One Mixed Nash Equiilbrium for DJ Newman Poker with number of cards from 2 to, 14, and max. bets until the saturation value By Shalosh B. Ekhad -------------------------------------------- The deck has, 2, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 2, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1, then he should bet, 0, dollars If Player 1 has card, 2, then he should bet, 0, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- The deck has, 3, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 3, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1 He should bet, 0, dollars with prob., 2/3 He should bet, 1, dollars with prob., 1/3 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 1, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- The deck has, 4, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 4, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1 He should bet, 0, dollars with prob., 2/3 He should bet, 1, dollars with prob., 1/3 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 1, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- The deck has, 5, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 5, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1 He should bet, 0, dollars with prob., 2/3 He should bet, 1, dollars with prob., 1/3 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 1, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 3 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 2/3, and fold with prob., 1/3 If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 5, cards and allowed bets in, {0, 1, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1 He should bet, 0, dollars with prob., 1/2 He should bet, 2, dollars with prob., 1/2 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 2, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} ----------------------------------- -------------------------------------------- The deck has, 6, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 6, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1 He should bet, 0, dollars with prob., 1/3 He should bet, 1, dollars with prob., 2/3 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 1, dollars If Player 1 has card, 6, then he should bet, 1, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 3 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 6, cards and allowed bets in, {0, 1, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1 He should bet, 0, dollars with prob., 1/6 He should bet, 1, dollars with prob., 1/3 He should bet, 2, dollars with prob., 1/2 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 1, dollars If Player 1 has card, 6, then he should bet, 2, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 3 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 4 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1/2, and fold with prob., 1/2 If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} ----------------------------------- -------------------------------------------- The deck has, 7, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 7, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1 He should bet, 0, dollars with prob., 1/3 He should bet, 1, dollars with prob., 2/3 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 1, dollars If Player 1 has card, 7, then he should bet, 1, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 7, cards and allowed bets in, {0, 1, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 5/42, and in decimal, 0.1190476190 if Player 1 has card, 1 He should bet, 0, dollars with prob., 1/6 He should bet, 1, dollars with prob., 1/3 He should bet, 2, dollars with prob., 1/2 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 1, dollars If Player 1 has card, 7, then he should bet, 2, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 7, cards and allowed bets in, {0, 1, 2, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1238095238 105 if Player 1 has card, 1 He should bet, 0, dollars with prob., 1/15 He should bet, 1, dollars with prob., 1/3 He should bet, 3, dollars with prob., 3/5 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 1, dollars If Player 1 has card, 7, then he should bet, 3, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 4 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1/10, and fold with prob., 9/10 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 5 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 2/5, and fold with prob., 3/5 If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} ----------------------------------- -------------------------------------------- The deck has, 8, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 8, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1 He should bet, 0, dollars with prob., 1/3 He should bet, 1, dollars with prob., 2/3 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 1, dollars If Player 1 has card, 8, then he should bet, 1, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 4 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 2/3, and fold with prob., 1/3 If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 8, cards and allowed bets in, {0, 1, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 5/42, and in decimal, 0.1190476190 if Player 1 has card, 1 He should bet, 0, dollars with prob., 1/6 He should bet, 1, dollars with prob., 1/3 He should bet, 2, dollars with prob., 1/2 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 1, dollars If Player 1 has card, 8, then he should bet, 2, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 4 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 2/3, and fold with prob., 1/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 5 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1/2, and fold with prob., 1/2 If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 8, cards and allowed bets in, {0, 1, 2, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1261904762 420 if Player 1 has card, 1 He should bet, 0, dollars with prob., 1/15 He should bet, 1, dollars with prob., 1/3 He should bet, 3, dollars with prob., 3/5 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 1, dollars If Player 