All the Pure Nash Equiilbria, and ONE Mixed One, for DJ Newman Poker with nu\ mber of cards from 2 to, 4, and maximum bet sizes from, 1, to, 2 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 bet any whole dollar amount from 0 to, 2 Now Player 2 can decide to have a counter bet (i,e, call) or fold, and conce\ de the pot to Player 1. If he decides to call then The cards are compared, and whoever has the larger card gets the pot, winnin\ g 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, 4, and bet sizes from 1 to, 2 -------------------------------------------- -------------------------------------------- If the deck has, 2, distinct cards and the alloawed bet sizes for Player 1 ar\ e from 0 dolalrs to, 1, dollars There are, 4, Nash equilibria here they are NE number, 1 Player 1 bets the following amounts according to his card value , [0, 0], while Player 2 calls if his card is in the b+1 entry of he list, [{2}, {2}], and folds otherwies , the value of the game is, 0 NE number, 2 Player 1 bets the following amounts according to his card value , [0, 0], while Player 2 calls if his card is in the b+1 entry of he list, [{1, 2}, {2}], and folds otherwies , the value of the game is, 0 NE number, 3 Player 1 bets the following amounts according to his card value , [0, 1], while Player 2 calls if his card is in the b+1 entry of he list, [{2}, {2}], and folds otherwies , the value of the game is, 0 NE number, 4 Player 1 bets the following amounts according to his card value , [0, 1], while Player 2 calls if his card is in the b+1 entry of he list, [{1, 2}, {2}], 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, have the betting table, according to his number cards, [0, 0] Player 2's strategy is With with probability, 1/3, have the calling table (according to the proposed bet, starting at 0), [{2}, {}] With with probability, 2/3, have the calling table (according to the proposed bet, starting at 0), [{2}, {2}] The value of this pair of strategies is, 0, and in decmimals, 0. -------------------------------------------- If the deck has, 2, distinct cards and the alloawed bet sizes for Player 1 ar\ e from 0 dolalrs to, 2, dollars There are, 6, Nash equilibria here they are NE number, 1 Player 1 bets the following amounts according to his card value , [0, 0], while Player 2 calls if his card is in the b+1 entry of he list, [{2}, {2}, {2}], and folds otherwies , the value of the game is, 0 NE number, 2 Player 1 bets the following amounts according to his card value , [0, 0], while Player 2 calls if his card is in the b+1 entry of he list, [{1, 2}, {2}, {2}], and folds otherwies , the value of the game is, 0 NE number, 3 Player 1 bets the following amounts according to his card value , [0, 1], while Player 2 calls if his card is in the b+1 entry of he list, [{2}, {2}, {2}], and folds otherwies , the value of the game is, 0 NE number, 4 Player 1 bets the following amounts according to his card value , [0, 1], while Player 2 calls if his card is in the b+1 entry of he list, [{1, 2}, {2}, {2}], and folds otherwies , the value of the game is, 0 NE number, 5 Player 1 bets the following amounts according to his card value , [0, 2], while Player 2 calls if his card is in the b+1 entry of he list, [{2}, {2}, {2}], and folds otherwies , the value of the game is, 0 NE number, 6 Player 1 bets the following amounts according to his card value , [0, 2], while Player 2 calls if his card is in the b+1 entry of he list, [{1, 2}, {2}, {2}], 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, have the betting table, according to his number cards, [0, 0] Player 2's strategy is With with probability, 1/3, have the calling table (according to the proposed bet, starting at 0), [{2}, {}, {2}] With with probability, 1/2, have the calling table (according to the proposed bet, starting at 0), [{2}, {2}, {}] With with probability, 1/6, have the calling table (according to the proposed bet, starting at 0), [{2}, {2}, {2}] The value of this pair of strategies is, 0, and in decmimals, 0. -------------------------------------------- -------------------------------------------- If the deck has, 3, distinct cards and the alloawed bet sizes for Player 1 ar\ e from 0 dolalrs to, 1, dollars There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 2/3, have the betting table, according to his number cards, [0, 0, 1] With with probability, 1/3, have the betting table, according to his number cards, [1, 0, 1] Player 2's strategy is With with probability, 2/3, have the calling table (according to the proposed bet, starting at 0), [{2, 3}, {3}] With with probability, 1/3, have the calling table (according to the proposed bet, starting at 0), [{2, 3}, {2, 3}] The value of this pair of strategies is, 1/18, and in decmimals, 0.05555555556 -------------------------------------------- If the deck has, 3, distinct cards and the alloawed bet sizes for Player 1 ar\ e from 0 dolalrs to, 2, dollars There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 2/3, have the betting table, according to his number cards, [0, 0, 1] With with probability, 1/3, have the betting table, according to his number cards, [1, 0, 1] Player 2's strategy is With with probability, 1/2, have the calling table (according to the proposed bet, starting at 0), [{2, 3}, {3}, {3}] With with probability, 1/6, have the calling table (according to the proposed bet, starting at 0), [{2, 3}, {3}, {2, 3}] With with probability, 1/6, have the calling table (according to the proposed bet, starting at 0), [{2, 3}, {2, 3}, {}] With with probability, 1/6, have the calling table (according to the proposed bet, starting at 0), [{2, 3}, {2, 3}, {3}] The value of this pair of strategies is, 1/18, and in decmimals, 0.05555555556 -------------------------------------------- -------------------------------------------- If the deck has, 4, distinct cards and the alloawed bet sizes for Player 1 ar\ e from 0 dolalrs to, 1, dollars There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 2/3, have the betting table, according to his number cards, [0, 0, 0, 1] With with probability, 1/3, have the betting table, according to his number cards, [1, 0, 0, 1] Player 2's strategy is With with probability, 2/3, have the calling table (according to the proposed bet, starting at 0), [{2, 3, 4}, {2, 4}] With with probability, 1/3, have the calling table (according to the proposed bet, starting at 0), [{2, 3, 4}, {3, 4}] The value of this pair of strategies is, 1/12, and in decmimals, 0.08333333333 -------------------------------------------- If the deck has, 4, distinct cards and the alloawed bet sizes for Player 1 ar\ e from 0 dolalrs to, 2, dollars There are no pure Nash Equilibria Let's find ONE Mixed Nash Equilibrium Player 1's strategy is With with probability, 2/3, have the betting table, according to his number cards, [0, 0, 0, 1] With with probability, 1/3, have the betting table, according to his number cards, [1, 0, 0, 1] Player 2's strategy is With with probability, 1/6, have the calling table (according to the proposed bet, starting at 0), [{2, 3, 4}, {2, 4}, {4}] With with probability, 1/2, have the calling table (according to the proposed bet, starting at 0), [{2, 3, 4}, {2, 4}, {2, 4}] With with probability, 1/3, have the calling table (according to the proposed bet, starting at 0), [{2, 3, 4}, {3, 4}, {4}] The value of this pair of strategies is, 1/12, and in decmimals, 0.08333333333 ----------------- This ends this paper that took, 435.141, seconds.