## Proof That a Two Player Zero Sum Game With Two Strategies Each and a Saddle Point Must Have Dominant Strategies

If the payoff matrix for a two player zero sum game with each player having two strategies has a saddle point, then one player has a dominant strategy. Consider the payoff matrix:A\B | \[B_1\] | \[B_2\] |

\[A_1\] | \[a_{11}\] | \[a_{12}\] |

\[A_2\] | \[a_{21}\] | \[a_{22}\] |

\[a_{11}\]

is the saddle point of the above game, then \[a_{11} \le MAX(a_{12}.a_{22})\]

. Also \[a_{11}\]

is the minimum entry in row \[A_1\]

so \[a_{11} \le a_{12}\]

$nbsp; (1)\[a_{11}\]

must be the column maximum of \[B_1\]

so \[a_{11} \ge a_{21}\]

. Also, \[a_{11} \le MAX(a_{12}, a_{22})\]

and \[a_{11} \ge MIN(a_{21}, a_{22})\]

. \[a_{12} \ge a_{22}\]

or \[a_{12} \le a_{22}\]

. If the first, then since \[a_{11} \ge a_{21}\]

, \[A_1\]

dominates \[A_2\]

. If the second, then since \[a_{11} \le a_{22}\]

but \[a_{11} \ge a_{21}\]

then \[a_{21} \le a_{22}\]

. Now use (1) to get \[B_1\]

dominates \[B_2\]

. The original payoff matrix reduces to the saddle point \[a_{11}\]

. It is not true in general that a two player zero sum game with a saddle point has a dominant strategy.