Payoff ($u_i(s_1,\dots ,s_n)$)

A payoff is the number assigned to a player's outcome at a strategy profile, with larger numbers preferred. In the usual expected utility model, expected payoffs are probability-weighted averages of these numbers. If a payoff is specified as money and a player has preferences involving risk, first apply the relevant utility function; maximizing expected money assumes risk neutrality. In a two-player zero-sum game, one player's payoff is the negative of the other's. General bimatrix games instead require a payoff matrix for each player.

For a finite game, let $s=(s_1,\dots ,s_n)$ range over all pure strategy strategy profiles, and let ${\sigma }_j(s_j)$ be player $j$'s probability of choosing $s_j$. Under independent mixed strategies, the probability of $s$ is ${\prod }_j{\sigma }_j(s_j)$. The displayed sum is the expected value of player $i$'s payoff. Holding everyone else's mixed strategy fixed makes $U_i$ a linear function of ${\sigma }_i$; this is why checking pure strategy deviations suffices to check all mixed strategy deviations.