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  , the worst that can happen is that he loses 3. If he chooses strategy  , the worst that can happen that he is paid 1, and if he plays strategy  , the worst that can happen is that he is paid 5. Player A will choose strategy    since he is guaranteed the highest payoff for this strategy. The Maximin of the rows is 5.
Player A will always choose strategy  , and player B will always choose strategy  . The value of the game is 5.