Symmetric Nash gain map ($T_i(x)=\frac{x_i+[(Px{)}_i-x^{T}Px{]}_{+}}{1+{\sum }_j[(Px{)}_j-x^{T}Px{]}_{+}}$)

The continuous gain map takes a probability simplex to itself. The Brouwer fixed-point theorem supplies a fixed point, whose positive gains would contradict the zero strategy-weighted average payoff excess. Thus all gains vanish, giving a strategy that is a best response to itself. In a symmetric bimatrix game, this produces a symmetric equilibrium.