All the Pure Nash Equiilbria, and ONE Mixed One, for von Neumann Poker with \ number of cards from 2 to, 11, and bet sizes from, 1, to, 3 By Shalosh B. Ekhad Player 1 and Player 2 each get (different) cards from a deck of n cards numb\ ered 1,..., n. They each see their own card, but not the other's They each put 1 dollar in the pot Player 1 can either check, in which case the cards are compared, and whoever\ has the larger card gets the pot (and hence wins 1 dollar), or bet an a\ dditional amount of b dollars Now Player 2 can decide to cut his losses, and fold, and the pot goes immedi\ ately to player 1, who is going to make a dollar, and player 2 is going \ to lose a dollar or else to also put b dollars in the pot Now the cards are compared, and whoever has the larger card gets the pot, wi\ nning b+1 dollars (and the other player lost b+1 dollars Here all the pure Nash Equilibria, and ONE Mixed one, for card sizes from 2 to, 11, and bet sizes from, 1, to, 3 -------------------------------------------- -------------------------------------------- If the deck has, 2, distinct cards and the bet size is, 1 There are, 2, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {}, and checks otherwise, while Player 2 calls if his card is in the set, {2}, and folds otherwies , the value of the game is, 0 NE number, 2 Player 1 bets if his card is in the set, {2}, and checks otherwise, while Player 2 calls if his card is in the set, {2}, and folds otherwies , the value of the game is, 0 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 0, and in decmimals, 0. -------------------------------------------- If the deck has, 2, distinct cards and the bet size is, 2 There are, 2, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {}, and checks otherwise, while Player 2 calls if his card is in the set, {2}, and folds otherwies , the value of the game is, 0 NE number, 2 Player 1 bets if his card is in the set, {2}, and checks otherwise, while Player 2 calls if his card is in the set, {2}, and folds otherwies , the value of the game is, 0 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 0, and in decmimals, 0. -------------------------------------------- If the deck has, 2, distinct cards and the bet size is, 3 There are, 2, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {}, and checks otherwise, while Player 2 calls if his card is in the set, {2}, and folds otherwies , the value of the game is, 0 NE number, 2 Player 1 bets if his card is in the set, {2}, and checks otherwise, while Player 2 calls if his card is in the set, {2}, and folds otherwies , the value of the game is, 0 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 0, and in decmimals, 0. -------------------------------------------- -------------------------------------------- If the deck has, 3, distinct cards and the bet size is, 1 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1/3, and check with probability, 2/3 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 1/3, and fold with probability, 2/3 If Player 2 got card, 3, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/18, and in decmimals, 0.05555555556 -------------------------------------------- If the deck has, 3, distinct cards and the bet size is, 2 There are, 2, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {}, and checks otherwise, while Player 2 calls if his card is in the set, {3}, and folds otherwies , the value of the game is, 0 NE number, 2 Player 1 bets if his card is in the set, {3}, and checks otherwise, while Player 2 calls if his card is in the set, {3}, and folds otherwies , the value of the game is, 0 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 0, and in decmimals, 0. -------------------------------------------- If the deck has, 3, distinct cards and the bet size is, 3 There are, 2, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {}, and checks otherwise, while Player 2 calls if his card is in the set, {3}, and folds otherwies , the value of the game is, 0 NE number, 2 Player 1 bets if his card is in the set, {3}, and checks otherwise, while Player 2 calls if his card is in the set, {3}, and folds otherwies , the value of the game is, 0 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 0, and in decmimals, 0. -------------------------------------------- -------------------------------------------- If the deck has, 4, distinct cards and the bet size is, 1 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1/3, and check with probability, 2/3 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 4, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/12, and in decmimals, 0.08333333333 -------------------------------------------- If the deck has, 4, distinct cards and the bet size is, 2 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1/2, and check with probability, 1/2 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 1/2, and fold with probability, 1/2 If Player 2 got card, 4, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/12, and in decmimals, 0.08333333333 -------------------------------------------- If the deck has, 4, distinct cards and the bet size is, 3 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 3/5, and check with probability, 2/5 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 1/5, and fold with probability, 4/5 If Player 2 got card, 4, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/20, and in decmimals, 0.05000000000 -------------------------------------------- -------------------------------------------- If the deck has, 5, distinct cards and the bet size is, 1 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1/3, and check with probability, 2/3 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 2/3, and fold with probability, 1/3 If Player 2 got card, 4, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 5, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/12, and in decmimals, 0.08333333333 -------------------------------------------- If the deck has, 5, distinct cards and the bet size is, 2 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1/2, and check with probability, 1/2 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 5, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/10, and in decmimals, 0.1000000000 -------------------------------------------- If the deck has, 5, distinct cards and the bet size is, 3 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 3/5, and check with probability, 2/5 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 3/5, and fold with probability, 2/5 If Player 2 got card, 5, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 9/100, and in decmimals, 0.09000000000 -------------------------------------------- -------------------------------------------- If the deck