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The set $E={\Delta }_2\times {\Delta }_2$ is a compact convex square. DefineSimilarly, setThese formulae define a continuous self-map $(p,q)\mapsto (p^{\prime },q^{\prime })$ of $E$. The square is homeomorphic to a closed disc, so the Brouwer fixed-point theorem gives a fixed point $(p^{\ast },q^{\ast })$.

Let $U=u_1+u_2$ at that fixed point. From $p^{\prime }=p$,If $U>0$, every $i$ with $p_i>0$ has $A(e_i,q)-A(p,q)>0$. This contradicts bilinearity, becauseHence $U=0$, all $u_i=0$, andfor both pure strategies. By linearity this holds for every mixed strategy $p$. The same argument with $v_j$ givesfor every $q$. Therefore

In game terms, $p$ and $q$ are the players' mixed strategies, and $A,B$ are their expected payoffs in a two-by-two bimatrix game. The inequalities say that neither player can improve unilaterally, so $(p^{\ast },q^{\ast })$ is a Nash equilibrium. This is the Brouwer proof of Nash equilibrium for a two-by-two game.