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# conservative strategy

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In the matrix game given by the matrix [aij], suppose that the players R and C use pure strategies. Let mi be equal to the minimum entry in the i-th row. A maximin strategy for R is to choose the r-th row, where mr=max{mi}. In doing so, R ensures that the smallest pay-off possible is as large as can be. Similarly, let Mj be equal to the maximum entry in the j-th column. A minimax strategy for C is to choose the s-th column, where Ms=min{Mj}. 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, miny 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 miny E(x, y). Similarly, a minimax strategy for C is a strategy y that minimizes maxx 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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