## 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 |

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.