All the Pure Nash Equiilbria, and ONE Mixed One, for von Neumann Poker with \ number of cards from 2 to, 27, and bet sizes from, 2, to, 2 USING SUGGESTED STRATEGIES ONLY 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, 27, and bet sizes from, 2, to, 2 -------------------------------------------- -------------------------------------------- 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, {1}, and checks otherwise, while Player 2 calls if his card is in the set, {2}, and folds otherwies , the value of the game is, -1 NE number, 2 Player 1 bets if his card is in the set, {1}, and checks otherwise, while Player 2 calls if his card is in the set, {1, 2}, and folds otherwies , the value of the game is, -1 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {2} The value of this pair of strategies is, -1, and in decmimals, -1. -------------------------------------------- -------------------------------------------- If the deck has, 3, distinct cards and the bet size is, 2 There are, 1, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 3}, and checks otherwise, while Player 2 calls if his card is in the set, {2, 3}, and folds otherwies , the value of the game is, -1/3 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1, 3} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {2, 3} The value of this pair of strategies is, -1/3, and in decmimals, -0.3333333333 -------------------------------------------- -------------------------------------------- If the deck has, 4, 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, {1, 4}, and checks otherwise, while Player 2 calls if his card is in the set, {2, 3, 4}, and folds otherwies , the value of the game is, -1/6 NE number, 2 Player 1 bets if his card is in the set, {1, 3, 4}, and checks otherwise, while Player 2 calls if his card is in the set, {2, 3, 4}, and folds otherwies , the value of the game is, -1/6 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1, 4} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {2, 3, 4} The value of this pair of strategies is, -1/6, and in decmimals, -0.1666666667 -------------------------------------------- -------------------------------------------- If the deck has, 5, 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, {1, 4, 5}, and checks otherwise, while Player 2 calls if his card is in the set, {3, 4, 5}, and folds otherwies , the value of the game is, 0 NE number, 2 Player 1 bets if his card is in the set, {1, 4, 5}, and checks otherwise, while Player 2 calls if his card is in the set, {2, 3, 4, 5}, and folds otherwies , the value of the game is, 0 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1, 4, 5} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {3, 4, 5} The value of this pair of strategies is, 0, and in decmimals, 0. -------------------------------------------- -------------------------------------------- If the deck has, 6, 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, 5, 6}, and checks otherwise, while Player 2 calls if his card is in the set, {4, 5, 6}, and folds otherwies , the value of the game is, 1/15 NE number, 2 Player 1 bets if his card is in the set, {1, 5, 6}, and checks otherwise, while Player 2 calls if his card is in the set, {3, 4, 5, 6}, and folds otherwies , the value of the game is, 1/15 NE number, 3 Player 1 bets if his card is in the set, {1, 5, 6}, and checks otherwise, while Player 2 calls if his card is in the set, {2, 3, 4, 5, 6}, and folds otherwies , the value of the game is, 1/15 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1, 5, 6} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {4, 5, 6} The value of this pair of strategies is, 1/15, and in decmimals, 0.06666666667 -------------------------------------------- -------------------------------------------- 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, {5, 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, 5, 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, {3, 4, 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 With with probability, 1, bet iff his card belongs to the set, {1, 6, 7} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {5, 6, 7} 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, {6, 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, 6, 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, {4, 5, 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 With with probability, 1, bet iff his card belongs to the set, {1, 7, 8} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {6, 7, 8} The value of this pair of strategies is, 3/28, and in decmimals, 0.1071428571 -------------------------------------------- -------------------------------------------- If the deck has, 9, 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, {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 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, {5, 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 With with probability, 1, bet iff his card belongs to the set, {1, 8, 9} Player 2's strategy is With with probability, 1/2, call iff his card belongs to the set, {7, 8, 9} With with probability, 1/2, call iff his card belongs to the set, {6, 7, 8, 9} 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, 2 There are, 2, 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, {7, 8, 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, {6, 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 With with probability, 1, bet iff his card belongs to the set, {1, 9, 10} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {7, 8, 9, 10} The value of this pair of strategies is, 1/9, and in decmimals, 0.1111111111 -------------------------------------------- -------------------------------------------- If the deck has, 11, distinct cards and the bet size is, 2 There are, 1, 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, {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 