| 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.