Approximate equilibria in strongly symmetric games

I study approximate equilibria in games with countably many players and finitely many pure strategies, with an emphasis on symmetric games. In a class of games called strongly symmetric tail function games, the following holds: existence of perfect ϵ-equilibrium (Solan and Vielle, 2001) for all ϵ>0 is equivalent to the existence of Nash equilibrium. In the larger class of strongly symmetric (not necessarily tail function) games, this equivalence no longer holds. The main result is that every strongly symmetric game has a symmetric ϵ proper equilibrium (Myerson, 1978) which is an ϵ-equilibrium (Radner, 1980). This existence result fails to hold in the larger class of weakly symmetric games.