## Stable Solutions to Zero Sum Games

Suppose we have the following payoff matrix.

5 | -2 | 3 | 3 |

5 | 6 | 5 | 4 |

2 | 3 | -1 | 2 |

We find the play safe strategies.

Row Minimum | |||||

5 | -2 | 3 | 3 | -2 | |

5 | 6 | 5 | 4 | 4 | |

2 | 3 | -1 | 2 | -1 | |

Column Maximum | 5 | 6 | 5 | 4 |

The row maximin is 4, so A to play it safe A should play strategy 2, and the column minimax is 4, so B should play strategy 4. This results in a win of 4 for A and a loss of 4 for B.

We now consider if it is worthwhile for A or B to change strategy.

If A assumes B will play strategy 4, then if A plays strategy 1, they will win 3 and if A plays strategy 3 they will win 2, but will win 8 by staying with strategy 2.

If B assumes A will play strategy 2, B will lose 5 if they play strategy 1, 6 if they play strategy 2, 5 if they play strategy 3 but only 4 if they stick with strategy 4.

It is not sensible for either player to change strategy, if they are both playing safe. The game has a stable equilibrium or saddle point, at (A2, B4) and the game has value 4 to A and -4 to B.