The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 5, cards and player1 raises (in case if he wants to) with the amount, 1, tokens First of all, the pay off for player 1 is, 3/40 -3 the pay off for player 2 is, -- 80 -3 and the pay off for player 3 is, -- 80 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [3/7, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 3/4, 1], for player 2. And, [0, 0, 0, 3/4, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 5, cards and player1 raises (in case if he wants to) with the amount, 2, tokens First of all, the pay off for player 1 is, 3/50 -3 the pay off for player 2 is, --- 100 -3 and the pay off for player 3 is, --- 100 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [3/4, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 3/10, 1], for player 2. And, [0, 0, 0, 3/10, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 5, cards and player1 raises (in case if he wants to) with the amount, 3, tokens The strategy is trivial. Player 1 will always check no matter what card he r\ ecieves. And it does not matter what strategies of player2 or 3 are. Eve\ rybody's payoffs are 0. -------------------------------- -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 6, cards and player1 raises (in case if he wants to) with the amount, 1, tokens First of all, the pay off for player 1 is, 1/12 -1 the pay off for player 2 is, -- 24 -1 and the pay off for player 3 is, -- 24 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [2/3, 0, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 1/4, 1, 1], for player 2. And, [0, 0, 0, 1/4, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 6, cards and player1 raises (in case if he wants to) with the amount, 2, tokens First of all, the pay off for player 1 is, 4/45 -2 the pay off for player 2 is, -- 45 -2 and the pay off for player 3 is, -- 45 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [8/13, 0, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 2/3, 1], for player 2. And, [0, 0, 0, 0, 2/3, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 6, cards and player1 raises (in case if he wants to) with the amount, 3, tokens First of all, the pay off for player 1 is, 1/15 -1 the pay off for player 2 is, -- 30 -1 and the pay off for player 3 is, -- 30 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [4/5, 0, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 1/3, 1], for player 2. And, [0, 0, 0, 0, 1/3, 1], for player 3. -------------------------------- -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 7, cards and player1 raises (in case if he wants to) with the amount, 1, tokens First of all, the pay off for player 1 is, 1/12 -1 the pay off for player 2 is, -- 24 -1 and the pay off for player 3 is, -- 24 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [1/2, 0, 0, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 3/4, 1, 1], for player 2. And, [0, 0, 0, 0, 3/4, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 7, cards and player1 raises (in case if he wants to) with the amount, 2, tokens First of all, the pay off for player 1 is, 2/21 -1 the pay off for player 2 is, -- 21 -1 and the pay off for player 3 is, -- 21 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [5/9, 0, 0, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 7, cards and player1 raises (in case if he wants to) with the amount, 3, tokens First of all, the pay off for player 1 is, 5/56 -5 the pay off for player 2 is, --- 112 -5 and the pay off for player 3 is, --- 112 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [5/7, 0, 0, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 5/8, 1], for player 2. And, [0, 0, 0, 0, 0, 5/8, 1], for player 3. -------------------------------- -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 8, cards and player1 raises (in case if he wants to) with the amount, 1, tokens 55 First of all, the pay off for player 1 is, --- 672 -55 the pay off for player 2 is, ---- 1344 -55 and the pay off for player 3 is, ---- 1344 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [1, 1/3, 0, 0, 0, 0, 1, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 1/4, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 1/4, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 8, cards and player1 raises (in case if he wants to) with the amount, 2, tokens First of all, the pay off for player 1 is, 1/10 -1 the pay off for player 2 is, -- 20 -1 and the pay off for player 3 is, -- 20 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [3/4, 0, 0, 0, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 2/5, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 2/5, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 8, cards and player1 raises (in case if he wants to) with the amount, 3, tokens 27 First of all, the pay off for player 1 is, --- 280 -27 the pay off for player 2 is, --- 560 -27 and the pay off for player 3 is, --- 560 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [2/3, 0, 0, 0, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 9/10, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 9/10, 1], for player 3. -------------------------------- -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 9, cards and player1 raises (in case if he wants to) with the amount, 1, tokens 13 First of all, the pay off for player 1 is, --- 144 -13 the pay off for player 2 is, --- 288 -13 and the pay off for player 3 is, --- 288 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [1, 1/13, 0, 0, 0, 0, 0, 1, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 3/4, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 3/4, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 9, cards and player1 raises (in case if he wants to) with the amount, 2, tokens 22 First of all, the pay off for player 1 is, --- 225 -11 the pay off for player 2 is, --- 225 -11 and the pay off for player 3 is, --- 225 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [2/3, 0, 0, 0, 0, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. 