and in decimal, 0.06944444444 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7} If Player 1's card is in the following set then he should DEFINITELY bet {8} If his card is, 1, then he should bet with probability, 7/9, and check with probability, 2/9 If Player 2's card is in the following set then he should DEFINITELY call {8} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} If his card is, 7, then he should call with probability, 5/9, and fold with probability, 4/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 8, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8}, . They can see their own cards but, of course, \ not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 2/35, and in decimal, 0.05714285714 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7} If Player 1's card is in the following set then he should DEFINITELY bet {8} If his card is, 1, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {8} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} If his card is, 7, then he should call with probability, 2/5, and fold with probability, 3/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 8, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8}, . They can see their own cards but, of course, \ not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 27 The value (to Player1) of the game is, ---, and in decimal, 0.04383116883 616 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7} If Player 1's card is in the following set then he should DEFINITELY bet {8} If his card is, 1, then he should bet with probability, 9/11, and check with probability, 2/11 If Player 2's card is in the following set then he should DEFINITELY call {8} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} If his card is, 7, then he should call with probability, 3/11, and fold with probability, 8/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 8, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8}, . They can see their own cards but, of course, \ not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 5/168, and in decimal, 0.02976190476 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7} If Player 1's card is in the following set then he should DEFINITELY bet {8} If his card is, 1, then he should bet with probability, 5/6, and check with probability, 1/6 If Player 2's card is in the following set then he should DEFINITELY call {8} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} If his card is, 7, then he should call with probability, 1/6, and fold with probability, 5/6 -------------------------------------------- The deck has, 9, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 9, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 7/72, and in decimal, 0.09722222222 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6} If Player 1's card is in the following set then he should DEFINITELY bet {1, 7, 8, 9} If Player 2's card is in the following set then he should DEFINITELY call {5, 6, 7, 8, 9} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3} If his card is, 4, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 9, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/9, and in decimal, 0.1111111111 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7} If Player 1's card is in the following set then he should DEFINITELY bet {1, 8, 9} If Player 2's card is in the following set then he should DEFINITELY call {6, 7, 8, 9} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 9, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 7/72, and in decimal, 0.09722222222 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7} If Player 1's card is in the following set then he should DEFINITELY bet {1, 9} If his card is, 8, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {7, 8, 9} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 9, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 5/54, and in decimal, 0.09259259259 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8} If Player 1's card is in the following set then he should DEFINITELY bet {9} If his card is, 1, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {8, 9} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} If his card is, 7, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 9, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 5/56, and in decimal, 0.08928571429 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8} If Player 1's card is in the following set then he should DEFINITELY bet {9} If his card is, 1, then he should bet with probability, 5/7, and check with probability, 2/7 If Player 2's card is in the following set then he should DEFINITELY call {8, 9} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} If his card is, 7, then he should call with probability, 2/7, and fold with probability, 5/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 9, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/12, and in decimal, 0.08333333333 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8} If Player 1's card is in the following set then he should DEFINITELY bet {9} If his card is, 1, then he should bet with probability, 3/4, and check with probability, 1/4 If Player 2's card is in the following set then he should DEFINITELY call {8, 9} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 9, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 49 The value (to Player1) of the game is, ---, and in decimal, 0.07561728395 648 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8} If Player 1's card is in the following set then he should DEFINITELY bet {9} If his card is, 1, then he should bet with probability, 7/9, and check with probability, 2/9 If Player 2's card is in the following set then he should DEFINITELY call {9} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} If his card is, 8, then he should call with probability, 7/9, and fold with probability, 2/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 9, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/15, and in decimal, 0.06666666667 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8} If Player 1's card is in the following set then he should DEFINITELY bet {9} If his card is, 1, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {9} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} If his card is, 8, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 9, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 5/88, and in decimal, 0.05681818182 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8} If Player 1's card is in the following set then he should DEFINITELY bet {9} If his card is, 1, then he should bet with probability, 9/11, and check with probability, 2/11 If Player 2's card is in the following set then he should DEFINITELY call {9} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} If his card is, 8, then he should call with probability, 5/11, and fold with probability, 6/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 9, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 5/108, and in decimal, 0.04629629630 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8} If Player 1's card is in the following set then he should DEFINITELY bet {9} If his card is, 1, then he should bet with probability, 5/6, and check with probability, 1/6 If Player 2's card is in the following set then he should DEFINITELY call {9} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} If his card is, 8, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- The deck has, 10, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 10, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7} If Player 1's card is in the following set then he should DEFINITELY bet {1, 8, 9, 10} If Player 2's card is in the following set then he should DEFINITELY call {5, 6, 7, 8, 9, 10} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 10, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/9, and in decimal, 0.1111111111 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8} If Player 1's card is in the following set then he should DEFINITELY bet {1, 9, 10} If Player 2's card is in the following set then he should DEFINITELY call {7, 8, 9, 10} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} If his card is, 6, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 10, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 23 The value (to Player1) of the game is, ---, and in decimal, 0.1022222222 225 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8} If Player 1's card is in the following set then he should DEFINITELY bet {1, 9, 10} If his card is, 2, then he should bet with probability, 1/5, and check with probability, 4/5 If Player 2's card is in the following set then he should DEFINITELY call {8, 9, 10} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} If his card is, 7, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 10, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 4/45, and in decimal, 0.08888888889 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9} If Player 1's card is in the following set then he should DEFINITELY bet {10} If his card is, 1, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {8, 9, 10} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 10, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 11 The value (to Player1) of the game is, ---, and in decimal, 0.08730158730 126 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9} If Player 1's card is in the following set then he should DEFINITELY bet {10} If his card is, 1, then he should bet with probability, 5/7, and check with probability, 2/7 If Player 2's card is in the following set then he should DEFINITELY call {9, 10} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} If his card is, 8, then he should call with probability, 4/7, and fold with probability, 3/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 10, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/12, and in decimal, 0.08333333333 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9} If Player 1's card is in the following set then he should DEFINITELY bet {10} If his card is, 1, then he should bet with probability, 3/4, and check with probability, 1/4 If Player 2's card is in the following set then he should DEFINITELY call {9, 10} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} If his card is, 8, then he should call with probability, 1/4, and fold with probability, 3/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 10, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 7/90, and in decimal, 0.07777777778 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9} If Player 1's card is in the following set then he should DEFINITELY bet {10} If his card is, 1, then he should bet with probability, 7/9, and check with probability, 2/9 If Player 2's card is in the following set then he should DEFINITELY call {9, 10} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 10, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 16 The value (to Player1) of the game is, ---, and in decimal, 0.07111111111 225 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9} If Player 1's card is in the following set then he should DEFINITELY bet {10} If his card is, 1, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {10} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} If his card is, 9, then he should call with probability, 4/5, and fold with probability, 1/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 10, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 7/110, and in decimal, 0.06363636364 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9} If Player 1's card is in the following set then he should DEFINITELY bet {10} If his card is, 1, then he should bet with probability, 9/11, and check with probability, 2/11 If Player 2's card is in the following set then he should DEFINITELY call {10} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} If his card is, 9, then he should call with probability, 7/11, and fold with probability, 4/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 10, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/18, and in decimal, 0.05555555556 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9} If Player 1's card is in the following set then he should DEFINITELY bet {10} If his card is, 1, then he should bet with probability, 5/6, and check with probability, 1/6 If Player 2's card is in the following set then he should DEFINITELY call {10} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} If his card is, 9, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- The deck has, 11, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 11, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8} If Player 1's card is in the following set then he should DEFINITELY bet {1, 9, 10, 11} If Player 2's card is in the following set then he should DEFINITELY call {6, 7, 8, 9, 10, 11} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4} If his card is, 5, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 11, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 6/55, and in decimal, 0.1090909091 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9} If Player 1's card is in the following set then he should DEFINITELY bet {1, 10, 11} If Player 2's card is in the following set then he should DEFINITELY call {7, 8, 9, 10, 11} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 11, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 29 The value (to Player1) of the game is, ---, and in decimal, 0.1054545455 275 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9} If Player 1's card is in the following set then he should DEFINITELY bet {1, 10, 11} If his card is, 2, then he should bet with probability, 1/5, and check with probability, 4/5 If Player 2's card is in the following set then he should DEFINITELY call {9, 10, 11} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} If his card is, 8, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 11, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/11, and in decimal, 0.09090909091 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9} If Player 1's card is in the following set then he should DEFINITELY bet {1, 11} If his card is, 10, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {9, 10, 11} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 11, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 13 The value (to Player1) of the game is, ---, and in decimal, 0.08441558442 154 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10} If Player 1's card is in the following set then he should DEFINITELY bet {11} If his card is, 1, then he should bet with probability, 5/7, and check with probability, 2/7 If Player 2's card is in the following set then he should DEFINITELY call {10, 11} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} If his card is, 9, then he should call with probability, 6/7, and fold with probability, 1/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 11, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 9/110, and in decimal, 0.08181818182 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10} If Player 1's card is in the following set then he should DEFINITELY bet {11} If his card is, 1, then he should bet with probability, 3/4, and check with probability, 1/4 If Player 2's card is in the following set then he should DEFINITELY call {10, 11} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} If his card is, 9, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 11, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 7/90, and in decimal, 0.07777777778 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10} If Player 1's card is in the following set then he should DEFINITELY bet {11} If his card is, 1, then he should bet with probability, 7/9, and check with probability, 2/9 If Player 2's card is in the following set then he should DEFINITELY call {10, 11} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} If his card is, 9, then he should call with probability, 2/9, and fold with probability, 7/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 11, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 4/55, and in decimal, 0.07272727273 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10} If Player 1's card is in the following set then he should DEFINITELY bet {11} If his card is, 1, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {10, 11} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 11, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 81 The value (to Player1) of the game is, ----, and in decimal, 0.06694214876 1210 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10} If Player 1's card is in the following set then he should DEFINITELY bet {11} If his card is, 1, then he should bet with probability, 9/11, and check with probability, 2/11 If Player 2's card is in the following set then he should DEFINITELY call {11} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} If his card is, 10, then he should call with probability, 9/11, and fold with probability, 2/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 11, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 2/33, and in decimal, 0.06060606061 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10} If Player 1's card is in the following set then he should DEFINITELY bet {11} If his card is, 1, then he should bet with probability, 5/6, and check with probability, 1/6 If Player 2's card is in the following set then he should DEFINITELY call {11} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} If his card is, 10, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- The deck has, 12, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 12, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 13 The value (to Player1) of the game is, ---, and in decimal, 0.09848484848 132 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9} If Player 1's card is in the following set then he should DEFINITELY bet {1, 10, 11, 12} If Player 2's card is in the following set then he should DEFINITELY call {6, 7, 8, 9, 10, 11, 12} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 12, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 7/66, and in decimal, 0.1060606061 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10} If Player 1's card is in the following set then he should DEFINITELY bet {1, 11, 12} If Player 2's card is in the following set then he should DEFINITELY call {8, 9, 10, 11, 12} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 12, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 7/66, and in decimal, 0.1060606061 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10} If Player 1's card is in the following set then he should DEFINITELY bet {1, 11, 12} If his card is, 2, then he should bet with probability, 1/5, and check with probability, 4/5 If Player 2's card is in the following set then he should DEFINITELY call {9, 10, 11, 12} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 12, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 19 The value (to Player1) of the game is, ---, and in decimal, 0.09595959596 198 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10} If Player 1's card is in the following set then he should DEFINITELY bet {1, 11, 12} If his card is, 2, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {10, 11, 12} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} If his card is, 9, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 12, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/12, and in decimal, 0.08333333333 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10} If Player 1's card is in the following set then he should DEFINITELY bet {1, 12} If his card is, 11, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {10, 11, 12} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 12, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 7/88, and in decimal, 0.07954545455 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If Player 1's card is in the following set then he should DEFINITELY bet {12} If his card is, 1, then he should bet with probability, 3/4, and check with probability, 1/4 If Player 2's card is in the following set then he should DEFINITELY call {11, 12} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} If his card is, 10, then he should call with probability, 3/4, and fold with probability, 1/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 12, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 91 The value (to Player1) of the game is, ----, and in decimal, 0.07659932660 1188 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If Player 1's card is in the following set then he should DEFINITELY bet {12} If his card is, 1, then he should bet with probability, 7/9, and check with probability, 2/9 If Player 2's card is in the following set then he should DEFINITELY call {11, 12} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} If his card is, 10, then he should call with probability, 4/9, and fold with probability, 5/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 12, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 4/55, and in decimal, 0.07272727273 