Uniformly supported approximate equilibria in families of games

This paper considers uniformly bounded classes of non-zero-sum strategic-form games with large finite or compact action spaces. The central class considered is assumed to be defined via a semi-algebraic condition. We show that for each ɛ>0, the support size required ɛ-equilibrium can taken uniform over entire class. As corollary, value zero-sum games, as function single-variable, well-behaved in limit. More generally, result only requires collection payoff functions considered, other players actions, have pseudo-dimension.