It is well known that for any symmetric game, $v(A)=0$.
That is, if $X^{*}$ is optimal for Player 1 (a.k.a row player), then it is also optimal for Player 2.
Proof: Let $X$ be any strategy for P1. Since $A=-A^{T}$, we obtain $E(X,X)=XAX^{T}=X(-A^{T})X^{T} \stackrel{?}{=}-(XA^{T}X^{T})^{T}=-XAX^{T}=-E(X,X)$.
That is, $2E(X,X)=0 \iff E(X,X)=0.$
We have shown that in a symmetric game, the payoff is zero when both players play the same strategy.
I need some help in understanding the equality $\stackrel{?}{=}$ in the proof. It seems that the transpose is intentionally applied so that we can obtain $-XAX^T$. Is that legal?
Yes, because $XA^TX^T$ is a scalar number