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If Player 1's card is in the following set then he should DEFINITELY bet {12} If his card is, 1, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {11, 12} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} If his card is, 10, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 12, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 3/44, and in decimal, 0.06818181818 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If Player 1's card is in the following set then he should DEFINITELY bet {12} If his card is, 1, then he should bet with probability, 9/11, and check with probability, 2/11 If Player 2's card is in the following set then he should DEFINITELY call {11, 12} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 12, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 25 The value (to Player1) of the game is, ---, and in decimal, 0.06313131313 396 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If Player 1's card is in the following set then he should DEFINITELY bet {12} If his card is, 1, then he should bet with probability, 5/6, and check with probability, 1/6 If Player 2's card is in the following set then he should DEFINITELY call {12} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} If his card is, 11, then he should call with probability, 5/6, and fold with probability, 1/6 -------------------------------------------- The deck has, 13, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 13, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 23 The value (to Player1) of the game is, ---, and in decimal, 0.09829059829 234 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9} If Player 1's card is in the following set then he should DEFINITELY bet {1, 10, 11, 12, 13} If his card is, 2, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {7, 8, 9, 10, 11, 12, 13} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5} If his card is, 6, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 13, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 17 The value (to Player1) of the game is, ---, and in decimal, 0.1089743590 156 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10} If Player 1's card is in the following set then he should DEFINITELY bet {1, 11, 12, 13} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {9, 10, 11, 12, 13} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} If his card is, 8, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 13, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 41 The value (to Player1) of the game is, ---, and in decimal, 0.1051282051 390 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11} If Player 1's card is in the following set then he should DEFINITELY bet {1, 12, 13} If his card is, 2, then he should bet with probability, 1/5, and check with probability, 4/5 If Player 2's card is in the following set then he should DEFINITELY call {10, 11, 12, 13} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} If his card is, 9, then he should call with probability, 2/5, and fold with probability, 3/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 13, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 23 The value (to Player1) of the game is, ---, and in decimal, 0.09829059829 234 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11} If Player 1's card is in the following set then he should DEFINITELY bet {1, 12, 13} If his card is, 2, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {11, 12, 13} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} If his card is, 10, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 13, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 47 The value (to Player1) of the game is, ---, and in decimal, 0.08608058608 546 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11} If Player 1's card is in the following set then he should DEFINITELY bet {1, 12, 13} If his card is, 2, then he should bet with probability, 3/7, and check with probability, 4/7 If Player 2's card is in the following set then he should DEFINITELY call {11, 12, 13} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} If his card is, 10, then he should call with probability, 1/7, and fold with probability, 6/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 13, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/13, and in decimal, 0.07692307692 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If Player 1's card is in the following set then he should DEFINITELY bet {13} If his card is, 1, then he should bet with probability, 3/4, and check with probability, 1/4 If Player 2's card is in the following set then he should DEFINITELY call {11, 12, 13} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 13, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 35 The value (to Player1) of the game is, ---, and in decimal, 0.07478632479 468 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If Player 1's card is in the following set then he should DEFINITELY bet {13} If his card is, 1, then he should bet with probability, 7/9, and check with probability, 2/9 If Player 2's card is in the following set then he should DEFINITELY call {12, 13} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} If his card is, 11, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 13, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 14 The value (to Player1) of the game is, ---, and in decimal, 0.07179487179 195 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If Player 1's card is in the following set then he should DEFINITELY bet {13} If his card is, 1, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {12, 13} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} If his card is, 11, then he should call with probability, 2/5, and fold with probability, 3/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 13, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 3/44, and in decimal, 0.06818181818 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If Player 1's card is in the following set then he should DEFINITELY bet {13} If his card is, 1, then he should bet with probability, 9/11, and check with probability, 2/11 If Player 2's card is in the following set then he should DEFINITELY call {12, 13} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} If his card is, 11, then he should call with probability, 2/11, and fold with probability, 9/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 13, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 5/78, and in decimal, 0.06410256410 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If Player 1's card is in the following set then he should DEFINITELY bet {13} If his card is, 1, then he should bet with probability, 5/6, and check with probability, 1/6 If Player 2's card is in the following set then he should DEFINITELY call {12, 13} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} -------------------------------------------- The deck has, 14, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 14, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 9/91, and in decimal, 0.09890109890 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10} If Player 1's card is in the following set then he should DEFINITELY bet {1, 11, 12, 13, 14} If his card is, 2, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {7, 8, 9, 10, 11, 12, 13, 14} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 14, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 10 The value (to Player1) of the game is, --, and in decimal, 0.1098901099 91 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11} If Player 1's card is in the following set then he should DEFINITELY bet {1, 12, 13, 14} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {9, 10, 11, 12, 13, 14} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 14, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 47 The value (to Player1) of the game is, ---, and in decimal, 0.1032967033 455 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If Player 1's card is in the following set then he should DEFINITELY bet {1, 13, 14} If his card is, 2, then he should bet with probability, 1/5, and check with probability, 4/5 If Player 2's card is in the following set then he should DEFINITELY call {11, 12, 13, 14} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} If his card is, 10, then he should call with probability, 4/5, and fold with probability, 1/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 14, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 9/91, and in decimal, 0.09890109890 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If Player 1's card is in the following set then he should DEFINITELY bet {1, 13, 14} If his card is, 2, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {11, 12, 13, 14} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 14, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 57 The value (to Player1) of the game is, ---, and in decimal, 0.08948194662 637 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If Player 1's card is in the following set then he should DEFINITELY bet {1, 13, 14} If his card is, 2, then he should bet with probability, 3/7, and check with probability, 4/7 If Player 2's card is in the following set then he should DEFINITELY call {12, 13, 14} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} If his card is, 11, then he should call with probability, 3/7, and fold with probability, 4/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 14, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/13, and in decimal, 0.07692307692 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If Player 1's card is in the following set then he should DEFINITELY bet {1, 14} If his card is, 13, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {12, 13, 14} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 14, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 17 The value (to Player1) of the game is, ---, and in decimal, 0.07264957265 234 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {14} If his card is, 1, then he should bet with probability, 7/9, and check with probability, 2/9 If Player 2's card is in the following set then he should DEFINITELY call {13, 14} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If his card is, 12, then he should call with probability, 8/9, and fold with probability, 1/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 14, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 32 The value (to Player1) of the game is, ---, and in decimal, 0.07032967033 455 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {14} If his card is, 1, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {13, 14} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If his card is, 12, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 14, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 135 The value (to Player1) of the game is, ----, and in decimal, 0.06743256743 2002 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {14} If his card is, 1, then he should bet with probability, 9/11, and check with probability, 2/11 If Player 2's card is in the following set then he should DEFINITELY call {13, 14} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If his card is, 12, then he should call with probability, 4/11, and fold with probability, 7/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 14, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 5/78, and in decimal, 0.06410256410 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {14} If his card is, 1, then he should bet with probability, 5/6, and check with probability, 1/6 If Player 2's card is in the following set then he should DEFINITELY call {13, 14} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If his card is, 12, then he should call with probability, 1/6, and fold with probability, 5/6 -------------------------------------------- The deck has, 15, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 15, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 31 The value (to Player1) of the game is, ---, and in decimal, 0.09841269841 315 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11} If Player 1's card is in the following set then he should DEFINITELY bet {1, 12, 13, 14, 15} If his card is, 2, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {8, 9, 10, 11, 12, 13, 14, 15} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} If his card is, 7, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 15, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 23 The value (to Player1) of the game is, ---, and in decimal, 0.1095238095 210 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If Player 1's card is in the following set then he should DEFINITELY bet {1, 13, 14, 15} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {10, 11, 12, 13, 14, 15} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} If his card is, 9, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 15, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 109 The value (to Player1) of the game is, ----, and in decimal, 0.1038095238 1050 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If Player 1's card is in the following set then he should DEFINITELY bet {1, 13, 14, 15} If his card is, 2, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {11, 12, 13, 14, 15} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} If his card is, 10, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 15, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 31 The value (to Player1) of the game is, ---, and in decimal, 0.09841269841 315 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {1, 14, 15} If his card is, 2, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {12, 13, 14, 15} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} If his card is, 11, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 15, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 67 The value (to Player1) of the game is, ---, and in decimal, 0.09115646258 735 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {1, 14, 15} If his card is, 2, then he should bet with probability, 3/7, and check with probability, 4/7 If Player 2's card is in the following set then he should DEFINITELY call {13, 14, 15} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If his card is, 12, then he should call with probability, 5/7, and fold with probability, 2/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 15, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 17 The value (to Player1) of the game is, ---, and in decimal, 0.08095238095 210 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {1, 14, 15} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {13, 14, 15} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If his card is, 12, then he should call with probability, 1/4, and fold with probability, 3/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 15, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/14, and in decimal, 0.07142857143 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {1, 15} If his card is, 14, then he should bet with probability, 2/7, and check with probability, 5/7 If Player 2's card is in the following set then he should DEFINITELY call {13, 14, 15} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 15, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 12 The value (to Player1) of the game is, ---, and in decimal, 0.06857142857 175 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If Player 1's card is in the following set then he should DEFINITELY bet {15} If his card is, 1, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {14, 15} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If his card is, 13, then he should call with probability, 4/5, and fold with probability, 1/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 15, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 51 The value (to Player1) of the game is, ---, and in decimal, 0.06623376623 770 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If Player 1's card is in the following set then he should DEFINITELY bet {15} If his card is, 1, then he should bet with probability, 9/11, and check with probability, 2/11 If Player 2's card is in the following set then he should DEFINITELY call {14, 15} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If his card is, 13, then he should call with probability, 6/11, and fold with probability, 5/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 15, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 4/63, and in decimal, 0.06349206349 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If Player 1's card is in the following set then he should DEFINITELY bet {15} If his card is, 1, then he should bet with probability, 5/6, and check with probability, 1/6 If Player 2's card is in the following set then he should DEFINITELY call {14, 15} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If his card is, 13, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- The deck has, 16, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 16, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 71 The value (to Player1) of the game is, ---, and in decimal, 0.09861111111 720 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11} If Player 1's card is in the following set then he should DEFINITELY bet {1, 12, 13, 14, 15, 16} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6} If his card is, 7, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 16, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 13 The value (to Player1) of the game is, ---, and in decimal, 0.1083333333 120 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {1, 14, 15, 16} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {10, 11, 12, 13, 14, 15, 16} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 16, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 127 The value (to Player1) of the game is, ----, and in decimal, 0.1058333333 1200 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {1, 14, 15, 16} If his card is, 2, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {12, 13, 14, 15, 16} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} If his card is, 11, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 16, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 7/72, and in decimal, 0.09722222222 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If Player 1's card is in the following set then he should DEFINITELY bet {1, 15, 16} If his card is, 2, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {13, 14, 15, 16} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If his card is, 12, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 16, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 11 The value (to Player1) of the game is, ---, and in decimal, 0.09166666667 120 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If Player 1's card is in the following set then he should DEFINITELY bet {1, 15, 16} If his card is, 2, then he should bet with probability, 3/7, and check with probability, 4/7 If Player 2's card is in the following set then he should DEFINITELY call {13, 14, 15, 16} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 16, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/12, and in decimal, 0.08333333333 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If Player 1's card is in the following set then he should DEFINITELY bet {1, 15, 16} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {14, 15, 16} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If his card is, 13, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 16, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 79 The value (to Player1) of the game is, ----, and in decimal, 0.07314814815 1080 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If Player 1's card is in the following set then he should DEFINITELY bet {1, 15, 16} If his card is, 2, then he should bet with probability, 5/9, and check with probability, 4/9 If Player 2's card is in the following set then he should DEFINITELY call {14, 15, 16} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If his card is, 13, then he should call with probability, 1/9, and fold with probability, 8/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 16, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/15, and in decimal, 0.06666666667 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If Player 1's card is in the following set then he should DEFINITELY bet {16} If his card is, 1, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {14, 15, 16} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 16, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 57 The value (to Player1) of the game is, ---, and in decimal, 0.06477272727 880 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If Player 1's card is in the following set then he should DEFINITELY bet {16} If his card is, 1, then he should bet with probability, 9/11, and check with probability, 2/11 If Player 2's card is in the following set then he should DEFINITELY call {15, 16} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If his card is, 14, then he should call with