With with probability, 1, bet iff his card belongs to the set, {1, 10, 11} Player 2's strategy is With with probability, 1/2, call iff his card belongs to the set, {8, 9, 10, 11} With with probability, 1/2, call iff his card belongs to the set, {7, 8, 9, 10, 11} The value of this pair of strategies is, 6/55, and in decmimals, 0.1090909091 -------------------------------------------- -------------------------------------------- If the deck has, 12, distinct cards and the bet size is, 2 There are, 1, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 11, 12}, and checks otherwise, while Player 2 calls if his card is in the set, {8, 9, 10, 11, 12}, and folds otherwies , the value of the game is, 7/66 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1, 11, 12} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {8, 9, 10, 11, 12} The value of this pair of strategies is, 7/66, and in decmimals, 0.1060606061 -------------------------------------------- -------------------------------------------- If the deck has, 13, distinct cards and the bet size is, 2 There are no pure Nash Equilibria, using the SUGGESTED STRATEGIES Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1/2, bet iff his card belongs to the set, {1, 11, 12, 13} With with probability, 1/2, bet iff his card belongs to the set, {1, 2, 11, 12, 13} Player 2's strategy is With with probability, 1/2, call iff his card belongs to the set, {9, 10, 11, 12, 13} With with probability, 1/2, call iff his card belongs to the set, {8, 9, 10, 11, 12, 13} 17 The value of this pair of strategies is, ---, and in decmimals, 0.1089743590 156 -------------------------------------------- -------------------------------------------- If the deck has, 14, distinct cards and the bet size is, 2 There are no pure Nash Equilibria, using the SUGGESTED STRATEGIES Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1/2, bet iff his card belongs to the set, {1, 12, 13, 14} With with probability, 1/2, bet iff his card belongs to the set, {1, 2, 12, 13, 14} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {9, 10, 11, 12, 13, 14} 10 The value of this pair of strategies is, --, and in decmimals, 0.1098901099 91 -------------------------------------------- -------------------------------------------- If the deck has, 15, distinct cards and the bet size is, 2 There are no pure Nash Equilibria, using the SUGGESTED STRATEGIES Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1/2, bet iff his card belongs to the set, {1, 13, 14, 15} With with probability, 1/2, bet iff his card belongs to the set, {1, 2, 13, 14, 15} Player 2's strategy is With with probability, 1/2, call iff his card belongs to the set, {10, 11, 12, 13, 14, 15} With with probability, 1/2, call iff his card belongs to the set, {9, 10, 11, 12, 13, 14, 15} 23 The value of this pair of strategies is, ---, and in decmimals, 0.1095238095 210 -------------------------------------------- -------------------------------------------- If the deck has, 16, distinct cards and the bet size is, 2 There are, 1, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 2, 13, 14, 15, 16}, and checks otherwise, while Player 2 calls if his card is in the set, {10, 11, 12, 13, 14, 15, 16}, 13 and folds otherwies , the value of the game is, --- 120 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1/2, bet iff his card belongs to the set, {1, 14, 15, 16} With with probability, 1/2, bet iff his card belongs to the set, {1, 2, 14, 15, 16} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {10, 11, 12, 13, 14, 15, 16} 13 The value of this pair of strategies is, ---, and in decmimals, 0.1083333333 120 -------------------------------------------- -------------------------------------------- If the deck has, 17, distinct cards and the bet size is, 2 There are, 1, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 2, 14, 15, 16, 17}, and checks otherwise, while Player 2 calls if his card is in the set, {11, 12, 13, 14, 15, 16, 17}, 15 and folds otherwies , the value of the game is, --- 136 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1, 2, 14, 15, 16, 17} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {11, 12, 13, 14, 15, 16, 17} 15 The value of this pair of strategies is, ---, and in decmimals, 0.1102941176 136 -------------------------------------------- -------------------------------------------- If the deck has, 18, distinct cards and the bet size is, 2 There are, 1, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 2, 15, 16, 17, 18}, and checks otherwise, while Player 2 calls if his card is in the set, {11, 12, 13, 14, 15, 16, 17, 18}, and folds otherwies , the value of the game is, 1/9 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1, 2, 15, 16, 17, 18} Player 2's strategy is With with probability, 1/2, call iff his card belongs to the set, {12, 13, 14, 15, 16, 17, 18} With with probability, 1/2, call iff his card belongs to the set, {11, 12, 13, 14, 15, 16, 17, 18} The value of this pair of strategies is, 1/9, and in decmimals, 0.1111111111 -------------------------------------------- -------------------------------------------- If the deck has, 19, distinct cards and the bet size is, 2 There are, 1, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 2, 16, 17, 18, 19}, and checks otherwise, while Player 2 calls if his card is in the set, {12, 13, 14, 15, 16, 17, 18, 19}, and folds otherwies , the value of the game is, 1/9 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1, 2, 16, 17, 18, 19} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {12, 13, 14, 15, 16, 17, 18, 19} The value of this pair of strategies is, 1/9, and in decmimals, 0.1111111111 -------------------------------------------- -------------------------------------------- If the deck has, 20, distinct cards and the bet size is, 2 There are, 1, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 