## A Twi Player Zero Sum Game Without a Saddle Point

Players A and B simultaneously call out the numbers 1 and 2.If the sum is even, B pays A the sum, and if odd, A pays B the sum.

Games may be classified according to the criteria:

1. Number of players

2. Number of moves

3. Whether or not they are zero sum

4. Whether or not each player is informed

Int the game above, there are two players. There are two moves to each game - A's move and B's move. The game is zero sum since A's gain is B's loss and vice versa, and a game is said to be of full information if at each stage, all previous moves are known to both players. The game above is not full information since the players do not take their turns one at a time.

The payoff matrix is:

A\B | 1 | 3 | Row Minimum |

1 | 2 | -3 | -3 |

2 | -3 | 4 | -3 |

Column Maximum | 2 | 4 | |

The optimal strategy for A is the maximum value of the game, the maximum of the row minima. By calling out either 1 or 2, A's maximum reward is -3. m. B optimum strategy is the column for the minimum value of the maximum - strategy 1. Thus A expects to lose 3 and B expects to lose 2- this is the payoff matrix for A - and the payoff for B are the negative of those in the matrix. Since this is a zero sum game, this conclusion cannot be true. The paradox arises because the game has no saddle point, and there is no predictable solution to a single game. Over many games, both players play their strategies randomly with probabilities of playing each strategy chosen to give each player their maximum payoff over many games. The players have moved from a pure strategy, always choosing the same strategy, to a mixed strategy. If either player chose a pure strategy, he would lose over many games. For example, if B persists in calling 1, then so will A and A will always win .