probability, 8/11, and fold with probability, 3/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 16, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/16, and in decimal, 0.06250000000 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If Player 1's card is in the following set then he should DEFINITELY bet {16} If his card is, 1, then he should bet with probability, 5/6, and check with probability, 1/6 If Player 2's card is in the following set then he should DEFINITELY call {15, 16} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If his card is, 14, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- The deck has, 17, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 17, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, . They can s\ ee their own cards but, of course, not their opponent's. They each put 1\ dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 27 The value (to Player1) of the game is, ---, and in decimal, 0.09926470588 272 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If Player 1's card is in the following set then he should DEFINITELY bet {1, 13, 14, 15, 16, 17} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 17, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, . They can s\ ee their own cards but, of course, not their opponent's. They each put 1\ dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 15 The value (to Player1) of the game is, ---, and in decimal, 0.1102941176 136 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 14, 15, 16, 17} If Player 2's card is in the following set then he should DEFINITELY call {11, 12, 13, 14, 15, 16, 17} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} If his card is, 10, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 17, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, . They can s\ ee their own cards but, of course, not their opponent's. They each put 1\ dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 29 The value (to Player1) of the game is, ---, and in decimal, 0.1066176471 272 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If Player 1's card is in the following set then he should DEFINITELY bet {1, 15, 16, 17} If his card is, 2, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {12, 13, 14, 15, 16, 17} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 17, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, . They can s\ ee their own cards but, of course, not their opponent's. They each put 1\ dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 13 The value (to Player1) of the game is, ---, and in decimal, 0.09558823529 136 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If Player 1's card is in the following set then he should DEFINITELY bet {1, 16, 17} If his card is, 2, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {13, 14, 15, 16, 17} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 17, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, . They can s\ ee their own cards but, of course, not their opponent's. They each put 1\ dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 87 The value (to Player1) of the game is, ---, and in decimal, 0.09138655462 952 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If Player 1's card is in the following set then he should DEFINITELY bet {1, 16, 17} If his card is, 2, then he should bet with probability, 3/7, and check with probability, 4/7 If Player 2's card is in the following set then he should DEFINITELY call {14, 15, 16, 17} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If his card is, 13, then he should call with probability, 2/7, and fold with probability, 5/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 17, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, . They can s\ ee their own cards but, of course, not their opponent's. They each put 1\ dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 23 The value (to Player1) of the game is, ---, and in decimal, 0.08455882353 272 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If Player 1's card is in the following set then he should DEFINITELY bet {1, 16, 17} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {15, 16, 17} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If his card is, 14, then he should call with probability, 3/4, and fold with probability, 1/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 17, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, . They can s\ ee their own cards but, of course, not their opponent's. They each put 1\ dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 31 The value (to Player1) of the game is, ---, and in decimal, 0.07598039216 408 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If Player 1's card is in the following set then he should DEFINITELY bet {1, 16, 17} If his card is, 2, then he should bet with probability, 5/9, and check with probability, 4/9 If Player 2's card is in the following set then he should DEFINITELY call {15, 16, 17} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If his card is, 14, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 17, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, . They can s\ ee their own cards but, of course, not their opponent's. They each put 1\ dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 9/136, and in decimal, 0.06617647059 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If Player 1's card is in the following set then he should DEFINITELY bet {1, 17} If his card is, 16, then he should bet with probability, 1/4, and check with probability, 3/4 If Player 2's card is in the following set then he should DEFINITELY call {15, 16, 17} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 17, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, . They can s\ ee their own cards but, of course, not their opponent's. They each put 1\ dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 189 The value (to Player1) of the game is, ----, and in decimal, 0.06316844920 2992 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {17} If his card is, 1, then he should bet with probability, 9/11, and check with probability, 2/11 If Player 2's card is in the following set then he should DEFINITELY call {16, 17} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} 10 If his card is, 15, then he should call with probability, --, 11 and fold with probability, 1/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 17, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17}, . They can s\ ee their own cards but, of course, not their opponent's. They each put 1\ dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 25 The value (to Player1) of the game is, ---, and in decimal, 0.06127450980 408 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {17} If his card is, 1, then he should bet with probability, 5/6, and check with probability, 1/6 If Player 2's card is in the following set then he should DEFINITELY call {16, 17} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If his card is, 15, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- The deck has, 18, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 18, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 91 The value (to Player1) of the game is, ---, and in decimal, 0.09912854030 918 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {1, 14, 15, 16, 17, 18} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} If his card is, 8, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 18, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/9, and in decimal, 0.1111111111 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 15, 16, 17, 18} If Player 2's card is in the following set then he should DEFINITELY call {11, 12, 13, 14, 15, 16, 17, 18} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 18, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 163 The value (to Player1) of the game is, ----, and in decimal, 0.1065359477 1530 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If Player 1's card is in the following set then he should DEFINITELY bet {1, 16, 17, 18} If his card is, 2, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {13, 14, 15, 16, 17, 18} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If his card is, 12, then he should call with probability, 2/5, and fold with probability, 3/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 18, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 5/51, and in decimal, 0.09803921569 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 16, 17, 18} If Player 2's card is in the following set then he should DEFINITELY call {14, 15, 16, 17, 18} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If his card is, 13, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 18, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 97 The value (to Player1) of the game is, ----, and in decimal, 0.09056956116 1071 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {1, 17, 18} If his card is, 2, then he should bet with probability, 3/7, and check with probability, 4/7 If Player 2's card is in the following set then he should DEFINITELY call {15, 16, 17, 18} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If his card is, 14, then he should call with probability, 4/7, and fold with probability, 3/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 18, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 13 The value (to Player1) of the game is, ---, and in decimal, 0.08496732026 153 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {1, 17, 18} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {15, 16, 17, 18} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 18, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 107 The value (to Player1) of the game is, ----, and in decimal, 0.07770515614 1377 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {1, 17, 18} If his card is, 2, then he should bet with probability, 5/9, and check with probability, 4/9 If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If his card is, 15, then he should call with probability, 5/9, and fold with probability, 4/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 18, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 53 The value (to Player1) of the game is, ---, and in decimal, 0.06928104575 765 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {1, 17, 18} If his card is, 2, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If his card is, 15, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 18, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 19 The value (to Player1) of the game is, ---, and in decimal, 0.06209150327 306 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {1, 18} If his card is, 17, then he should bet with probability, 2/9, and check with probability, 7/9 If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 18, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18}, . They c\ an see their own cards but, of course, not their opponent's. They each p\ ut 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 55 The value (to Player1) of the game is, ---, and in decimal, 0.05991285403 918 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 1's card is in the following set then he should DEFINITELY bet {18} If his card is, 1, then he should bet with probability, 5/6, and check with probability, 1/6 If Player 2's card is in the following set then he should DEFINITELY call {17, 18} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If his card is, 16, then he should call with probability, 5/6, and fold with probability, 1/6 -------------------------------------------- The deck has, 19, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 19, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, . Th\ ey can see their own cards but, of course, not their opponent's. They ea\ ch put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 17 The value (to Player1) of the game is, ---, and in decimal, 0.09941520468 171 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 14, 15, 16, 17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY call {9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7} If his card is, 8, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 19, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, . Th\ ey can see their own cards but, of course, not their opponent's. They ea\ ch put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/9, and in decimal, 0.1111111111 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 16, 17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY call {12, 13, 14, 15, 16, 17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} If his card is, 11, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 19, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, . Th\ ey can see their own cards but, of course, not their opponent's. They ea\ ch put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 181 The value (to Player1) of the game is, ----, and in decimal, 0.1058479532 1710 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {1, 17, 18, 19} If his card is, 2, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {14, 15, 16, 17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If his card is, 13, then he should call with probability, 4/5, and fold with probability, 1/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 19, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, . Th\ ey can see their own cards but, of course, not their opponent's. They ea\ ch put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 17 The value (to Player1) of the game is, ---, and in decimal, 0.09941520468 171 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY call {15, 16, 17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If his card is, 14, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 19, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, . Th\ ey can see their own cards but, of course, not their opponent's. They ea\ ch put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 107 The value (to Player1) of the game is, ----, and in decimal, 0.08939014202 1197 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 1's card is in the following set then he should DEFINITELY bet {1, 18, 19} If his card is, 2, then he should bet with probability, 3/7, and check with probability, 4/7 If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If his card is, 15, then he should call with probability, 6/7, and fold with probability, 1/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 19, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, . Th\ ey can see their own cards but, of course, not their opponent's. They ea\ ch put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 29 The value (to Player1) of the game is, ---, and in decimal, 0.08479532164 342 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 1's card is in the following set then he should DEFINITELY bet {1, 18, 19} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If his card is, 15, then he should call with probability, 1/4, and fold with probability, 3/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 19, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, . Th\ ey can see their own cards but, of course, not their opponent's. They ea\ ch put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 121 The value (to Player1) of the game is, ----, and in decimal, 0.07862248213 1539 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 1's card is in the following set then he should DEFINITELY bet {1, 18, 19} If his card is, 2, then he should bet with probability, 5/9, and check with probability, 4/9 If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If his card is, 16, then he should call with probability, 7/9, and fold with probability, 2/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 19, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, . Th\ ey can see their own cards but, of course, not their opponent's. They ea\ ch put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 61 The value (to Player1) of the game is, ---, and in decimal, 0.07134502924 855 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 1's card is in the following set then he should DEFINITELY bet {1, 18, 19} If his card is, 2, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If his card is, 16, then he should call with probability, 2/5, and fold with probability, 3/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 19, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, . Th\ ey can see their own cards but, of course, not their opponent's. They ea\ ch put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 119 The value (to Player1) of the game is, ----, and in decimal, 0.06326422116 1881 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 1's card is in the following set then he should DEFINITELY bet {1, 18, 19} If his card is, 2, then he should bet with probability, 7/11, and check with probability, 4/11 If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If his card is, 16, then he should call with probability, 1/11, 10 and fold with probability, -- 11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 19, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19}, . Th\ ey can see their own cards but, of course, not their opponent's. They ea\ ch put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 10 The value (to Player1) of the game is, ---, and in decimal, 0.05847953216 171 If Player 1's card is in the following set then he should DEFINITELY check {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 1's card is in the following set then he should DEFINITELY bet {19} If his card is, 1, then he should bet with probability, 5/6, and check with probability, 1/6 If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} -------------------------------------------- The deck has, 20, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 20, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 15, 16, 17, 18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY call {9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 20, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 21 The value (to Player1) of the game is, ---, and in decimal, 0.1105263158 190 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 17, 18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY call {12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 20, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 2/19, and in decimal, 0.1052631579 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 18, 19, 20} If his card is, 17, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {14, 15, 16, 17, 18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 20, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY call {15, 16, 17, 18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 20, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 17 The value (to Player1) of the game is, ---, and in decimal, 0.08947368421 190 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 19, 20} If his card is, 18, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 20, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 8/95, and in decimal, 0.08421052632 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 1's card is in the following set then he should DEFINITELY bet {1, 19, 20} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If his card is, 16, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 20, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 3/38, and in decimal, 0.07894736842 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 1's card is in the following set then he should DEFINITELY bet {1, 19, 20} If his card is, 2, then he should bet with probability, 5/9, and check with probability, 4/9 If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 20, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 69 The value (to Player1) of the game is, ---, and in decimal, 0.07263157895 950 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 1's card is in the following set then he should DEFINITELY bet {1, 19, 20} If his card is, 2, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If his card is, 17, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 20, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 137 The value (to Player1) of the game is, ----, and in decimal, 0.06555023923 2090 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 1's card is in the following set then he should DEFINITELY bet {1, 19, 20} If his card is, 2, then he should bet with probability, 7/11, and check with probability, 4/11 If Player 2's card is in the following set then he should DEFINITELY call {18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If his card is, 17, then he should call with probability, 3/11, and fold with probability, 8/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 20, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 11 The value (to Player1) of the game is, ---, and in decimal, 0.05789473684 190 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 1's card is in the following set then he should DEFINITELY bet {1, 20} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If his card is, 