has, 6, distinct cards and the bet size is, 1 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 2/3, and check with probability, 1/3 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 6, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 1/3, and fold with probability, 2/3 If Player 2 got card, 4, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 5, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 6, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 4/45, and in decmimals, 0.08888888889 -------------------------------------------- If the deck has, 6, distinct cards and the bet size is, 2 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1/2, and check with probability, 1/2 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 1/2, and fold with probability, 1/2 If Player 2 got card, 5, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 6, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/10, and in decmimals, 0.1000000000 -------------------------------------------- If the deck has, 6, distinct cards and the bet size is, 3 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 3/5, and check with probability, 2/5 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 6, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/10, and in decmimals, 0.1000000000 -------------------------------------------- -------------------------------------------- If the deck has, 7, distinct cards and the bet size is, 1 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 2/3, and check with probability, 1/3 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 7, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 5, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 6, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 2/21, and in decmimals, 0.09523809524 -------------------------------------------- If the deck has, 7, distinct cards and the bet size is, 2 There are, 3, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 6, 7}, and checks otherwise, while Player 2 calls if his card is in the set, {3, 6, 7}, and folds otherwies , the value of the game is, 2/21 NE number, 2 Player 1 bets if his card is in the set, {1, 6, 7}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 6, 7}, and folds otherwies , the value of the game is, 2/21 NE number, 3 Player 1 bets if his card is in the set, {1, 6, 7}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 6, 7}, and folds otherwies , the value of the game is, 2/21 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1/2, and check with probability, 1/2 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 6, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 2/21, and in decmimals, 0.09523809524 -------------------------------------------- If the deck has, 7, distinct cards and the bet size is, 3 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 3/5, and check with probability, 2/5 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 2/5, and fold with probability, 3/5 If Player 2 got card, 6, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/10, and in decmimals, 0.1000000000 -------------------------------------------- -------------------------------------------- If the deck has, 8, distinct cards and the bet size is, 1 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 2/3, and check with probability, 1/3 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 8, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 2/3, and fold with probability, 1/3 If Player 2 got card, 5, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 6, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 8, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 2/21, and in decmimals, 0.09523809524 -------------------------------------------- If the deck has, 8, distinct cards and the bet size is, 2 There are, 3, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 7, 8}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 7, 8}, and folds otherwies , the value of the game is, 3/28 NE number, 2 Player 1 bets if his card is in the set, {1, 7, 8}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 7, 8}, and folds otherwies , the value of the game is, 3/28 NE number, 3 Player 1 bets if his card is in the set, {1, 7, 8}, and checks otherwise, while Player 2 calls if his card is in the set, {6, 7, 8}, and folds otherwies , the value of the game is, 3/28 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 8, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 1/2, and fold with probability, 1/2 If Player 2 got card, 6, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 8, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 3/28, and in decmimals, 0.1071428571 -------------------------------------------- If the deck has, 8, distinct cards and the bet size is, 3 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 3/5, and check with probability, 2/5 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 8, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 6, he should call With probability, 4/5, and fold with probability, 1/5 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 8, he should call With probability, 1, and fold with probability, 0 27 The value of this pair of strategies is, ---, and in decmimals, 0.09642857143 280 -------------------------------------------- -------------------------------------------- If the deck has, 9, distinct cards and the bet size is, 1 There are, 5, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 7, 8, 9}, and checks otherwise, while Player 2 calls if his card is in the set, {3, 5, 7, 8, 9}, and folds otherwies , the value of the game is, 7/72 NE number, 2 Player 1 bets if his card is in the set, {1, 7, 8, 9}, and checks otherwise, while Player 2 calls if his card is in the set, {3, 6, 7, 8, 9}, and folds otherwies , the value of the game is, 7/72 NE number, 3 Player 1 bets if his card is in the set, {1, 7, 8, 9}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 5, 7, 8, 9}, and folds otherwies , the value of the game is, 7/72 NE number, 4 Player 1 bets if his card is in the set, {1, 7, 8, 9}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 6, 7, 8, 9}, and folds otherwies , the value of the game is, 7/72 NE number, 5 Player 1 bets if his card is in the set, {1, 7, 8, 9}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 6, 7, 8, 9}, and folds otherwies , the value of the game is, 7/72 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 8, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 9, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 1/3, and fold with probability, 2/3 If Player 2 got card, 5, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 6, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 8, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 9, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 7/72, and in decmimals, 0.09722222222 -------------------------------------------- If the deck has, 9, distinct cards and the bet size is, 2 There are, 7, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 8, 9}, and checks otherwise, while Player 2 calls if his card is in the set, {3, 6, 8, 9}, and folds otherwies , the value of the game is, 1/9 NE number, 2 Player 1 bets if his card is in the set, {1, 8, 9}, and checks otherwise, while Player 2 calls if his card is in the set, {3, 7, 8, 9}, and folds otherwies , the value of the game is, 1/9 NE number, 3 Player 1 bets if his card is in the set, {1, 8, 9}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 6, 8, 9}, and folds otherwies , the value of the game is, 1/9 NE number, 4 Player 1 bets if his card is in the set, {1, 8, 9}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 7, 8, 9}, and folds otherwies , the value of the game is, 1/9 NE number, 5 Player 1 bets if his card is in the set, {1, 8, 9}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 6, 8, 9}, and folds otherwies , the value of the game is, 1/9 NE number, 6 Player 1 bets if his card is in the set, {1, 8, 9}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 7, 8, 9}, and folds otherwies , the value of the game is, 1/9 NE number, 7 Player 1 bets if his card is in the set, {1, 8, 9}, and checks otherwise, while Player 2 calls if his card is in the set, {6, 7, 8, 9}, and folds otherwies , the value of the game is, 1/9 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 8, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 9, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 6, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 8, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 9, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/9, and in decmimals, 0.1111111111 -------------------------------------------- If the deck has, 9, distinct cards and the bet size is, 3 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 8, he should bet With probability, 2/3, and check with probability, 1/3 If Player 1 got card, 9, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 6, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 8, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 9, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 7/72, and in decmimals, 0.09722222222 -------------------------------------------- -------------------------------------------- If the deck has, 10, distinct cards and the bet size is, 1 There are, 5, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 8, 9, 10}, and checks otherwise, while Player 2 calls if his card is in the set, {3, 5, 7, 8, 9, 10}, and folds otherwies , the value of the game is, 1/10 NE number, 2 Player 1 bets if his card is in the set, {1, 8, 9, 10}, and checks otherwise, while Player 2 calls if his card is in the set, {3, 6, 7, 8, 9, 10}, and folds otherwies , the value of the game is, 1/10 NE number, 3 Player 1 bets if his card is in the set, {1, 8, 9, 10}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 5, 7, 8, 9, 10}, and folds otherwies , the value of the game is, 1/10 NE number, 4 Player 1 bets if his card is in the set, {1, 8, 9, 10}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 6, 7, 8, 9, 10}, and folds otherwies , the value of the game is, 1/10 NE number, 5 Player 1 bets if his card is in the set, {1, 8, 9, 10}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 6, 7, 8, 9, 10}, and folds otherwies , the value of the game is, 1/10 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 8, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 9, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 10, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 6, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 8, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 9, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 10, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/10, and in decmimals, 0.1000000000 -------------------------------------------- If the deck has, 10, distinct cards and the bet size is, 2 There are, 7, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 9, 10}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 7, 9, 10}, and folds otherwies , the value of the game is, 1/9 NE number, 2 Player 1 bets if his card is in the set, {1, 9, 10}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 8, 9, 10}, and folds otherwies , the value of the game is, 1/9 NE number, 3 Player 1 bets if his card is in the set, {1, 9, 10}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 7, 9, 10}, and folds otherwies , the value of the game is, 1/9 NE number, 4 Player 1 bets if his card is in the set, {1, 9, 10}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 8, 9, 10}, and folds otherwies , the value of the game is, 1/9 NE number, 5 Player 1 bets if his card is in the set, {1, 9, 10}, and checks otherwise, while Player 2 calls if his card is in the set, {6, 7, 9, 10}, and folds otherwies , the value of the game is, 1/9 NE number, 6 Player 1 bets if his card is in the set, {1, 9, 10}, and checks otherwise, while Player 2 calls if his card is in the set, {6, 8, 9, 10}, and folds otherwies , the value of the game is, 1/9 NE number, 7 Player 1 bets if his card is in the set, {1, 9, 10}, and checks otherwise, while Player 2 calls if his card is in the set, {7, 8, 9, 10}, and folds otherwies , the value of the game is, 1/9 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 8, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 9, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 10, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 6, he should call With probability, 1/2, and fold with probability, 1/2 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 8, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 9, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 10, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/9, and in decmimals, 0.1111111111 -------------------------------------------- If the deck has, 10, distinct cards and the bet size is, 3 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 2, he should bet With probability, 1/5, and check with probability, 4/5 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 8, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 9, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 10, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 6, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 7, he should call With probability, 1/5, and fold with probability, 4/5 If Player 2 got card, 8, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 9, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 