2, 17, 18, 19, 20}, and checks otherwise, while Player 2 calls if his card is in the set, {12, 13, 14, 15, 16, 17, 18, 19, 20}, 21 and folds otherwies , the value of the game is, --- 190 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1, 2, 17, 18, 19, 20} Player 2's strategy is With with probability, 1/2, call iff his card belongs to the set, {13, 14, 15, 16, 17, 18, 19, 20} With with probability, 1/2, call iff his card belongs to the set, {12, 13, 14, 15, 16, 17, 18, 19, 20} 21 The value of this pair of strategies is, ---, and in decmimals, 0.1105263158 190 -------------------------------------------- -------------------------------------------- If the deck has, 21, distinct cards and the bet size is, 2 There are, 1, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 2, 18, 19, 20, 21}, and checks otherwise, while Player 2 calls if his card is in the set, {13, 14, 15, 16, 17, 18, 19, 20, 21}, 23 and folds otherwies , the value of the game is, --- 210 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1, 2, 18, 19, 20, 21} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {13, 14, 15, 16, 17, 18, 19, 20, 21} 23 The value of this pair of strategies is, ---, and in decmimals, 0.1095238095 210 -------------------------------------------- -------------------------------------------- If the deck has, 22, distinct cards and the bet size is, 2 There are no pure Nash Equilibria, using the SUGGESTED STRATEGIES Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1/2, bet iff his card belongs to the set, {1, 2, 18, 19, 20, 21, 22} With with probability, 1/2, bet iff his card belongs to the set, {1, 2, 3, 18, 19, 20, 21, 22} Player 2's strategy is With with probability, 1/2, call iff his card belongs to the set, {14, 15, 16, 17, 18, 19, 20, 21, 22} With with probability, 1/2, call iff his card belongs to the set, {13, 14, 15, 16, 17, 18, 19, 20, 21, 22} 17 The value of this pair of strategies is, ---, and in decmimals, 0.1103896104 154 -------------------------------------------- -------------------------------------------- If the deck has, 23, distinct cards and the bet size is, 2 There are no pure Nash Equilibria, using the SUGGESTED STRATEGIES Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1/2, bet iff his card belongs to the set, {1, 2, 19, 20, 21, 22, 23} With with probability, 1/2, bet iff his card belongs to the set, {1, 2, 3, 19, 20, 21, 22, 23} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {14, 15, 16, 17, 18, 19, 20, 21, 22, 23} 28 The value of this pair of strategies is, ---, and in decmimals, 0.1106719368 253 -------------------------------------------- -------------------------------------------- If the deck has, 24, distinct cards and the bet size is, 2 There are no pure Nash Equilibria, using the SUGGESTED STRATEGIES Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1/2, bet iff his card belongs to the set, {1, 2, 20, 21, 22, 23, 24} With with probability, 1/2, bet iff his card belongs to the set, {1, 2, 3, 20, 21, 22, 23, 24} Player 2's strategy is With with probability, 1/2, call iff his card belongs to the set, {15, 16, 17, 18, 19, 20, 21, 22, 23, 24} With with probability, 1/2, call iff his card belongs to the set, {14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24} 61 The value of this pair of strategies is, ---, and in decmimals, 0.1105072464 552 -------------------------------------------- -------------------------------------------- If the deck has, 25, distinct cards and the bet size is, 2 There are, 1, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 2, 3, 20, 21, 22, 23, 24, 25}, and checks otherwise, while Player 2 calls if his card is in the set, {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, 11 and folds otherwies , the value of the game is, --- 100 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1/2, bet iff his card belongs to the set, {1, 2, 21, 22, 23, 24, 25} With with probability, 1/2, bet iff his card belongs to the set, {1, 2, 3, 21, 22, 23, 24, 25} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25} 11 The value of this pair of strategies is, ---, and in decmimals, 0.1100000000 100 -------------------------------------------- -------------------------------------------- If the deck has, 26, distinct cards and the bet size is, 2 There are, 1, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 2, 3, 21, 22, 23, 24, 25, 26}, and checks otherwise, while Player 2 calls if his card is in the set, {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26}, 36 and folds otherwies , the value of the game is, --- 325 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1, 2, 3, 21, 22, 23, 24, 25, 26} Player 2's strategy is With with probability, 1, call iff his card belongs to the set, {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26} 36 The value of this pair of strategies is, ---, and in decmimals, 0.1107692308 325 -------------------------------------------- -------------------------------------------- If the deck has, 27, distinct cards and the bet size is, 2 There are, 1, Nash equilibria here they are NE number, 1 Player 1 bets if his card is in the set, {1, 2, 3, 22, 23, 24, 25, 26, 27}, and checks otherwise, while Player 2 calls if his card is in the set, {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27}, and folds otherwies , the value of the game is, 1/9 Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 1, bet iff his card belongs to the set, {1, 2, 3, 22, 23, 24, 25, 26, 27} Player 2's strategy is With with probability, 1/2, call iff his card belongs to the set, {17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} With with probability, 1/2, call iff his card belongs to the set, {16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27} The value of this pair of strategies is, 1/9, and in decmimals, 0.1111111111 ----------------- This ends this paper that took, 50.537, seconds.