19, then he should bet with probability, 4/5, and check with probability, 1/5 If Player 2's card is in the following set then he should DEFINITELY call {18, 19, 20} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} -------------------------------------------- The deck has, 21, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 21, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} , . They can see their own cards but, of course, not their opponent's. Th\ ey each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 16, 17, 18, 19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY call {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8} If his card is, 9, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 21, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} , . They can see their own cards but, of course, not their opponent's. Th\ ey each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 23 The value (to Player1) of the game is, ---, and in decimal, 0.1095238095 210 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 18, 19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY call {13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 21, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} , . They can see their own cards but, of course, not their opponent's. Th\ ey each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 37 The value (to Player1) of the game is, ---, and in decimal, 0.1057142857 350 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 18, 19, 20, 21} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {15, 16, 17, 18, 19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If his card is, 14, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 21, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} , . They can see their own cards but, of course, not their opponent's. Th\ ey each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18, 19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If his card is, 15, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 21, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} , . They can see their own cards but, of course, not their opponent's. Th\ ey each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 89 The value (to Player1) of the game is, ---, and in decimal, 0.09081632653 980 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 19, 20, 21} If his card is, 3, then he should bet with probability, 1/7, and check with probability, 6/7 If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If his card is, 16, then he should call with probability, 1/7, and fold with probability, 6/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 21, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} , . They can see their own cards but, of course, not their opponent's. Th\ ey each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/12, and in decimal, 0.08333333333 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If Player 1's card is in the following set then he should DEFINITELY bet {1, 20, 21} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {18, 19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If his card is, 17, then he should call with probability, 3/4, and fold with probability, 1/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 21, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} , . They can see their own cards but, of course, not their opponent's. Th\ ey each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 149 The value (to Player1) of the game is, ----, and in decimal, 0.07883597884 1890 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If Player 1's card is in the following set then he should DEFINITELY bet {1, 20, 21} If his card is, 2, then he should bet with probability, 5/9, and check with probability, 4/9 If Player 2's card is in the following set then he should DEFINITELY call {18, 19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If his card is, 17, then he should call with probability, 2/9, and fold with probability, 7/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 21, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} , . They can see their own cards but, of course, not their opponent's. Th\ ey each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 11 The value (to Player1) of the game is, ---, and in decimal, 0.07333333333 150 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If Player 1's card is in the following set then he should DEFINITELY bet {1, 20, 21} If his card is, 2, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If his card is, 18, then he should call with probability, 4/5, and fold with probability, 1/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 21, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} , . They can see their own cards but, of course, not their opponent's. Th\ ey each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 31 The value (to Player1) of the game is, ---, and in decimal, 0.06709956710 462 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If Player 1's card is in the following set then he should DEFINITELY bet {1, 20, 21} If his card is, 2, then he should bet with probability, 7/11, and check with probability, 4/11 If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If his card is, 18, then he should call with probability, 5/11, and fold with probability, 6/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 21, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} , . They can see their own cards but, of course, not their opponent's. Th\ ey each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 19 The value (to Player1) of the game is, ---, and in decimal, 0.06031746032 315 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If Player 1's card is in the following set then he should DEFINITELY bet {1, 20, 21} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If his card is, 18, then he should call with probability, 1/6, and fold with probability, 5/6 -------------------------------------------- The deck has, 22, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 22, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, . They can see\ their own cards but, of course, not their opponent's. They each put 1 d\ ollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 23 The value (to Player1) of the game is, ---, and in decimal, 0.09956709957 231 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 17, 18, 19, 20, 21, 22} If Player 2's card is in the following set then he should DEFINITELY call {10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 22, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, . They can see\ their own cards but, of course, not their opponent's. They each put 1 d\ ollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 17 The value (to Player1) of the game is, ---, and in decimal, 0.1103896104 154 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 18, 19, 20, 21, 22} If his card is, 3, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If his card is, 13, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 22, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, . They can see\ their own cards but, of course, not their opponent's. They each put 1 d\ ollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 41 The value (to Player1) of the game is, ---, and in decimal, 0.1064935065 385 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 19, 20, 21, 22} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18, 19, 20, 21, 22} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If his card is, 15, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 22, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, . They can see\ their own cards but, of course, not their opponent's. They each put 1 d\ ollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 23 The value (to Player1) of the game is, ---, and in decimal, 0.09956709957 231 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 20, 21, 22} If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19, 20, 21, 22} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If his card is, 16, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 22, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, . They can see\ their own cards but, of course, not their opponent's. They each put 1 d\ ollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 9/98, and in decimal, 0.09183673469 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 20, 21, 22} If his card is, 3, then he should bet with probability, 1/7, and check with probability, 6/7 If Player 2's card is in the following set then he should DEFINITELY call {18, 19, 20, 21, 22} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If his card is, 17, then he should call with probability, 3/7, and fold with probability, 4/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 22, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, . They can see\ their own cards but, of course, not their opponent's. They each put 1 d\ ollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 19 The value (to Player1) of the game is, ---, and in decimal, 0.08225108225 231 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 21, 22} If his card is, 2, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {18, 19, 20, 21, 22} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 22, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, . They can see\ their own cards but, of course, not their opponent's. They each put 1 d\ ollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 163 The value (to Player1) of the game is, ----, and in decimal, 0.07840307840 2079 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 21, 22} If his card is, 2, then he should bet with probability, 5/9, and check with probability, 4/9 If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21, 22} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If his card is, 18, then he should call with probability, 4/9, and fold with probability, 5/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 22, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, . They can see\ their own cards but, of course, not their opponent's. They each put 1 d\ ollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 17 The value (to Player1) of the game is, ---, and in decimal, 0.07359307359 231 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 21, 22} If his card is, 2, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21, 22} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 22, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, . They can see\ their own cards but, of course, not their opponent's. They each put 1 d\ ollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 173 The value (to Player1) of the game is, ----, and in decimal, 0.06808343172 2541 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 21, 22} If his card is, 2, then he should bet with probability, 7/11, and check with probability, 4/11 If Player 2's card is in the following set then he should DEFINITELY call {20, 21, 22} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If his card is, 19, then he should call with probability, 7/11, and fold with probability, 4/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 22, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22}, . They can see\ their own cards but, of course, not their opponent's. They each put 1 d\ ollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 43 The value (to Player1) of the game is, ---, and in decimal, 0.06204906205 693 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 21, 22} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {20, 21, 22} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If his card is, 19, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- The deck has, 23, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 23, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}, . They can\ see their own cards but, of course, not their opponent's. They each put\ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 151 The value (to Player1) of the game is, ----, and in decimal, 0.09947299078 1518 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 17, 18, 19, 20, 21, 22, 23} If his card is, 3, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9} If his card is, 10, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 23, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}, . They can\ see their own cards but, of course, not their opponent's. They each put\ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 28 The value (to Player1) of the game is, ---, and in decimal, 0.1106719368 253 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 19, 20, 21, 22, 23} If his card is, 3, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 23, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}, . They can\ see their own cards but, of course, not their opponent's. They each put\ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 27 The value (to Player1) of the game is, ---, and in decimal, 0.1067193676 253 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 20, 21, 22, 23} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18, 19, 20, 21, 22, 23} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 23, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}, . They can\ see their own cards but, of course, not their opponent's. They each put\ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 25 The value (to Player1) of the game is, ---, and in decimal, 0.09881422925 253 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 21, 22, 23} If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19, 20, 21, 22, 23} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 23, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}, . They can\ see their own cards but, of course, not their opponent's. They each put\ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 327 The value (to Player1) of the game is, ----, and in decimal, 0.09232072276 3542 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 21, 22, 23} If his card is, 3, then he should bet with probability, 1/7, and check with probability, 6/7 If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21, 22, 23} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If his card is, 18, then he should call with probability, 5/7, and fold with probability, 2/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 23, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}, . They can\ see their own cards but, of course, not their opponent's. They each put\ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 21 The value (to Player1) of the game is, ---, and in decimal, 0.08300395257 253 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 22, 23} If his card is, 21, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21, 22, 23} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 23, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}, . They can\ see their own cards but, of course, not their opponent's. They each put\ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 59 The value (to Player1) of the game is, ---, and in decimal, 0.07773386034 759 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 1's card is in the following set then he should DEFINITELY bet {1, 22, 23} If his card is, 2, then he should bet with probability, 5/9, and check with probability, 4/9 If Player 2's card is in the following set then he should DEFINITELY call {20, 21, 22, 23} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If his card is, 19, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 23, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}, . They can\ see their own cards but, of course, not their opponent's. They each put\ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 93 The value (to Player1) of the game is, ----, and in decimal, 0.07351778656 1265 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 1's card is in the following set then he should DEFINITELY bet {1, 22, 23} If his card is, 2, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {20, 21, 22, 23} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If his card is, 19, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 23, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}, . They can\ see their own cards but, of course, not their opponent's. They each put\ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 191 The value (to Player1) of the game is, ----, and in decimal, 0.06863097377 2783 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 1's card is in the following set then he should DEFINITELY bet {1, 22, 23} If his card is, 2, then he should bet with probability, 7/11, and check with probability, 4/11 If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If his card is, 20, then he should call with probability, 9/11, and fold with probability, 2/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 23, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23}, . They can\ see their own cards but, of course, not their opponent's. They each put\ 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 16 The value (to Player1) of the game is, ---, and in decimal, 0.06324110672 253 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 1's card is in the following set then he should DEFINITELY bet {1, 22, 23} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If his card is, 20, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- The deck has, 24, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 24, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, . They\ can see their own cards but, of course, not their opponent's. They each\ put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 55 The value (to Player1) of the game is, ---, and in decimal, 0.09963768116 552 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 18, 19, 20, 21, 22, 23, 24} If his card is, 3, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 24, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, . They\ can see their own cards but, of course, not their opponent's. They each\ put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 61 The value (to Player1) of the game is, ---, and in decimal, 0.1105072464 552 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 20, 21, 22, 23, 24} If his card is, 3, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If his card is, 14, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 24, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, . They\ can see their own cards but, of course, not their opponent's. They each\ put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 49 The value (to Player1) of the game is, ---, and in decimal, 0.1065217391 460 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 21, 22, 23, 24} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19, 20, 21, 22, 23, 24} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If his card is, 16, then he should call with probability, 2/5, and fold with probability, 3/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 24, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, . They\ can see their own cards but, of course, not their opponent's. They each\ put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 9/92, and in decimal, 0.09782608696 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 22, 23, 24} If his card is, 3, then he should bet with probability, 1/2, and check with probability, 1/2 If his card is, 21, then he should bet with probability, 3/4, and check with probability, 1/4 If Player 2's card is in the following set then he should DEFINITELY call {18, 19, 20, 21, 22, 23, 24} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 24, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, . They\ can see their own cards but, of course, not their opponent's. They each\ put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 17 The value (to Player1) of the game is, ---, and in decimal, 0.09239130435 184 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 22, 23, 24} If his card is, 3, then he should bet with probability, 1/7, and check with probability, 6/7 If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21, 22, 23, 24} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 24, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, . They\ can see their own cards but, of course, not their opponent's. They each\ put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 31 The value (to Player1) of the game is, ---, and in decimal, 0.08423913043 368 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 22, 23, 24} If his card is, 3, then he should bet with probability, 1/4, and check with probability, 3/4 If Player 2's card is in the following set then he should DEFINITELY call {20, 21, 22, 23, 24} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If his card is, 19, then he should call with probability, 1/4, and fold with probability, 3/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 24, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, . They\ can see their own cards but, of course, not their opponent's. They each\ put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 