The Value of a Zero Sum Game

The payoff matrix below is for a two player game between players A and B who have 3 and possible strategies respectively.
A\B  
\[B_1\]
 		  
\[B_2\]
 		  
\[B_3\]
 		  
\[B_4\]
 
 
\[A_1\]
 		 2 3 -3 2
 
\[A_2\]
 		 1 3 5 2
 
\[A_3\]
 		 9 5 8 10
Each entry in the matrix is a payment from B to A, so this is a zero sum game. Assuming each player behaves rationally, they reason as follows:
Player A thinks that if he picks strategy  
\[A_1\]
, the worst that can happen is that he loses 3. If he chooses strategy  
\[A_2\]
, the worst that can happen that he is paid 1, and if he plays strategy  
\[A_3\]
, the worst that can happen is that he is paid 5. Player A will choose strategy  
\[A_3\]
  since he is guaranteed the highest payoff for this strategy. The Maximin of the rows is 5.
Player A will always choose strategy  
\[A_3\]
, and player B will always choose strategy  
\[B_2\]
. The value of the game is 5.

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