I am the aforementioned economist. 
I figured out a counterexample for the non-singular case:

Let 
$$
A = \left( \begin{array}{ccc}
0 & 3 & 1 \\
3 & 0 & 1 \\
2 & 2 & 2 \end{array} \right)
$$

$$
z = \left( \begin{array}{ccc}
\frac{1}{3}, & \frac{1}{3}, & \frac{1}{3} \end{array} \right)
$$

$$
x = \left( \begin{array}{ccc}
0, & \frac{1}{2}, & \frac{1}{2} \end{array} \right)
$$

$$
y = \left( \begin{array}{ccc}
\frac{1}{2}, & 0, & \frac{1}{2} \end{array} \right)
$$

The expected payoff is 2 in the symmetric equilibrium and $\frac{3}{2}$  in the other one. As far as I can tell there is no 2x2 non-singular counterexample.