An incomplete information justification of symmetric equilibrium in symmetric games

Consider a symmetric 2-player game of complete information. Consider an arbitrary Bayesian extension of that game with payoff-irrelevant types, independent random matching, and anonymity (private types). We show that, in this setting, while strategies in a Bayesian Nash equilibrium of that game can differ across types, aggregate play in any such equilibrium must coincide with a symmetric Nash equilibrium of the complete information game. This justifies the interpretation of certain data, including many laboratory experiments, as arising from a symmetric equilibrium, even when asymmetric equilibria exist and, in addition, subjects may be heterogeneous. JEL codes: C72, C90