191 The value (to Player1) of the game is, ----, and in decimal, 0.07689210950 2484 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 1's card is in the following set then he should DEFINITELY bet {1, 23, 24} If his card is, 2, then he should bet with probability, 5/9, and check with probability, 4/9 If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23, 24} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If his card is, 20, then he should call with probability, 8/9, and fold with probability, 1/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 24, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, . They\ can see their own cards but, of course, not their opponent's. They each\ put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 101 The value (to Player1) of the game is, ----, and in decimal, 0.07318840580 1380 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 1's card is in the following set then he should DEFINITELY bet {1, 23, 24} If his card is, 2, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23, 24} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If his card is, 20, then he should call with probability, 2/5, and fold with probability, 3/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 24, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, . They\ can see their own cards but, of course, not their opponent's. They each\ put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 19 The value (to Player1) of the game is, ---, and in decimal, 0.06884057971 276 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 1's card is in the following set then he should DEFINITELY bet {1, 23, 24} If his card is, 2, then he should bet with probability, 7/11, and check with probability, 4/11 If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23, 24} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 24, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24}, . They\ can see their own cards but, of course, not their opponent's. They each\ put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 53 The value (to Player1) of the game is, ---, and in decimal, 0.06400966184 828 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 1's card is in the following set then he should DEFINITELY bet {1, 23, 24} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If his card is, 21, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- The deck has, 25, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 25, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 179 The value (to Player1) of the game is, ----, and in decimal, 0.09944444444 1800 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 19, 20, 21, 22, 23, 24, 25} If his card is, 3, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} If his card is, 11, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 25, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 11 The value (to Player1) of the game is, ---, and in decimal, 0.1100000000 100 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 21, 22, 23, 24, 25} If his card is, 3, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 25, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 53 The value (to Player1) of the game is, ---, and in decimal, 0.1060000000 500 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 22, 23, 24, 25} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {18, 19, 20, 21, 22, 23, 24, 25} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If his card is, 17, then he should call with probability, 4/5, and fold with probability, 1/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 25, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 89 The value (to Player1) of the game is, ---, and in decimal, 0.09888888889 900 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 22, 23, 24, 25} If his card is, 3, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21, 22, 23, 24, 25} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If his card is, 18, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 25, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 129 The value (to Player1) of the game is, ----, and in decimal, 0.09214285714 1400 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 23, 24, 25} If his card is, 3, then he should bet with probability, 1/7, and check with probability, 6/7 If Player 2's card is in the following set then he should DEFINITELY call {20, 21, 22, 23, 24, 25} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If his card is, 19, then he should call with probability, 2/7, and fold with probability, 5/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 25, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 17 The value (to Player1) of the game is, ---, and in decimal, 0.08500000000 200 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 23, 24, 25} If his card is, 3, then he should bet with probability, 1/4, and check with probability, 3/4 If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23, 24, 25} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If his card is, 20, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 25, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 23 The value (to Player1) of the game is, ---, and in decimal, 0.07666666667 300 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 24, 25} If his card is, 23, then he should bet with probability, 4/7, and check with probability, 3/7 If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23, 24, 25} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 25, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 109 The value (to Player1) of the game is, ----, and in decimal, 0.07266666667 1500 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 1's card is in the following set then he should DEFINITELY bet {1, 24, 25} If his card is, 2, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24, 25} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If his card is, 21, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 25, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 227 The value (to Player1) of the game is, ----, and in decimal, 0.06878787879 3300 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 1's card is in the following set then he should DEFINITELY bet {1, 24, 25} If his card is, 2, then he should bet with probability, 7/11, and check with probability, 4/11 If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24, 25} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If his card is, 21, then he should call with probability, 2/11, and fold with probability, 9/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 25, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, . \ They can see their own cards but, of course, not their opponent's. They \ each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 29 The value (to Player1) of the game is, ---, and in decimal, 0.06444444444 450 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 1's card is in the following set then he should DEFINITELY bet {1, 24, 25} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {23, 24, 25} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If his card is, 22, then he should call with probability, 5/6, and fold with probability, 1/6 -------------------------------------------- The deck has, 26, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 26, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 97 The value (to Player1) of the game is, ---, and in decimal, 0.09948717949 975 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 19, 20, 21, 22, 23, 24, 25, 26} If his card is, 3, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} If his card is, 11, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 26, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 36 The value (to Player1) of the game is, ---, and in decimal, 0.1107692308 325 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 21, 22, 23, 24, 25, 26} If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If his card is, 15, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 26, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 69 The value (to Player1) of the game is, ---, and in decimal, 0.1061538462 650 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 22, 23, 24, 25, 26} If Player 2's card is in the following set then he should DEFINITELY call {18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} If his card is, 17, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 26, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 97 The value (to Player1) of the game is, ---, and in decimal, 0.09948717949 975 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 23, 24, 25, 26} If his card is, 3, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {20, 21, 22, 23, 24, 25, 26} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If his card is, 19, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 26, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 417 The value (to Player1) of the game is, ----, and in decimal, 0.09164835165 4550 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 24, 25, 26} If his card is, 3, then he should bet with probability, 1/7, and check with probability, 6/7 If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23, 24, 25, 26} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If his card is, 20, then he should call with probability, 4/7, and fold with probability, 3/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 26, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 111 The value (to Player1) of the game is, ----, and in decimal, 0.08538461538 1300 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 24, 25, 26} If his card is, 3, then he should bet with probability, 1/4, and check with probability, 3/4 If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24, 25, 26} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If his card is, 21, then he should call with probability, 3/4, and fold with probability, 1/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 26, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 151 The value (to Player1) of the game is, ----, and in decimal, 0.07743589744 1950 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 24, 25, 26} If his card is, 3, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24, 25, 26} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If his card is, 21, then he should call with probability, 1/9, and fold with probability, 8/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 26, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 9/125, and in decimal, 0.07200000000 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 1's card is in the following set then he should DEFINITELY bet {1, 25, 26} If his card is, 2, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {23, 24, 25, 26} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If his card is, 22, then he should call with probability, 4/5, and fold with probability, 1/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 26, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 49 The value (to Player1) of the game is, ---, and in decimal, 0.06853146853 715 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 1's card is in the following set then he should DEFINITELY bet {1, 25, 26} If his card is, 2, then he should bet with probability, 7/11, and check with probability, 4/11 If Player 2's card is in the following set then he should DEFINITELY call {23, 24, 25, 26} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If his card is, 22, then he should call with probability, 4/11, and fold with probability, 7/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 26, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, . They can see their own cards but, of course, not their opponent's. The\ y each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 21 The value (to Player1) of the game is, ---, and in decimal, 0.06461538462 325 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 1's card is in the following set then he should DEFINITELY bet {1, 25, 26} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {23, 24, 25, 26} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} -------------------------------------------- The deck has, 27, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 27, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}, . They can see their own cards but, of course, not their opponent's.\ They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 35 The value (to Player1) of the game is, ---, and in decimal, 0.09971509972 351 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 20, 21, 22, 23, 24, 25, 26, 27} If his card is, 3, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 27, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}, . They can see their own cards but, of course, not their opponent's.\ They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/9, and in decimal, 0.1111111111 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 22, 23, 24, 25, 26, 27} If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 27, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}, . They can see their own cards but, of course, not their opponent's.\ They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 25 The value (to Player1) of the game is, ---, and in decimal, 0.1068376068 234 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 23, 24, 25, 26, 27} If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If his card is, 18, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 27, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}, . They can see their own cards but, of course, not their opponent's.\ They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 35 The value (to Player1) of the game is, ---, and in decimal, 0.09971509972 351 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 24, 25, 26, 27} If his card is, 3, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {20, 21, 22, 23, 24, 25, 26, 27} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 27, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}, . They can see their own cards but, of course, not their opponent's.\ They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 149 The value (to Player1) of the game is, ----, and in decimal, 0.09096459096 1638 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 25, 26, 27} If his card is, 3, then he should bet with probability, 1/7, and check with probability, 6/7 If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24, 25, 26, 27} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If his card is, 21, then he should call with probability, 6/7, and fold with probability, 1/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 27, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}, . They can see their own cards but, of course, not their opponent's.\ They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 10 The value (to Player1) of the game is, ---, and in decimal, 0.08547008547 117 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 25, 26, 27} If his card is, 3, then he should bet with probability, 1/4, and check with probability, 3/4 If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24, 25, 26, 27} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 27, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}, . They can see their own cards but, of course, not their opponent's.\ They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 55 The value (to Player1) of the game is, ---, and in decimal, 0.07834757835 702 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 25, 26, 27} If his card is, 3, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {23, 24, 25, 26, 27} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If his card is, 22, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 27, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}, . They can see their own cards but, of course, not their opponent's.\ They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 25 The value (to Player1) of the game is, ---, and in decimal, 0.07122507122 351 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 1's card is in the following set then he should DEFINITELY bet {1, 26, 27} If his card is, 2, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {23, 24, 25, 26, 27} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 27, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}, . They can see their own cards but, of course, not their opponent's.\ They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 263 The value (to Player1) of the game is, ----, and in decimal, 0.06811706812 3861 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 1's card is in the following set then he should DEFINITELY bet {1, 26, 27} If his card is, 2, then he should bet with probability, 7/11, and check with probability, 4/11 If Player 2's card is in the following set then he should DEFINITELY call {24, 25, 26, 27} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If his card is, 23, then he should call with probability, 6/11, and fold with probability, 5/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 27, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}, . They can see their own cards but, of course, not their opponent's.\ They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 68 The value (to Player1) of the game is, ----, and in decimal, 0.06457739791 1053 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 1's card is in the following set then he should DEFINITELY bet {1, 26, 27} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {24, 25, 26, 27} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If his card is, 23, then he should call with probability, 1/6, and fold with probability, 5/6 -------------------------------------------- The deck has, 28, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 28, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}, . They can see their own cards but, of course, not their opponen\ t's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 113 The value (to Player1) of the game is, ----, and in decimal, 0.09964726631 1134 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 21, 22, 23, 24, 25, 26, 27, 28} If his card is, 3, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If his card is, 12, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 28, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}, . They can see their own cards but, of course, not their opponen\ t's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/9, and in decimal, 0.1111111111 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 23, 24, 25, 26, 27, 28} If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If his card is, 16, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 28, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}, . They can see their own cards but, of course, not their opponen\ t's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 3/28, and in decimal, 0.1071428571 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 24, 25, 26, 27, 28} If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 28, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}, . They can see their own cards but, of course, not their opponen\ t's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 113 The value (to Player1) of the game is, ----, and in decimal, 0.09964726631 1134 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 25, 26, 27, 28} If his card is, 3, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23, 24, 25, 26, 27, 28} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If his card is, 20, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 28, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}, . They can see their own cards but, of course, not their opponen\ t's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 241 The value (to Player1) of the game is, ----, and in decimal, 0.09108087680 2646 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 25, 26, 27, 28} If his card is, 3, then he should bet with probability, 6/7, and check with probability, 1/7 If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24, 25, 26, 27, 28} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If his card is, 21, then he should call with probability, 1/7, and fold with probability, 6/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 28, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}, . They can see their own cards but, of course, not their opponen\ t's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 43 The value (to Player1) of the game is, ---, and in decimal, 0.08531746032 504 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 26, 27, 28} If his card is, 3, then he should bet with probability, 1/4, and check with probability, 3/4 If Player 2's card is in the following set then he should DEFINITELY call {23, 24, 25, 26, 27, 28} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If his card is, 22, then he should call with probability, 1/4, and fold with probability, 3/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 28, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}, . They