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.

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