Symmetric Nash gain map
= Symmetric Nash gain map
{title2=$T_i(x)=\frac{x_i+[(Px)_i-x^TPx]_+}{1+\sum_j[(Px)_j-x^TPx]_+}$}
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>.