Could there be multiple symmetric equilibriums in a symmetric games?

Question

Given a finite symmetric 2 player game with a strategy space $S$, a (mixed-strategy) symmetric equilibrium is a distribution $d\in \Delta(S)$ such that $(d,d)$ is a Nash equilibrium.

A known result is that in such game there always exists a symmetric equilibrium (see here).

Could there be multiple (extreme) symmetric equilibriums? that is two symmetric equilibriums $d_1,d_2$ such that $d=\frac{d_1+d_2}{2}$ is not an equilibrium?

Answer 1

Yes. Consider the coordination game $(A,B) = (I,I)$, where $I$ is the $n \times n$ identity matrix.

For this game there are $2^{n}-1$ symmetric equilibria of the form $(x,x)$ where $x=(1/|S|,\ldots,1/|S|)$ for non-empty $S \subseteq \{1,\ldots,n\}$. I.e., the equilibria comprise all identical uniform mixtures. These are all equilibria.