When a saddle point is present, the strategy each player should follow will always be the same, regardless of the other player’s strategy. This is called a pure strategy. A saddle point is a situation in which both players are facing pure strategies.

Using the minimax criterion, we saw that the game in Table M4.2 had a saddle point and thus is an example of a pure strategy game. It is beneficial for player X and for player Y to always choose one strategy. Simple logic would lead us to this same conclusion. Player X will always select $X_1$, since the payoffs for $X_1$ are better than the payoffs for $X_2,$ regardless of what player Y does. Knowing that player X will select $X_1,$ player Y will always select strategy $Y_1$ and only lose 3 rather than 5. Note that the saddle point in this example, 3, is the largest number in its column and the smallest number in its row. This is true of all saddle points.

Another example of a pure strategy game is shown in Table M4.3. Notice that the value 6 is the lowest number in its row and the highest number in its column. Thus, it is a saddle point and indicates that strategy $X_1$ will be selected by player X and strategy $Y_2$ will be selected by player Y. The value of this game is 6.

Table M4.3 Example of a Pure Strategy Game