For two-person, zero-sum games, there is a logical approach to finding the solution: In a zero-sum game, each person should choose the strategy that minimizes the maximum loss, called the minimax criterion. This is identical to maximizing one’s minimum gains, so for one player, this could be called the maximin criterion.

Let us use the example in Table M4.1 to illustrate the minimax criterion. This is a two-person, zero-sum game with the strategies for player Y given as the columns of the table. The values are gains for player X and losses for player Y. Player Y is looking at a maximum loss of 3 if strategy $Y_1$ is selected and a maximum loss of 5 if strategy $Y_2$ is selected. Thus, player Y should select strategy $Y_1,$ which results in a maximum loss of 3 (the minimum of the maximum possible losses). This is called the upper value of the game. Table M4.2 illustrates this minimax approach.

Table M4.2 Minimax Solution

In considering the maximin strategy for player X (whose strategies correspond to the rows of the table), let us look at the minimum payoff for each row. The payoffs are $\mathrm{+}3$ for strategy $X_1$ and $\mathrm{-}2$ for strategy $X_2.$ The maximum of these minimums is $\mathrm{+}3,$ which means strategy $X_1$ will be selected. This value $\left(\mathrm{+}3\right)$ is called the lower value of the game.

If the upper and lower values of a game are the same, this number is called the value of the game, and an equilibrium or saddle point condition exists. For the game presented in Table M4.2, the value of the game is 3 because this is the value for both the upper and the lower values. The value of the game is the average or expected game outcome if the game is played an infinite number of times.

In implementing the minimax strategy, player Y will find the maximum value in each column and select the minimum of these maximums. In implementing the maximin strategy, player X will find the minimum value in each row and select the maximum of these minimums. When a saddle point is present, this approach will result in pure strategies for each player. Otherwise, the solution to the game will involve mixed strategies. These concepts are discussed in the following sections.