## Quick Reference

In the matrix game given by the matrix [*a*_{ij}], suppose that the players R and C use pure strategies. Let *m*_{i} be equal to the minimum entry in the *i*-th row. A maximin strategy for R is to choose the *r*-th row, where *m*_{r}=max{*m*_{i}}. In doing so, R ensures that the smallest pay-off possible is as large as can be. Similarly, let *M*_{j} be equal to the maximum entry in the *j*-th column. A minimax strategy for C is to choose the *s*-th column, where *M*_{s}=min{*M*_{j}}. These are called conservative strategies for the two players.

Now let E(**x**, **y**) be the expectation when R and C use mixed strategies **x** and **y**. Then, for any **x**, min_{y} E(**x**, **y**) is the smallest expectation possible, for all mixed strategies **y** that C may use. A maximin strategy for R is a strategy **x** that maximizes min_{y} E(**x**, **y**). Similarly, a minimax strategy for C is a strategy **y** that minimizes max_{x} E(**x**, **y**). By the Fundamental Theorem of Game Theory, when R and C use such strategies the expectation takes a certain value, the value of the game.

*Subjects:*
Mathematics.

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