## Quick Reference

The following theorem, also known as the ‘Minimax Theorem’, due to Von Neumann:

Theorem

Suppose that, in a matrix game, E(**x**, **y**) is the expectation, where **x** and **y** are mixed strategies for the two players. Then

max_{x} min_{y} E(**x**, **y**)=min_{y} max_{x} E(**x**, **y**).

By using a maximin strategy, one player, R, ensures that the expectation is at least as large as the left-hand side of the equation appearing in the theorem. Similarly, by using a minimax strategy, the other player, C, ensures that the expectation is less than or equal to the right-hand side of the equation. Such strategies may be called optimal strategies for R and C. Since, by the theorem, the two sides of the equation are equal, then if R and C use optimal strategies the expectation is equal to the common value, which is called the value of the game.

For example, consider the game given by the matrixif **x***= it can be shown that E(**x***, **y**)≥10/3 for all **y**. Also, if then E(**x**, **y***)≤ 10/3 for all **x**. It follows that the value of the game is 10/3, and **x*** and **y*** are optimal strategies for the two players.

*Subjects:*
Mathematics.

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