 Solution

For a finite normal-form game, a mixed strategy ${\sigma }_i$ is a probability distribution over player $i$'s pure strategies. Independent randomization by the players defines an expected payoff $u_i(\sigma )$. A strategy profile $\sigma$ is a Nash equilibrium iffor every player $i$ and every alternative mixed strategy ${\tau }_i$. By linearity in a player's own distribution, it is enough to check all pure deviations. Equivalently, every pure strategy in the support of a mixed strategy must be a best response to the other players' choices.