10, he should call With probability, 1, and fold with probability, 0 23 The value of this pair of strategies is, ---, and in decmimals, 0.1022222222 225 -------------------------------------------- -------------------------------------------- If the deck has, 11, distinct cards and the bet size is, 1 There are, 5, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 9, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 6, 8, 9, 10, 11}, and folds otherwies , the value of the game is, 1/10 NE number, 2 Player 1 bets if his card is in the set, {1, 9, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 7, 8, 9, 10, 11}, and folds otherwies , the value of the game is, 1/10 NE number, 3 Player 1 bets if his card is in the set, {1, 9, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 6, 8, 9, 10, 11}, and folds otherwies , the value of the game is, 1/10 NE number, 4 Player 1 bets if his card is in the set, {1, 9, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 7, 8, 9, 10, 11}, and folds otherwies , the value of the game is, 1/10 NE number, 5 Player 1 bets if his card is in the set, {1, 9, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {6, 7, 8, 9, 10, 11}, and folds otherwies , the value of the game is, 1/10 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 8, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 9, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 10, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 11, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 2/3, and fold with probability, 1/3 If Player 2 got card, 6, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 8, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 9, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 10, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 11, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 1/10, and in decmimals, 0.1000000000 -------------------------------------------- If the deck has, 11, distinct cards and the bet size is, 2 There are, 12, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {3, 6, 9, 10, 11}, and folds otherwies , the value of the game is, 6/55 NE number, 2 Player 1 bets if his card is in the set, {1, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {3, 7, 9, 10, 11}, and folds otherwies , the value of the game is, 6/55 NE number, 3 Player 1 bets if his card is in the set, {1, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {3, 8, 9, 10, 11}, and folds otherwies , the value of the game is, 6/55 NE number, 4 Player 1 bets if his card is in the set, {1, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 6, 9, 10, 11}, and folds otherwies , the value of the game is, 6/55 NE number, 5 Player 1 bets if his card is in the set, {1, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 7, 9, 10, 11}, and folds otherwies , the value of the game is, 6/55 NE number, 6 Player 1 bets if his card is in the set, {1, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 8, 9, 10, 11}, and folds otherwies , the value of the game is, 6/55 NE number, 7 Player 1 bets if his card is in the set, {1, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 6, 9, 10, 11}, and folds otherwies , the value of the game is, 6/55 NE number, 8 Player 1 bets if his card is in the set, {1, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 7, 9, 10, 11}, and folds otherwies , the value of the game is, 6/55 NE number, 9 Player 1 bets if his card is in the set, {1, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {5, 8, 9, 10, 11}, and folds otherwies , the value of the game is, 6/55 NE number, 10 Player 1 bets if his card is in the set, {1, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {6, 7, 9, 10, 11}, and folds otherwies , the value of the game is, 6/55 NE number, 11 Player 1 bets if his card is in the set, {1, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {6, 8, 9, 10, 11}, and folds otherwies , the value of the game is, 6/55 NE number, 12 Player 1 bets if his card is in the set, {1, 10, 11}, and checks otherwise, while Player 2 calls if his card is in the set, {7, 8, 9, 10, 11}, and folds otherwies , the value of the game is, 6/55 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 2, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 8, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 9, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 10, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 11, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 6, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 7, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 8, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 9, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 10, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 11, he should call With probability, 1, and fold with probability, 0 The value of this pair of strategies is, 6/55, and in decmimals, 0.1090909091 -------------------------------------------- If the deck has, 11, distinct cards and the bet size is, 3 There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is If Player 1 got card, 1, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 2, he should bet With probability, 1/5, and check with probability, 4/5 If Player 1 got card, 3, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 4, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 5, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 6, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 7, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 8, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 9, he should bet With probability, 0, and check with probability, 1 If Player 1 got card, 10, he should bet With probability, 1, and check with probability, 0 If Player 1 got card, 11, he should bet With probability, 1, and check with probability, 0 Player 2's strategy is If Player 2 got card, 1, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 2, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 3, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 4, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 5, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 6, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 7, he should call With probability, 0, and fold with probability, 1 If Player 2 got card, 8, he should call With probability, 3/5, and fold with probability, 2/5 If Player 2 got card, 9, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 10, he should call With probability, 1, and fold with probability, 0 If Player 2 got card, 11, he should call With probability, 1, and fold with probability, 0 29 The value of this pair of strategies is, ---, and in decmimals, 0.1054545455 275 ----------------- This ends this paper that took, 4391.697, seconds.