19 The lists are, [0, 0, 0, 0, 0, 0, --, 1, 1], for player 2. And, 25 19 [0, 0, 0, 0, 0, 0, --, 1, 1], for player 3. 25 -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 9, cards and player1 raises (in case if he wants to) with the amount, 3, tokens First of all, the pay off for player 1 is, 1/10 -1 the pay off for player 2 is, -- 20 -1 and the pay off for player 3 is, -- 20 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [7/8, 0, 0, 0, 0, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 1/5, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 1/5, 1, 1], for player 3. -------------------------------- -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 10, cards and player1 raises (in case if he wants to) with the amount, 1, tokens First of all, the pay off for player 1 is, 3/32 -3 the pay off for player 2 is, -- 64 -3 and the pay off for player 3 is, -- 64 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [1, 1/3, 0, 0, 0, 0, 0, 0, 1, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 1/4, 1, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 1/4, 1, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 10, cards and player1 raises (in case if he wants to) with the amount, 2, tokens 106 First of all, the pay off for player 1 is, ---- 1125 -53 the pay off for player 2 is, ---- 1125 -53 and the pay off for player 3 is, ---- 1125 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. 16 The list is, [--, 0, 0, 0, 0, 0, 0, 0, 0, 1] 19 The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 3/25, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 3/25, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 10, cards and player1 raises (in case if he wants to) with the amount, 3, tokens First of all, the pay off for player 1 is, 1/10 -1 the pay off for player 2 is, -- 20 -1 and the pay off for player 3 is, -- 20 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [4/5, 0, 0, 0, 0, 0, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1], for player 3. -------------------------------- -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 11, cards and player1 raises (in case if he wants to) with the amount, 1, tokens 17 First of all, the pay off for player 1 is, --- 180 -17 the pay off for player 2 is, --- 360 -17 and the pay off for player 3 is, --- 360 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [1, 1/8, 0, 0, 0, 0, 0, 0, 0, 1, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 11, cards and player1 raises (in case if he wants to) with the amount, 2, tokens 17 First of all, the pay off for player 1 is, --- 165 -17 the pay off for player 2 is, --- 330 -17 and the pay off for player 3 is, --- 330 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 1, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 0, 1/2, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 11, cards and player1 raises (in case if he wants to) with the amount, 3, tokens 15 First of all, the pay off for player 1 is, --- 154 -15 the pay off for player 2 is, --- 308 -15 and the pay off for player 3 is, --- 308 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [3/4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. 11 The lists are, [0, 0, 0, 0, 0, 0, 0, 0, --, 1, 1], for player 2. And, 14 11 [0, 0, 0, 0, 0, 0, 0, 0, --, 1, 1], for player 3. 14 -------------------------------- -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 12, cards and player1 raises (in case if he wants to) with the amount, 1, tokens 247 First of all, the pay off for player 1 is, ---- 2640 -247 the pay off for player 2 is, ---- 5280 -247 and the pay off for player 3 is, ---- 5280 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [1, 1/3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 1/4, 1, 1, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 1/4, 1, 1, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 12, cards and player1 raises (in case if he wants to) with the amount, 2, tokens 247 First of all, the pay off for player 1 is, ---- 2310 -247 the pay off for player 2 is, ---- 4620 -247 and the pay off for player 3 is, ---- 4620 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. 11 The list is, [1, --, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1] 29 The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 0, 0, 6/7, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 0, 0, 6/7, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 12, cards and player1 raises (in case if he wants to) with the amount, 3, tokens 29 First of all, the pay off for player 1 is, --- 308 -29 the pay off for player 2 is, --- 616 -29 and the pay off for player 3 is, --- 616 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. 