Brouwer gain-map proof of bimatrix equilibrium ($T_i(p,q)=\frac{p_i+[(Aq{)}_i-p^{T}Aq{]}_{+}}{1+{\sum }_j[(Aq{)}_j-p^{T}Aq{]}_{+}}$)

For each player in a finite bimatrix game, add the positive pure-strategy gains to its current mixed strategy and normalize. This gives a continuous self-map of the product of probability simplices. At a Brouwer fixed-point theorem fixed point, the gain coordinates equal the current strategy times their total gain. If that total were positive, every supported action would beat the current average payoff, contradicting that average. Thus every pure deviation has nonpositive gain for both players, proving a Nash equilibrium.