Proving the existence of mixed-strategy NE for 2-player zero-sum symmetric game

Question

I am trying to prove the existence of mixed-strategy NE for 2-player zero-sum symmetric game, under the condition that given they have $I$ pure strategies and for the pay-off matrix $A$, $\exists x\in R_+^I$ such that $xA\ge0$. I know that I can use Nash's theorem directly, but I want to prove it explicitly.

So far what I have done is that I proved the payoff must be 0 since $A=-A^T$. And if $xA$ always has negative element(s), the best response of each players must be some pure strategy (by decomposing $xAy$). But it seems that all these are irrelevant to a direct proof the existence of NE. Please share with me if you have any idea!

Answer 1

An nice little proof is contained in Brown and von Neumann (1950), see also the (slightly more general) third proof in Hofbauer (2000). This is the idea:

For a mixed strategy $x$ (a column vector in the $I$-simplex) define a pure strategy $i$'s excess payoff against $x$ as $k_i(x)=\max\{(Ax)_i,\,0\}$. Let $\bar k(x)=\sum_{j\in I} k_j(x)$ be the sum of excess payoffs.

Now consider the system of differential equations known as the Brown-von Neumann-Nash dynamics (BNN dynamics), $\dot x_i=k_i(x)-x_i\bar k(x)$.

Then it is fairly straightforward to show: The function $V(x)=\sum_{j\in I} k_j(x)^2$ is a global Ljapunov function for the BNN dynamics and therefore all solutions converge to the set where $V(x)=0$. This set is the set of Nash equilibria, which is therefore nonempty.