## 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 $B_1$ $B_2$ $A_1$ $a_{11}$ $a_{12}$ $A_2$ $a_{21}$ $a_{22}$
The game and the payoff matrix has a saddle point if the maximum of the minimum payoffs for A equals the minimum of the maximum payoffs for B. Suppose
$a_{11}$
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.