can see their own cards but, of course, not their opponen\ t's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 179 The value (to Player1) of the game is, ----, and in decimal, 0.07892416226 2268 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 26, 27, 28} If his card is, 3, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {24, 25, 26, 27, 28} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If his card is, 23, then he should call with probability, 5/9, and fold with probability, 4/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 28, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}, . They can see their own cards but, of course, not their opponen\ t's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/14, and in decimal, 0.07142857143 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 26, 27, 28} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {24, 25, 26, 27, 28} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 28, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}, . They can see their own cards but, of course, not their opponen\ t's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 281 The value (to Player1) of the game is, ----, and in decimal, 0.06758056758 4158 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 1's card is in the following set then he should DEFINITELY bet {1, 27, 28} If his card is, 2, then he should bet with probability, 7/11, and check with probability, 4/11 If Player 2's card is in the following set then he should DEFINITELY call {25, 26, 27, 28} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If his card is, 24, then he should call with probability, 8/11, and fold with probability, 3/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 28, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28}, . They can see their own cards but, of course, not their opponen\ t's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 73 The value (to Player1) of the game is, ----, and in decimal, 0.06437389771 1134 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 1's card is in the following set then he should DEFINITELY bet {1, 27, 28} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {25, 26, 27, 28} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If his card is, 24, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- The deck has, 29, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 29, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, . They can see their own cards but, of course, not their opp\ onent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 81 The value (to Player1) of the game is, ---, and in decimal, 0.09975369458 812 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY call {13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11} If his card is, 12, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 29, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, . They can see their own cards but, of course, not their opp\ onent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 45 The value (to Player1) of the game is, ---, and in decimal, 0.1108374384 406 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 24, 25, 26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 29, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, . They can see their own cards but, of course, not their opp\ onent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 3/28, and in decimal, 0.1071428571 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 25, 26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY call {20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If his card is, 19, then he should call with probability, 2/5, and fold with probability, 3/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 29, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, . They can see their own cards but, of course, not their opp\ onent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 121 The value (to Player1) of the game is, ----, and in decimal, 0.09934318555 1218 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 26, 27, 28, 29} If his card is, 3, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24, 25, 26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If his card is, 21, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 29, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, . They can see their own cards but, of course, not their opp\ onent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 9/98, and in decimal, 0.09183673469 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 26, 27, 28, 29} If his card is, 3, then he should bet with probability, 6/7, and check with probability, 1/7 If Player 2's card is in the following set then he should DEFINITELY call {23, 24, 25, 26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If his card is, 22, then he should call with probability, 3/7, and fold with probability, 4/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 29, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, . They can see their own cards but, of course, not their opp\ onent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 69 The value (to Player1) of the game is, ---, and in decimal, 0.08497536946 812 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 27, 28, 29} If his card is, 3, then he should bet with probability, 1/4, and check with probability, 3/4 If Player 2's card is in the following set then he should DEFINITELY call {24, 25, 26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If his card is, 23, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 29, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, . They can see their own cards but, of course, not their opp\ onent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 193 The value (to Player1) of the game is, ----, and in decimal, 0.07922824302 2436 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 27, 28, 29} If his card is, 3, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {25, 26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If his card is, 24, then he should call with probability, 7/9, and fold with probability, 2/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 29, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, . They can see their own cards but, of course, not their opp\ onent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 21 The value (to Player1) of the game is, ---, and in decimal, 0.07241379310 290 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 27, 28, 29} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {25, 26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If his card is, 24, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 29, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, . They can see their own cards but, of course, not their opp\ onent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 299 The value (to Player1) of the game is, ----, and in decimal, 0.06695029109 4466 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 28, 29} If his card is, 2, then he should bet with probability, 7/11, and check with probability, 4/11 If Player 2's card is in the following set then he should DEFINITELY call {26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} 10 If his card is, 25, then he should call with probability, --, 11 and fold with probability, 1/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 29, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29}, . They can see their own cards but, of course, not their opp\ onent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 13 The value (to Player1) of the game is, ---, and in decimal, 0.06403940887 203 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 28, 29} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {26, 27, 28, 29} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If his card is, 25, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- The deck has, 30, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 30, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, . They can see their own cards but, of course, not their\ opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY call {13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 30, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, . They can see their own cards but, of course, not their\ opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 16 The value (to Player1) of the game is, ---, and in decimal, 0.1103448276 145 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 25, 26, 27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY call {18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 30, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, . They can see their own cards but, of course, not their\ opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 31 The value (to Player1) of the game is, ---, and in decimal, 0.1068965517 290 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 26, 27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If his card is, 20, then he should call with probability, 4/5, and fold with probability, 1/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 30, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, . They can see their own cards but, of course, not their\ opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 43 The value (to Player1) of the game is, ---, and in decimal, 0.09885057471 435 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 27, 28, 29, 30} If his card is, 3, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 30, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, . They can see their own cards but, of course, not their\ opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 281 The value (to Player1) of the game is, ----, and in decimal, 0.09228243021 3045 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 27, 28, 29, 30} If his card is, 3, then he should bet with probability, 6/7, and check with probability, 1/7 If Player 2's card is in the following set then he should DEFINITELY call {24, 25, 26, 27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If his card is, 23, then he should call with probability, 5/7, and fold with probability, 2/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 30, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, . They can see their own cards but, of course, not their\ opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 49 The value (to Player1) of the game is, ---, and in decimal, 0.08448275862 580 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 28, 29, 30} If his card is, 3, then he should bet with probability, 1/4, and check with probability, 3/4 If Player 2's card is in the following set then he should DEFINITELY call {25, 26, 27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If his card is, 24, then he should call with probability, 3/4, and fold with probability, 1/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 30, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, . They can see their own cards but, of course, not their\ opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 23 The value (to Player1) of the game is, ---, and in decimal, 0.07931034483 290 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 28, 29, 30} If his card is, 3, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {25, 26, 27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 30, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, . They can see their own cards but, of course, not their\ opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 53 The value (to Player1) of the game is, ---, and in decimal, 0.07310344828 725 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 28, 29, 30} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {26, 27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If his card is, 25, then he should call with probability, 2/5, and fold with probability, 3/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 30, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, . They can see their own cards but, of course, not their\ opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/15, and in decimal, 0.06666666667 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 29, 30} If his card is, 28, then he should bet with probability, 4/9, and check with probability, 5/9 If Player 2's card is in the following set then he should DEFINITELY call {26, 27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 30, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, . They can see their own cards but, of course, not their\ opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 83 The value (to Player1) of the game is, ----, and in decimal, 0.06360153257 1305 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 1's card is in the following set then he should DEFINITELY bet {1, 29, 30} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {27, 28, 29, 30} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If his card is, 26, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- The deck has, 31, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 31, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}, . They can see their own cards but, of course, not t\ heir opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 2's card is in the following set then he should DEFINITELY call {14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} If his card is, 13, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 31, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}, . They can see their own cards but, of course, not t\ heir opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 103 The value (to Player1) of the game is, ---, and in decimal, 0.1107526882 930 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 25, 26, 27, 28, 29, 30, 31} If his card is, 4, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17} If his card is, 18, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 31, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}, . They can see their own cards but, of course, not t\ heir opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 33 The value (to Player1) of the game is, ---, and in decimal, 0.1064516129 310 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 27, 28, 29, 30, 31} If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 31, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}, . They can see their own cards but, of course, not t\ heir opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 46 The value (to Player1) of the game is, ---, and in decimal, 0.09892473118 465 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 28, 29, 30, 31} If his card is, 27, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 31, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}, . They can see their own cards but, of course, not t\ heir opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 43 The value (to Player1) of the game is, ---, and in decimal, 0.09247311828 465 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 28, 29, 30, 31} If his card is, 3, then he should bet with probability, 6/7, and check with probability, 1/7 If Player 2's card is in the following set then he should DEFINITELY call {24, 25, 26, 27, 28, 29, 30, 31} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 31, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}, . They can see their own cards but, of course, not t\ heir opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 13 The value (to Player1) of the game is, ---, and in decimal, 0.08387096774 155 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 29, 30, 31} If his card is, 3, then he should bet with probability, 1/4, and check with probability, 3/4 If Player 2's card is in the following set then he should DEFINITELY call {25, 26, 27, 28, 29, 30, 31} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 31, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}, . They can see their own cards but, of course, not t\ heir opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 221 The value (to Player1) of the game is, ----, and in decimal, 0.07921146953 2790 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 29, 30, 31} If his card is, 3, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {26, 27, 28, 29, 30, 31} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If his card is, 25, then he should call with probability, 2/9, and fold with probability, 7/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 31, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}, . They can see their own cards but, of course, not t\ heir opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 57 The value (to Player1) of the game is, ---, and in decimal, 0.07354838710 775 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 29, 30, 31} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {27, 28, 29, 30, 31} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If his card is, 26, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 31, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}, . They can see their own cards but, of course, not t\ heir opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 229 The value (to Player1) of the game is, ----, and in decimal, 0.06715542522 3410 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 29, 30, 31} If his card is, 3, then he should bet with probability, 5/11, and check with probability, 6/11 If Player 2's card is in the following set then he should DEFINITELY call {27, 28, 29, 30, 31} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If his card is, 26, then he should call with probability, 1/11, 10 and fold with probability, -- 11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 31, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31}, . They can see their own cards but, of course, not t\ heir opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 88 The value (to Player1) of the game is, ----, and in decimal, 0.06308243728 1395 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 1's card is in the following set then he should DEFINITELY bet {1, 30, 31} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {28, 29, 30, 31} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If his card is, 27, then he should call with probability, 5/6, and fold with probability, 1/6 -------------------------------------------- The deck has, 32, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 32, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, . They can see their own cards but, of course, n\ ot their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 99 The value (to Player1) of the game is, ---, and in decimal, 0.09979838710 992 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 2's card is in the following set then he should DEFINITELY call {14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 32, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, . They can see their own cards but, of course, n\ ot their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 55 The value (to Player1) of the game is, ---, and in decimal, 0.1108870968 496 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 26, 27, 28, 29, 30, 31, 32} If his card is, 4, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 32, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, . They can see their own cards but, of course, n\ ot their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 33 The value (to Player1) of the game is, ---, and in decimal, 0.1064516129 310 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 27, 28, 29, 30, 31, 32} If his card is, 4, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If his card is, 21, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 32, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, . They can see their own cards but, of course, n\ ot their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 37 The value (to Player1) of the game is, ---, and in decimal, 0.09946236559 372 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 28, 29, 30, 31, 32} If his card is, 4, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If his card is, 23, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 32, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, . They can see their own cards but, of course, n\ ot their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 321 The value (to Player1) of the game is, ----, and in decimal, 0.09245391705 3472 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 29, 30, 31, 32} If his card is, 3, then he should bet with probability, 6/7, and check with probability, 1/7 If Player 2's card is in the following set then he should DEFINITELY call {25, 26, 27, 28, 29, 30, 31, 32} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If his card is, 24, then he should call with probability, 2/7, and fold with probability, 5/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 32, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, . They can see their own cards but, of course, n\ ot their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 21 The value (to Player1) of the game is, ---, and in decimal, 0.08467741935 248 