10 The list is, [--, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 11 The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 0, 0, 1/14, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 0, 0, 1/14, 1, 1, 1], for player 3. -------------------------------- -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 13, cards and player1 raises (in case if he wants to) with the amount, 1, tokens First of all, the pay off for player 1 is, 5/52 -5 the pay off for player 2 is, --- 104 -5 and the pay off for player 3 is, --- 104 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. 14 The list is, [1, --, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1] 19 The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 13, cards and player1 raises (in case if he wants to) with the amount, 2, tokens First of all, the pay off for player 1 is, 6/55 -3 the pay off for player 2 is, -- 55 -3 and the pay off for player 3 is, -- 55 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. 17 The list is, [1, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1] 27 The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 0, 0, 8/35, 1, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 0, 0, 8/35, 1, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 13, cards and player1 raises (in case if he wants to) with the amount, 3, tokens 21 First of all, the pay off for player 1 is, --- 208 -21 the pay off for player 2 is, --- 416 -21 and the pay off for player 3 is, --- 416 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [1, 9/13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 0, 0, 0, 3/8, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 0, 0, 0, 3/8, 1, 1, 1], for player 3. -------------------------------- -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 14, cards and player1 raises (in case if he wants to) with the amount, 1, tokens 11 First of all, the pay off for player 1 is, --- 112 -11 the pay off for player 2 is, --- 224 -11 and the pay off for player 3 is, --- 224 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 0, 1/4, 1, 1, 1, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 0, 1/4, 1, 1, 1, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 14, cards and player1 raises (in case if he wants to) with the amount, 2, tokens 23 First of all, the pay off for player 1 is, --- 210 -23 the pay off for player 2 is, --- 420 -23 and the pay off for player 3 is, --- 420 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [1, 1/2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 0, 0, 0, 3/5, 1, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 0, 0, 0, 3/5, 1, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 14, cards and player1 raises (in case if he wants to) with the amount, 3, tokens 115 First of all, the pay off for player 1 is, ---- 1092 -115 the pay off for player 2 is, ---- 2184 -115 and the pay off for player 3 is, ---- 2184 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [1, 3/5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2/3, 1, 1, 1], for player 3. -------------------------------- -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 15, cards and player1 raises (in case if he wants to) with the amount, 1, tokens First of all, the pay off for player 1 is, 9/91 -9 the pay off for player 2 is, --- 182 -9 and the pay off for player 3 is, --- 182 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. 17 The list is, [1, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1] 22 The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. The lists are, [0, 0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1, 1, 1, 1], for player 2. And, [0, 0, 0, 0, 0, 0, 0, 0, 3/4, 1, 1, 1, 1, 1, 1], for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 15, cards and player1 raises (in case if he wants to) with the amount, 2, tokens 38 First of all, the pay off for player 1 is, --- 351 -19 the pay off for player 2 is, --- 351 -19 and the pay off for player 3 is, --- 351 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. 15 The list is, [1, --, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1] 37 The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. 43 The lists are, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 1, 1, 1, 1], 45 43 for player 2. And, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 1, 1, 1, 1], 45 for player 3. -------------------------------- The solution for von Neumann poker: Three-person version Each player put 1 token to the pot. Player 1 can check or raise. In case of raise, player 2 and player 3 can choose to call or fold. We will present ONE Nash Equilibrium of this game. The game is played with, 15, cards and player1 raises (in case if he wants to) with the amount, 3, tokens First of all, the pay off for player 1 is, 3/28 -3 the pay off for player 2 is, -- 56 -3 and the pay off for player 3 is, -- 56 The strategies for player 1 is a list of n probabilities P1 meaning if he ge\ ts card x he should bet with prob. P1[x] and check with prob. 1-P1[x]. The list is, [1, 9/17, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1] The strategies for player 2,3 are lists of prob. P2,P3 meaning if he gets ca\ rd y,z, he should call with prob. P2[y], P3[z] and fold with prob. 1-P2[\ y], 1-P3[z]. 19 The lists are, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 1, 1, 1], 20 19 for player 2. And, [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, --, 1, 1, 1], 20 for player 3. This took, 13.830, seconds.