Consider the first two games given as examples of normal form games.

In Prisoner's Dilemma the only Nash equilibrium is for both players to play : it's apparent that, no matter what player plays, player does better playing , and vice-versa for .

Battle of the Sexes has three Nash equilibria. Both and are Nash equilibria, since it should be clear that if player expects player to play , player does best by playing , and vice-versa, while the same situation holds if player expects player to play . The third is a mixed equilibrium; player plays with probability and player plays with probability. We confirm that these are equilibria by testing the first derivatives (if 0 then the strategy is either maximal or minimal). Technically we also need to check the second derivative to make sure that it is a maximum, but with simple games this is not really necessary.

Let player play with probability and player plays with probability .

And indeed the derivatives are 0 at and .