The following theorem, also known as the ‘Minimax Theorem’, due to Von Neumann:
Suppose that, in a matrix game, E(x, y) is the expectation, where x and y are mixed strategies for the two players. Then
maxx miny E(x, y)=miny maxx 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.