1 has card, 8, then he should bet, 3, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 4 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 2/3, and fold with prob., 1/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 5 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 7/10, and fold with prob., 3/10 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 6 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 4/5, and fold with prob., 1/5 If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} ----------------------------------- -------------------------------------------- The deck has, 9, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 9, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1, then he should bet, 1, dollars If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 1, dollars If Player 1 has card, 8, then he should bet, 1, dollars If Player 1 has card, 9, then he should bet, 1, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 4 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 9, cards and allowed bets in, {0, 1, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1180555556 144 if Player 1 has card, 1 He should bet, 1, dollars with prob., 1/2 He should bet, 2, dollars with prob., 1/2 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars if Player 1 has card, 7 He should bet, 0, dollars with prob., 1/2 He should bet, 1, dollars with prob., 1/2 If Player 1 has card, 8, then he should bet, 1, dollars If Player 1 has card, 9, then he should bet, 2, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2} If Player 2 has card, 6 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 3/4, and fold with prob., 1/4 If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 9, cards and allowed bets in, {0, 1, 2, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 1/8, and in decimal, 0.1250000000 if Player 1 has card, 1 He should bet, 1, dollars with prob., 2/5 He should bet, 3, dollars with prob., 3/5 If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars if Player 1 has card, 7 He should bet, 0, dollars with prob., 4/5 He should bet, 1, dollars with prob., 1/5 If Player 1 has card, 8, then he should bet, 1, dollars If Player 1 has card, 9, then he should bet, 3, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} Player 2 should definitely fold if Player 1's bet belongs to, {3} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} ----------------------------------- -------------------------------------------- The deck has, 10, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 10, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1, then he should bet, 1, dollars If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 1, dollars If Player 1 has card, 9, then he should bet, 1, dollars If Player 1 has card, 10, then he should bet, 1, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 10, cards and allowed bets in, {0, 1, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1185185185 135 if Player 1 has card, 1 He should bet, 1, dollars with prob., 1/2 He should bet, 2, dollars with prob., 1/2 if Player 1 has card, 2 He should bet, 0, dollars with prob., 5/6 He should bet, 1, dollars with prob., 1/6 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 1, dollars If Player 1 has card, 9, then he should bet, 1, dollars If Player 1 has card, 10, then he should bet, 2, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 5 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 10, cards and allowed bets in, {0, 1, 2, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re compared, and whoever has the larger card takes the whole pot. We will describe ONE mixed strategy that is a Nash Equilibrium. 169 The value (to Player1) of the game is, ----, and in decimal, 0.1251851852 1350 if Player 1 has card, 1 He should bet, 1, dollars with prob., 2/5 He should bet, 3, dollars with prob., 3/5 if Player 1 has card, 2 11 He should bet, 0, dollars with prob., -- 15 He should bet, 1, dollars with prob., 4/15 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 1, dollars If Player 1 has card, 9, then he should bet, 1, dollars If Player 1 has card, 10, then he should bet, 3, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 5 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 6 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1/6, and fold with prob., 5/6 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 7 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 1/5, and fold with prob., 4/5 If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 10, cards and allowed bets in, {0, 1, 2, 3, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1259259259 135 if Player 1 has card, 1 He should bet, 1, dollars with prob., 1/3 He should bet, 4, dollars with prob., 2/3 if Player 1 has card, 2 He should bet, 0, dollars with prob., 2/3 He should bet, 1, dollars with prob., 1/3 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 1, dollars If Player 1 has card, 9, then he should bet, 1, dollars If Player 1 has card, 10, then he should bet, 4, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 5 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 6 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1/6, and fold with prob., 5/6 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 7 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 2/9, and fold with prob., 7/9 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 8 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 4, then Player 2 should call with prob., 2/3, and fold with prob., 1/3 If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} ----------------------------------- -------------------------------------------- The deck has, 11, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 11, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1, then he should bet, 1, dollars If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 1, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 1, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 5 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 2/3, and fold with prob., 1/3 If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 11, cards