## Zero Sum, Fair Games

Two players, A and B, each call out one of the numbers 1 and 2,simultaneously. If they both call 1, no payments are made. If they both call 2, B pays A £3.I A calls 1 and B calls 2, B pays A £1. If A calls 2 and B calls 1, A pays B £1.
We can construct the payoff matrix with A's strategies as rows, and B's strategies as columns. The convention is to show the payoffs to the row player.
The payoff matrix for this game is:
 A\B Calls 1 Calls 2 Calls 1 0 1 Calls 2 -1 3
The game is a zero sum game, since if one wins, the other must lose.
The game is said to be fair if the value of the game is the same to both players. To find the value of the game for A, we assume that he behave rationally. He will reason that if he calls 1, his worst profit is 0. If he calls 2, his worst profit is -£1. Hence A will always call 1, so not to lose anything. B reasons that if he calls 1,his worst profit is 0. If B calls 2, his worst profit is -£3. Hence B should also call 1. The value of the game to both players is 0, and the game is fair.