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 29, 30, 31, 32} If Player 2's card is in the following set then he should DEFINITELY call {26, 27, 28, 29, 30, 31, 32} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If his card is, 25, then he should call with probability, 1/4, and fold with probability, 3/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 32, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, . They can see their own cards but, of course, n\ ot their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 235 The value (to Player1) of the game is, ----, and in decimal, 0.07896505376 2976 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 30, 31, 32} If his card is, 3, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {27, 28, 29, 30, 31, 32} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If his card is, 26, then he should call with probability, 4/9, and fold with probability, 5/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 32, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, . They can see their own cards but, of course, n\ ot their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 183 The value (to Player1) of the game is, ----, and in decimal, 0.07379032258 2480 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 30, 31, 32} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {28, 29, 30, 31, 32} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If his card is, 27, then he should call with probability, 4/5, and fold with probability, 1/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 32, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, . They can see their own cards but, of course, n\ ot their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 741 The value (to Player1) of the game is, -----, and in decimal, 0.06790689150 10912 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 30, 31, 32} If his card is, 3, then he should bet with probability, 5/11, and check with probability, 6/11 If Player 2's card is in the following set then he should DEFINITELY call {28, 29, 30, 31, 32} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If his card is, 27, then he should call with probability, 3/11, and fold with probability, 8/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 32, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32}, . They can see their own cards but, of course, n\ ot their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/16, and in decimal, 0.06250000000 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 1's card is in the following set then he should DEFINITELY bet {1, 31, 32} If his card is, 2, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {28, 29, 30, 31, 32} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} -------------------------------------------- The deck has, 33, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 33, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 79 The value (to Player1) of the game is, ---, and in decimal, 0.09974747475 792 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If his card is, 4, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13} If his card is, 14, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 33, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 39 The value (to Player1) of the game is, ---, and in decimal, 0.1107954545 352 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 27, 28, 29, 30, 31, 32, 33} If his card is, 4, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18} If his card is, 19, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 33, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 47 The value (to Player1) of the game is, ---, and in decimal, 0.1068181818 440 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 28, 29, 30, 31, 32, 33} If his card is, 4, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} If his card is, 22, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 33, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 79 The value (to Player1) of the game is, ---, and in decimal, 0.09974747475 792 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 29, 30, 31, 32, 33} If his card is, 4, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {25, 26, 27, 28, 29, 30, 31, 32, 33} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If his card is, 24, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 33, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 31 The value (to Player1) of the game is, ---, and in decimal, 0.09226190476 336 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 30, 31, 32, 33} If his card is, 3, then he should bet with probability, 6/7, and check with probability, 1/7 If Player 2's card is in the following set then he should DEFINITELY call {26, 27, 28, 29, 30, 31, 32, 33} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If his card is, 25, then he should call with probability, 4/7, and fold with probability, 3/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 33, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 15 The value (to Player1) of the game is, ---, and in decimal, 0.08522727273 176 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 30, 31, 32, 33} If Player 2's card is in the following set then he should DEFINITELY call {27, 28, 29, 30, 31, 32, 33} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If his card is, 26, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 33, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 83 The value (to Player1) of the game is, ----, and in decimal, 0.07859848485 1056 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 31, 32, 33} If his card is, 3, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {28, 29, 30, 31, 32, 33} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If his card is, 27, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 33, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 13 The value (to Player1) of the game is, ---, and in decimal, 0.07386363636 176 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 31, 32, 33} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {28, 29, 30, 31, 32, 33} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 33, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 265 The value (to Player1) of the game is, ----, and in decimal, 0.06844008264 3872 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 31, 32, 33} If his card is, 3, then he should bet with probability, 5/11, and check with probability, 6/11 If Player 2's card is in the following set then he should DEFINITELY call {29, 30, 31, 32, 33} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If his card is, 28, then he should call with probability, 5/11, and fold with probability, 6/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 33, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33}, . They can see their own cards but, of cours\ e, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/16, and in decimal, 0.06250000000 If Player 1's card is in the following set then he should DEFINITELY check {3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 32, 33} If his card is, 31, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {29, 30, 31, 32, 33} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} -------------------------------------------- The deck has, 34, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 34, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 56 The value (to Player1) of the game is, ---, and in decimal, 0.09982174688 561 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34} If his card is, 4, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34 } If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 34, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 62 The value (to Player1) of the game is, ---, and in decimal, 0.1105169340 561 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 28, 29, 30, 31, 32, 33, 34} If his card is, 4, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 34, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 20 The value (to Player1) of the game is, ---, and in decimal, 0.1069518717 187 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 29, 30, 31, 32, 33, 34} If his card is, 4, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 34, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 56 The value (to Player1) of the game is, ---, and in decimal, 0.09982174688 561 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 30, 31, 32, 33, 34} If his card is, 4, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {25, 26, 27, 28, 29, 30, 31, 32, 33, 34} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 34, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 361 The value (to Player1) of the game is, ----, and in decimal, 0.09192768016 3927 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 31, 32, 33, 34} If his card is, 3, then he should bet with probability, 6/7, and check with probability, 1/7 If Player 2's card is in the following set then he should DEFINITELY call {27, 28, 29, 30, 31, 32, 33, 34} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If his card is, 26, then he should call with probability, 6/7, and fold with probability, 1/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 34, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 16 The value (to Player1) of the game is, ---, and in decimal, 0.08556149733 187 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 31, 32, 33, 34} If Player 2's card is in the following set then he should DEFINITELY call {28, 29, 30, 31, 32, 33, 34} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If his card is, 27, then he should call with probability, 3/4, and fold with probability, 1/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 34, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 263 The value (to Player1) of the game is, ----, and in decimal, 0.07813428402 3366 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 32, 33, 34} If his card is, 3, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {29, 30, 31, 32, 33, 34} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If his card is, 28, then he should call with probability, 8/9, and fold with probability, 1/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 34, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 69 The value (to Player1) of the game is, ---, and in decimal, 0.07379679144 935 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 32, 33, 34} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {29, 30, 31, 32, 33, 34} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If his card is, 28, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 34, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 283 The value (to Player1) of the game is, ----, and in decimal, 0.06878949927 4114 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 32, 33, 34} If his card is, 3, then he should bet with probability, 5/11, and check with probability, 6/11 If Player 2's card is in the following set then he should DEFINITELY call {30, 31, 32, 33, 34} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If his card is, 29, then he should call with probability, 7/11, and fold with probability, 4/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 34, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34}, . They can see their own cards but, of c\ ourse, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 71 The value (to Player1) of the game is, ----, and in decimal, 0.06327985740 1122 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 32, 33, 34} If his card is, 3, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {30, 31, 32, 33, 34} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If his card is, 29, then he should call with probability, 1/6, and fold with probability, 5/6 -------------------------------------------- The deck has, 35, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 35, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 178 The value (to Player1) of the game is, ----, and in decimal, 0.09971988796 1785 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35} If his card is, 4, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35 } If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If his card is, 15, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 35, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 66 The value (to Player1) of the game is, ---, and in decimal, 0.1109243697 595 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 28, 29, 30, 31, 32, 33, 34, 35} If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19} If his card is, 20, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 35, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 318 The value (to Player1) of the game is, ----, and in decimal, 0.1068907563 2975 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 30, 31, 32, 33, 34, 35} If his card is, 4, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If his card is, 23, then he should call with probability, 2/5, and fold with probability, 3/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 35, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 178 The value (to Player1) of the game is, ----, and in decimal, 0.09971988796 1785 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 31, 32, 33, 34, 35} If his card is, 4, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {26, 27, 28, 29, 30, 31, 32, 33, 34, 35} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If his card is, 25, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 35, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 109 The value (to Player1) of the game is, ----, and in decimal, 0.09159663866 1190 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 32, 33, 34, 35} If his card is, 31, then he should bet with probability, 1/5, and check with probability, 4/5 If Player 2's card is in the following set then he should DEFINITELY call {27, 28, 29, 30, 31, 32, 33, 34, 35} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 35, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 3/35, and in decimal, 0.08571428571 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 32, 33, 34, 35} If Player 2's card is in the following set then he should DEFINITELY call {28, 29, 30, 31, 32, 33, 34, 35} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 35, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 93 The value (to Player1) of the game is, ----, and in decimal, 0.07815126050 1190 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 33, 34, 35} If his card is, 32, then he should bet with probability, 6/7, and check with probability, 1/7 If Player 2's card is in the following set then he should DEFINITELY call {29, 30, 31, 32, 33, 34, 35} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 35, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 219 The value (to Player1) of the game is, ----, and in decimal, 0.07361344538 2975 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 33, 34, 35} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {30, 31, 32, 33, 34, 35} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If his card is, 29, then he should call with probability, 2/5, and fold with probability, 3/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 35, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 129 The value (to Player1) of the game is, ----, and in decimal, 0.06898395722 1870 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 33, 34, 35} If his card is, 3, then he should bet with probability, 5/11, and check with probability, 6/11 If Player 2's card is in the following set then he should DEFINITELY call {31, 32, 33, 34, 35} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If his card is, 30, then he should call with probability, 9/11, and fold with probability, 2/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 35, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35}, . They can see their own cards but, \ of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 38 The value (to Player1) of the game is, ---, and in decimal, 0.06386554622 595 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 33, 34, 35} If his card is, 3, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {31, 32, 33, 34, 35} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If his card is, 30, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- The deck has, 36, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 36, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 377 The value (to Player1) of the game is, ----, and in decimal, 0.09973544974 3780 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} If his card is, 4, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14} If his card is, 15, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 36, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/9, and in decimal, 0.1111111111 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 29, 30, 31, 32, 33, 34, 35, 36} If Player 2's card is in the following set then he should DEFINITELY call {21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 36, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 8/75, and in decimal, 0.1066666667 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 31, 32, 33, 34, 35, 36} If his card is, 4, then he should bet with probability, 3/5, and check with probability, 2/5 If Player 2's card is in the following set then he should DEFINITELY call {25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23} If his card is, 24, then he should call with probability, 4/5, and fold with probability, 1/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 36, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 94 The value (to Player1) of the game is, ---, and in decimal, 0.09947089947 945 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 32, 33, 34, 35, 36} If his card is, 4, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {27, 28, 29, 30, 31, 32, 33, 34, 35, 36} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If his card is, 26, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 36, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 809 The value (to Player1) of the game is, ----, and in decimal, 0.09172335601 8820 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 32, 33, 34, 35, 36} If his card is, 4, then he should bet with probability, 4/7, and check with probability, 3/7 If Player 2's card is in the following set then he should DEFINITELY call {28, 29, 30, 31, 32, 33, 34, 35, 36} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If his card is, 27, then he should call with probability, 1/7, and fold with probability, 6/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 36, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 3/35, and in decimal, 0.08571428571 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 33, 34, 35, 36} If Player 2's card is in the following set then he should DEFINITELY call {29, 30, 31, 32, 33, 34, 35, 36} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If his card is, 28, then he should call with probability, 1/4, and fold with probability, 3/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 36, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 223 The value (to Player1) of the game is, ----, and in decimal, 0.07865961199 2835 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 33, 34, 35, 36} If his card is, 4, then he should bet with probability, 1/9, and check with probability, 8/9 If Player 2's card is in the following set then he should DEFINITELY call {30, 31, 32, 33, 34, 35, 36} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If his card is, 29, then he should call with probability, 1/9, and fold with probability, 8/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 36, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 11 The value (to Player1) of the game is, ---, and in decimal, 0.07333333333 150 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 34, 35, 36} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {31, 32, 33, 34, 35, 36} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If his card is, 30, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 36, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 29 The value (to Player1) of the game is, ---, and in decimal, 0.06904761905 420 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 34, 35, 36} If his card is, 3, then he should bet with probability, 5/11, and check with probability, 6/11 If Player 2's card is in the following set then he should DEFINITELY call {31, 32, 33, 34, 35, 36} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 36, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36}, . They can see their own cards b\ ut, of course, not their opponent's. They each put 1 dollar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 9/140, and in decimal, 0.06428571429 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 34, 35, 36} If his card is, 3, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {32, 33, 34, 35, 36} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If his card is, 31, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- The deck has, 37, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 37, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 133 The value (to Player1) of the game is, ----, and in decimal, 0.09984984985 1332 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37} If his card is, 4, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 37, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/9, and in decimal, 0.1111111111 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 30, 31, 32, 33, 34, 35, 36, 37} If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20} If his card is, 21, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 37, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 71 The value (to Player1) of the game is, ---, and in decimal, 0.1066066066 666 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 32, 33, 34, 35, 36, 37} If his card is, 31, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 37, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 11 The value (to Player1) of the game is, ---, and in decimal, 0.09909909910 111 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 33, 34, 35, 36, 37} If his card is, 4, then he should bet with probability, 1/3, and check with probability, 2/3 If Player 2's card is in the following set then he should DEFINITELY call {27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 37, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 859 The value (to Player1) of the game is, ----, and in decimal, 0.09212784213 9324 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 33, 34, 35, 36, 37} If his card is, 4, then he should bet with probability, 4/7, and check with probability, 3/7 If Player 2's card is in the following set then he should DEFINITELY call {29, 30, 31, 32, 33, 34, 35, 36, 37} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If his card is, 28, then he should call with probability, 3/7, and fold with probability, 4/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 37, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 19 The value (to Player1) of the game is, ---, and in decimal, 0.08558558559 222 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 34, 35, 36, 37} If Player 2's card is in the following set then he should DEFINITELY call {30, 31, 32, 33, 34, 35, 36, 37} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If his card is, 29, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 37, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 79 The value (to Player1) of the game is, ---, and in decimal, 0.07907907908 999 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 34, 35, 36, 37} If his card is, 4, then he should bet with probability, 1/9, and check with probability, 8/9 If Player 2's card is in the following set then he should DEFINITELY call {31, 32, 33, 34, 35, 36, 37} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If his card is, 30, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 37, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 27 The value (to Player1) of the game is, ---, and in decimal, 0.07297297297 370 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 35, 36, 37} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {32, 33, 34, 35, 36, 37} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If his card is, 31, then he should call with probability, 4/5, and fold with probability, 1/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 37, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 337 The value (to Player1) of the game is, ----, and in decimal, 0.06900081900 4884 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 35, 36, 37} If his card is, 3, then he should bet with probability, 5/11, and check with probability, 6/11 If Player 2's card is in the following set then he should DEFINITELY call {32, 33, 34, 35, 36, 37} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If his card is, 31, then he should call with probability, 2/11, and fold with probability, 9/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 37, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37}, . They can see their own car\ ds but, of course, not their opponent's. They each put 1 dollar in the p\ ot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 43 The value (to Player1) of the game is, ---, and in decimal, 0.06456456456 666 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 35, 36, 37} If his card is, 3, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {33, 34, 35, 36, 37} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If his card is, 32, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- The deck has, 38, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 38, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 421 The value (to Player1) of the game is, ----, and in decimal, 0.09981033665 4218 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38} If his card is, 4, then he should bet with probability, 2/3, and check with probability, 1/3 If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If his card is, 16, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 38, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 78 The value (to Player1) of the game is, ---, and in decimal, 0.1109530583 703 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 31, 32, 33, 34, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY call {22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 38, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 751 The value (to Player1) of the game is, ----, and in decimal, 0.1068278805 7030 If Player 1's card is in the following set then he should DEFINITELY check {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 32, 33, 34, 35, 36, 37, 38} If his card is, 5, then he should bet with probability, 1/5, and check with probability, 4/5 If Player 2's card is in the following set then he should DEFINITELY call {26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} If his card is, 25, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 38, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 70 The value (to Player1) of the game is, ---, and in decimal, 0.09957325747 703 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 33, 34, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY call {28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} If his card is, 27, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 38, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 909 The value (to Player1) of the game is, ----, and in decimal, 0.09235927657 9842 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 34, 35, 36, 37, 38} If his card is, 4, then he should bet with probability, 4/7, and check with probability, 3/7 If Player 2's card is in the following set then he should DEFINITELY call {30, 31, 32, 33, 34, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If his card is, 29, then he should call with probability, 5/7, and fold with probability, 2/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 38, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 60 The value (to Player1) of the game is, ---, and in decimal, 0.08534850640 703 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY call {31, 32, 33, 34, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If his card is, 30, then he should call with probability, 3/4, and fold with probability, 1/4 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 38, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 502 The value (to Player1) of the game is, ----, and in decimal, 0.07934250040 6327 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 35, 36, 37, 38} If his card is, 4, then he should bet with probability, 1/9, and check with probability, 8/9 If Player 2's card is in the following set then he should DEFINITELY call {32, 33, 34, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} If his card is, 31, then he should call with probability, 5/9, and fold with probability, 4/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 38, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 51 The value (to Player1) of the game is, ---, and in decimal, 0.07254623044 703 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 36, 37, 38} If his card is, 3, then he should bet with probability, 2/5, and check with probability, 3/5 If Player 2's card is in the following set then he should DEFINITELY call {32, 33, 34, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 38, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 1065 The value (to Player1) of the game is, -----, and in decimal, 0.06886072676 15466 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 36, 37, 38} If his card is, 3, then he should bet with probability, 5/11, and check with probability, 6/11 If Player 2's card is in the following set then he should DEFINITELY call {33, 34, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If his card is, 32, then he should call with probability, 4/11, and fold with probability, 7/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 38, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38}, . They can see their own\ cards but, of course, not their opponent's. They each put 1 dollar in t\ he pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 91 The value (to Player1) of the game is, ----, and in decimal, 0.06472261735 1406 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 36, 37, 38} If his card is, 3, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {34, 35, 36, 37, 38} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If his card is, 33, then he should call with probability, 5/6, and fold with probability, 1/6 -------------------------------------------- The deck has, 39, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 39, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 74 The value (to Player1) of the game is, ---, and in decimal, 0.09986504723 741 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15} If his card is, 16, then he should call with probability, 1/3, and fold with probability, 2/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 39, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 82 The value (to Player1) of the game is, ---, and in decimal, 0.1106612686 741 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 32, 33, 34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY call {23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 39, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 61 The value (to Player1) of the game is, ---, and in decimal, 0.1070175439 570 If Player 1's card is in the following set then he should DEFINITELY check {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 33, 34, 35, 36, 37, 38, 39} If his card is, 5, then he should bet with probability, 1/5, and check with probability, 4/5 If Player 2's card is in the following set then he should DEFINITELY call {27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} If his card is, 26, then he should call with probability, 3/5, and fold with probability, 2/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 39, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 74 The value (to Player1) of the game is, ---, and in decimal, 0.09986504723 741 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY call {29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} If his card is, 28, then he should call with probability, 2/3, and fold with probability, 1/3 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 39, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 137 The value (to Player1) of the game is, ----, and in decimal, 0.09244264507 1482 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 35, 36, 37, 38, 39} If his card is, 4, then he should bet with probability, 4/7, and check with probability, 3/7 If Player 2's card is in the following set then he should DEFINITELY call {30, 31, 32, 33, 34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 39, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 21 The value (to Player1) of the game is, ---, and in decimal, 0.08502024291 247 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY call {31, 32, 33, 34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 39, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 530 The value (to Player1) of the game is, ----, and in decimal, 0.07947218474 6669 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 36, 37, 38, 39} If his card is, 4, then he should bet with probability, 1/9, and check with probability, 8/9 If Player 2's card is in the following set then he should DEFINITELY call {33, 34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If his card is, 32, then he should call with probability, 7/9, and fold with probability, 2/9 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 39, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 18 The value (to Player1) of the game is, ---, and in decimal, 0.07287449393 247 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 37, 38, 39} If his card is, 36, then he should bet with probability, 3/4, and check with probability, 1/4 If Player 2's card is in the following set then he should DEFINITELY call {33, 34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 39, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 373 The value (to Player1) of the game is, ----, and in decimal, 0.06864188443 5434 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 37, 38, 39} If his card is, 3, then he should bet with probability, 5/11, and check with probability, 6/11 If Player 2's card is in the following set then he should DEFINITELY call {34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If his card is, 33, then he should call with probability, 6/11, and fold with probability, 5/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 39, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39}, . They can see their\ own cards but, of course, not their opponent's. They each put 1 dollar \ in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 16 The value (to Player1) of the game is, ---, and in decimal, 0.06477732794 247 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 37, 38, 39} If his card is, 3, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {34, 35, 36, 37, 38, 39} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} -------------------------------------------- The deck has, 40, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 40, cards and fixed bet-size, 1 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 1, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 1, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY call {17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 40, cards and fixed bet-size, 2 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 2, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 2, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 173 The value (to Player1) of the game is, ----, and in decimal, 0.1108974359 1560 If Player 1's card is in the following set then he should DEFINITELY check {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 32, 33, 34, 35, 36, 37, 38, 39, 40} If his card is, 5, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22} If his card is, 23, then he should call with probability, 1/2, and fold with probability, 1/2 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 40, cards and fixed bet-size, 3 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 3, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 3, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 167 The value (to Player1) of the game is, ----, and in decimal, 0.1070512821 1560 If Player 1's card is in the following set then he should DEFINITELY check {6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 34, 35, 36, 37, 38, 39, 40} If his card is, 5, then he should bet with probability, 1/5, and check with probability, 4/5 If Player 2's card is in the following set then he should DEFINITELY call {27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 40, cards and fixed bet-size, 4 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 4, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 4, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. The value (to Player1) of the game is, 1/10, and in decimal, 0.1000000000 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 4, 35, 36, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY call {29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 40, cards and fixed bet-size, 5 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 5, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 5, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 1009 The value (to Player1) of the game is, -----, and in decimal, 0.09239926740 10920 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 36, 37, 38, 39, 40} If his card is, 4, then he should bet with probability, 4/7, and check with probability, 3/7 If Player 2's card is in the following set then he should DEFINITELY call {31, 32, 33, 34, 35, 36, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29} If his card is, 30, then he should call with probability, 2/7, and fold with probability, 5/7 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 40, cards and fixed bet-size, 6 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 6, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 6, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 11 The value (to Player1) of the game is, ---, and in decimal, 0.08461538462 130 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY call {32, 33, 34, 35, 36, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 40, cards and fixed bet-size, 7 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 7, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 7, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 31 The value (to Player1) of the game is, ---, and in decimal, 0.07948717949 390 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 37, 38, 39, 40} If his card is, 4, then he should bet with probability, 1/9, and check with probability, 8/9 If Player 2's card is in the following set then he should DEFINITELY call {33, 34, 35, 36, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 40, cards and fixed bet-size, 8 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 8, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 8, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 11 The value (to Player1) of the game is, ---, and in decimal, 0.07333333333 150 If Player 1's card is in the following set then he should DEFINITELY check {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 3, 37, 38, 39, 40} If his card is, 4, then he should bet with probability, 1/5, and check with probability, 4/5 If Player 2's card is in the following set then he should DEFINITELY call {34, 35, 36, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32} If his card is, 33, then he should call with probability, 1/5, and fold with probability, 4/5 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 40, cards and fixed bet-size, 9 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 9, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 9, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 391 The value (to Player1) of the game is, ----, and in decimal, 0.06835664336 5720 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 38, 39, 40} If his card is, 3, then he should bet with probability, 5/11, and check with probability, 6/11 If Player 2's card is in the following set then he should DEFINITELY call {35, 36, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If his card is, 34, then he should call with probability, 8/11, and fold with probability, 3/11 -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for von Neumann Poker with, 40, cards and fixed bet-size, 10 Player 1 and Player 2 are each dealt diffrent cards from, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40}, . They can see t\ heir own cards but, of course, not their opponent's. They each put 1 dol\ lar in the pot. Player 1 can decide to check, in which case the cards are compared and whoev\ er has the larger card wins the pot, and the game is over. Player 1 can also decide to bet, in which he places, 10, dollars in the pot, and the game goes to Player 2. Player 2 can decide to fold, and forfeit his money, losing one dollar, and t\ he game ends, or else he can decide to call, putting his own, 10, dollars, and the cards are compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 101 The value (to Player1) of the game is, ----, and in decimal, 0.06474358974 1560 If Player 1's card is in the following set then he should DEFINITELY check {4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37} If Player 1's card is in the following set then he should DEFINITELY bet {1, 2, 38, 39, 40} If his card is, 3, then he should bet with probability, 1/2, and check with probability, 1/2 If Player 2's card is in the following set then he should DEFINITELY call {35, 36, 37, 38, 39, 40} If Player 2's card is in the following set then he should DEFINITELY fold {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33} If his card is, 34, then he should call with probability, 1/6, and fold with probability, 5/6 This ends this paper that took, 1172.062, seconds to generate.