## 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.