and allowed bets in, {0, 1, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1181818182 110 if Player 1 has card, 1 He should bet, 1, dollars with prob., 1/2 He should bet, 2, dollars with prob., 1/2 if Player 1 has card, 2 He should bet, 0, dollars with prob., 5/6 He should bet, 1, dollars with prob., 1/6 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 1, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 2, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2} If Player 2 has card, 7 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1/2, and fold with prob., 1/2 If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 11, cards and allowed bets in, {0, 1, 2, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1254545455 550 if Player 1 has card, 1 He should bet, 1, dollars with prob., 2/5 He should bet, 3, dollars with prob., 3/5 if Player 1 has card, 2 11 He should bet, 0, dollars with prob., -- 15 He should bet, 1, dollars with prob., 4/15 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 1, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 3, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3} If Player 2 has card, 7 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1/2, and fold with prob., 1/2 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 8 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 3/5, and fold with prob., 2/5 If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 11, cards and allowed bets in, {0, 1, 2, 3, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 7/55, and in decimal, 0.1272727273 if Player 1 has card, 1 He should bet, 1, dollars with prob., 1/3 He should bet, 4, dollars with prob., 2/3 if Player 1 has card, 2 He should bet, 0, dollars with prob., 2/3 He should bet, 1, dollars with prob., 1/3 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 1, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 4, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3, 4} If Player 2 has card, 7 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1/2, and fold with prob., 1/2 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 8 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 2/3, and fold with prob., 1/3 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} ----------------------------------- -------------------------------------------- The deck has, 12, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 12, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1, then he should bet, 1, dollars If Player 1 has card, 2, then he should bet, 0, dollars If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 1, dollars If Player 1 has card, 12, then he should bet, 1, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 12, cards and allowed bets in, {0, 1, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1186868687 396 If Player 1 has card, 1, then he should bet, 2, dollars if Player 1 has card, 2 He should bet, 0, dollars with prob., 2/3 He should bet, 1, dollars with prob., 1/3 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 2, dollars If Player 1 has card, 12, then he should bet, 2, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 6 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 2/3, and fold with prob., 1/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 12, cards and allowed bets in, {0, 1, 2, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1262626263 198 if Player 1 has card, 1 He should bet, 2, dollars with prob., 2/5 He should bet, 3, dollars with prob., 3/5 if Player 1 has card, 2 17 He should bet, 0, dollars with prob., -- 30 He should bet, 1, dollars with prob., 1/3 He should bet, 2, dollars with prob., 1/10 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 2, dollars If Player 1 has card, 12, then he should bet, 3, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 6 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 2/3, and fold with prob., 1/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} Player 2 should definitely fold if Player 1's bet belongs to, {3} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 12, cards and allowed bets in, {0, 1, 2, 3, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1287878788 132 if Player 1 has card, 1 He should bet, 2, dollars with prob., 1/3 He should bet, 4, dollars with prob., 2/3 if Player 1 has card, 2 He should bet, 0, dollars with prob., 1/2 He should bet, 1, dollars with prob., 1/3 He should bet, 2, dollars with prob., 1/6 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 2, dollars If Player 1 has card, 12, then he should bet, 4, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 6 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 2/3, and fold with prob., 1/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3, 4} If Player 2 has card, 8 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 1/9, and fold with prob., 8/9 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 9 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 4, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} ----------------------------------- -------------------------------------------- The deck has, 13, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 13, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1, then he should bet, 1, dollars if Player 1 has card, 2 He should bet, 0, dollars with prob., 2/3 He should bet, 1, dollars with prob., 1/3 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 1, dollars If Player 1 has card, 12, then he should bet, 1, dollars If Player 1 has card, 13, then he should bet, 1, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 6 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 13 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 13, cards and allowed bets in, {0, 1, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1196581197 117 If Player 1 has card, 1, then he should bet, 2, dollars if Player 1 has card, 2 He should bet, 0, dollars with prob., 1/3 He should bet, 1, dollars with prob., 2/3 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 1, dollars If Player 1 has card, 12, then he should bet, 2, dollars If Player 1 has card, 13, then he should bet, 2, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 6 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2} If Player 2 has card, 8 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1/2, and fold with prob., 1/2 If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 13 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 13, cards and allowed bets in, {0, 1, 2, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1273504274 1170 if Player 1 has card, 1 He should bet, 2, dollars with prob., 2/5 He should bet, 3, dollars with prob., 3/5 if Player 1 has card, 2 He should bet, 0, dollars with prob., 7/30 He should bet, 1, dollars with prob., 2/3 He should bet, 2, dollars with prob., 1/10 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 1, dollars If Player 1 has card, 12, then he should bet, 2, dollars If Player 1 has card, 13, then he should bet, 3, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 6 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3} If Player 2 has card, 8 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1/2, and fold with prob., 1/2 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 9 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 2/5, and fold with prob., 3/5 If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 13 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 13, cards and allowed bets in, {0, 1, 2, 3, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1303418803 468 if Player 1 has card, 1 He should bet, 2, dollars with prob., 1/3 He should bet, 4, dollars with prob., 2/3 if Player 1 has card, 2 He should bet, 0, dollars with prob., 1/6 He should bet, 1, dollars with prob., 2/3 He should bet, 2, dollars with prob., 1/6 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 1, dollars If Player 1 has card, 12, then he should bet, 2, dollars If Player 1 has card, 13, then he should bet, 4, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 6 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3, 4} If Player 2 has card, 8 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1/2, and fold with prob., 1/2 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 9 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 5/9, and fold with prob., 4/9 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 10 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 4, then Player 2 should call with prob., 2/3, and fold with prob., 1/3 If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} If Player 2 has card, 13 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 13, cards and allowed bets in, {0, 1, 2, 3, 4, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1306471306 819 if Player 1 has card, 1 He should bet, 2, dollars with prob., 2/7 He should bet, 5, dollars with prob., 5/7 if Player 1 has card, 2 He should bet, 0, dollars with prob., 5/42 He should bet, 1, dollars with prob., 2/3 He should bet, 2, dollars with prob., 3/14 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 1, dollars If Player 1 has card, 11, then he should bet, 1, dollars If Player 1 has card, 12, then he should bet, 2, dollars If Player 1 has card, 13, then he should bet, 5, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4, 5} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4, 5} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4, 5} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4, 5} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4, 5} If Player 2 has card, 6 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1/3, and fold with prob., 2/3 If the bet size is, 2, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 5, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3, 4, 5} If Player 2 has card, 8 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1/2, and fold with prob., 1/2 If the bet size is, 3, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 5, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 9 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 4/7, and fold with prob., 3/7 If the bet size is, 4, then Player 2 should call with prob., 0, and fold with prob., 1 If the bet size is, 5, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 10 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 1, and fold with prob., 0 19 If the bet size is, 4, then Player 2 should call with prob., --, 28 and fold with prob., 9/28 If the bet size is, 5, then Player 2 should call with prob., 1/7, and fold with prob., 6/7 If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4, 5} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4, 5} If Player 2 has card, 13 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4, 5} ----------------------------------- -------------------------------------------- The deck has, 14, cards -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 14, cards and allowed bets in, {0, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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 has card, 1, then he should bet, 1, dollars if Player 1 has card, 2 He should bet, 0, dollars with prob., 2/3 He should bet, 1, dollars with prob., 1/3 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 0, dollars If Player 1 has card, 11, then he should bet, 1, dollars If Player 1 has card, 12, then he should bet, 1, dollars If Player 1 has card, 13, then he should bet, 1, dollars If Player 1 has card, 14, then he should bet, 1, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 13 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} If Player 2 has card, 14 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 14, cards and allowed bets in, {0, 1, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1208791209 91 If Player 1 has card, 1, then he should bet, 2, dollars if Player 1 has card, 2 He should bet, 0, dollars with prob., 1/3 He should bet, 1, dollars with prob., 2/3 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 0, dollars If Player 1 has card, 11, then he should bet, 1, dollars If Player 1 has card, 12, then he should bet, 1, dollars If Player 1 has card, 13, then he should bet, 2, dollars If Player 1 has card, 14, then he should bet, 2, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 13 