# Game theory problem: Three man poker game

Mine question is from Three-Man Poker Game discussed in Nash's paper.

This is a greatly simplified version of poker. The cards are of only two kinds, high and low. The three players A, B, and C ante up two chips each to start. Then each player is dealt one card. We assume that each of the eight possible hands is equally likely. Starting with player A, each player is given a chance to “open”, i.e.,to place the first bet (two chips are always used to bet). If no one does so, the players retrieve their antes from the pot. Once a player has opened, the other two players are again given a chance to bet, i.e., they may “call”. Finally, the cards are revealed and those players with the highest cards among those who placed bets share the pot equally.

Once the game is open, one should call if one has a high card and pass if one has a low card. So the only question is whether to open the game. Player C should obviously open if he has a high card. Player A should never open if he has a low card. Thus player A has two pure strategies: when he has a high card, to open or not to open. We denote his probability of opening in this case by a. (His subsequent moves, and his moves in case he has a low card, are determined.) Player C also has two pure strategies: when he has a low card, to open or not to open. We denote his probability of opening in this case by c. Player B has four pure strategies: for each of his possible cards, to open or not to open. We denote his probability of opening when he has a high card by d, and his probability of opening when he has a low card by e. It turns out that the equilibrium strategy is totally mixed in these four parameters.

The payoff matrix (where by payoff we mean the expected value of the payoff) contains 48=3*2*4*2 rational eneteries.

Here is the left (a=0) block:

Here is the right (a=1) block:

For instance B is B’s payoff when player A does not open on a high card (so a1 = 1), player B does open on a high card (so d0 = 1) and does not open on a low card (so e1 = 1), and player C does open on a low card (so c0 = 1). In general, X[ijkl] is player X’s payoff when ai = 1, dj = 1, ek = 1, and cl = 1.

Mine question is: How we get the numbers in the matrix?

• Why should one pass if one has a low card? The other(s) might be bluffing ... Why should A never open if he has a low card? Bluffing might pay out ... – Hagen von Eitzen Dec 5 '17 at 22:21
• For the first question my literature says: Once the game is open, one should call if one has a high card and pass if one has a low card. The former is obvious; the latter follows because it might be the strategy of the player who opened the game, to only open on a high card. In this case one would definitely lose one’s bet as well as the ante. So the only question is whether to open the game. And why should A never open if he has a low card? It requires a proof, but i don't have it. – Petra Dec 6 '17 at 10:32
• You don't have proof buy you assert should never open with low card. – paparazzo Dec 6 '17 at 19:52
• Yes, but that's not what I'm asking here. I'm just asking about the numbers in the matrix. – Petra Dec 6 '17 at 21:04