Uniformly supported approximate equilibria in families of games

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