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} If Player 2 has card, 14 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 14, cards and allowed bets in, {0, 1, 2, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 9/70, and in decimal, 0.1285714286 if Player 1 has card, 1 He should bet, 2, dollars with prob., 2/5 He should bet, 3, dollars with prob., 3/5 if Player 1 has card, 2 He should bet, 0, dollars with prob., 7/30 He should bet, 1, dollars with prob., 2/3 He should bet, 2, dollars with prob., 1/10 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 0, dollars If Player 1 has card, 11, then he should bet, 1, dollars If Player 1 has card, 12, then he should bet, 1, dollars If Player 1 has card, 13, then he should bet, 2, dollars If Player 1 has card, 14, then he should bet, 3, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} Player 2 should definitely fold if Player 1's bet belongs to, {3} If Player 2 has card, 10 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 4/5, and fold with prob., 1/5 If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 13 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} If Player 2 has card, 14 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 14, cards and allowed bets in, {0, 1, 2, 3, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1318681319 91 if Player 1 has card, 1 He should bet, 2, dollars with prob., 1/3 He should bet, 4, dollars with prob., 2/3 if Player 1 has card, 2 He should bet, 0, dollars with prob., 1/6 He should bet, 1, dollars with prob., 2/3 He should bet, 2, dollars with prob., 1/6 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 0, dollars If Player 1 has card, 11, then he should bet, 1, dollars If Player 1 has card, 12, then he should bet, 1, dollars If Player 1 has card, 13, then he should bet, 2, dollars If Player 1 has card, 14, then he should bet, 4, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3, 4} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3, 4} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} Player 2 should definitely fold if Player 1's bet belongs to, {3, 4} If Player 2 has card, 10 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3} Player 2 should definitely fold if Player 1's bet belongs to, {4} If Player 2 has card, 11 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} If Player 2 has card, 13 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} If Player 2 has card, 14 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4} ----------------------------------- -------------------------------------------- ----------------------------------------------------------------- (Mixed) Nash Equlibrium for DJ Newman Poker with, 14, cards and allowed bets in, {0, 1, 2, 3, 4, 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. Looking at his card, Player 1 decides what amount to bet (including 0 dollar\ s, that is the same as checking) Looking at his card, and Player1's bet size, Player 2 can decide to fold, a\ nd forfeit his money, losing one dollar, and the game ends, or else he c\ an decide to call, putting the same amount as Player 1, and the cards a\ re 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, 0.1326530612 98 if Player 1 has card, 1 He should bet, 2, dollars with prob., 2/7 He should bet, 5, dollars with prob., 5/7 if Player 1 has card, 2 He should bet, 0, dollars with prob., 5/42 He should bet, 1, dollars with prob., 2/3 He should bet, 2, dollars with prob., 3/14 If Player 1 has card, 3, then he should bet, 0, dollars If Player 1 has card, 4, then he should bet, 0, dollars If Player 1 has card, 5, then he should bet, 0, dollars If Player 1 has card, 6, then he should bet, 0, dollars If Player 1 has card, 7, then he should bet, 0, dollars If Player 1 has card, 8, then he should bet, 0, dollars If Player 1 has card, 9, then he should bet, 0, dollars If Player 1 has card, 10, then he should bet, 0, dollars If Player 1 has card, 11, then he should bet, 1, dollars If Player 1 has card, 12, then he should bet, 1, dollars If Player 1 has card, 13, then he should bet, 2, dollars If Player 1 has card, 14, then he should bet, 5, dollars ------------------ If Player 2 has card, 1 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4, 5} If Player 2 has card, 2 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4, 5} If Player 2 has card, 3 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4, 5} If Player 2 has card, 4 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4, 5} If Player 2 has card, 5 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4, 5} If Player 2 has card, 6 Player 2 should definitely call if Player 1's bet belongs to, {0} Player 2 should definitely fold if Player 1's bet belongs to, {1, 2, 3, 4, 5} If Player 2 has card, 7 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3, 4, 5} If Player 2 has card, 8 Player 2 should definitely call if Player 1's bet belongs to, {0, 1} Player 2 should definitely fold if Player 1's bet belongs to, {2, 3, 4, 5} If Player 2 has card, 9 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2} Player 2 should definitely fold if Player 1's bet belongs to, {3, 4, 5} If Player 2 has card, 10 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 4, then Player 2 should call with prob., 1/28, 27 and fold with prob., -- 28 If the bet size is, 5, then Player 2 should call with prob., 0, and fold with prob., 1 If Player 2 has card, 11 If the bet size is, 0, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 1, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 2, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 3, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 4, then Player 2 should call with prob., 1, and fold with prob., 0 If the bet size is, 5, then Player 2 should call with prob., 3/7, and fold with prob., 4/7 If Player 2 has card, 12 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4, 5} If Player 2 has card, 13 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4, 5} If Player 2 has card, 14 Player 2 should definitely call if Player 1's bet belongs to, {0, 1, 2, 3, 4, 5} ----------------------------------- This ends this paper that took, 103307.358, seconds to generate.