= Expected payoff
{title2=$U_i(\sigma)=\sum_s u_i(s)\prod_j\sigma_j(s_j)$}
For a <finite game>, let $s=(s_1,\ldots,s_n)$ 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.
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