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 ith row. A maximin strategy for R is to choose the rth row, where m_r = max{m_i}. In doing so, R ensures that the smallest payoff possible is as large as can be. Similarly, let M_j be equal to the maximum entry in the jth column. A minimax strategy for C is to choose the sth 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.