Do Symmetric Games with Nash Equilibria always have a symmetric Equilbrium?

Solution 1:

The answer is yes for finite games and for zero-sum games. In general, however, the answer is no: http://www.rochester.edu/college/faculty/markfey/papers/SymmGame3.pdf

Solution 2:

The answer is yes for finite games and mixed strategies and this was already shown in the Ph.D thesis of John Nash, where it occurs as Theorem 4. Nash considered actually slightly more invariances in his theorem.

The proof amounts to the verification that one can do the usual fixed-point argument used for the proof that every finite game has a Nash equilibrium in mixed strategies, restricted to the set of symmetric strategy profiles and to symmetric best responses.