Asymmetric Equilibria in Symmetric Games with Many Players

It is often the case in symmetric games in normal form (i.e., games with a symmetric payoff matrix), that the only existing pure-strategy equilibria are asymmetric. Examples of this are models of entry into an industry [cf., Dixit and Shapiro (1986)], models of price-dispersion [cf. Salop and Stiglitz (1976)], and models of information about prices [cf. Grossman and Stiglitz (1976)]. If there is an asymmetric equilibrium for a model with N ‘equal’ players, then there are multiple equilibria, only differing on the ‘name’ of the players ‘assigned’ to each one of the actions which together form an equilibrium. A natural question to ask is, then, how to select among these equilibria. If there are few players, one can assume that there is some communication and coordination mechanism which will lead the players to a specified equilibrium. [See for example Farrell (1987).] However, if there are many players, communication and coordination are not so simple, and another mechanism should be found. In this note, we show that in symmetric games with many players, an asymmetric pure-strategy equilibrium can be thought of as the approximate outcome of the play of a specific symmetric mixed-strategy equilibrium. In this mixed-strategy equilibrium, each player chases action a, with probability close to the fraction of players choosing that action in the asymmetric pure-strategy equilibrium. The idea is that with large numbers ex-ante probability and ex-post frequency are approximately the same. Schneidler (1973) presents a result (Theorem 2) similar to the one in this note, but for the case of non-atomic games, whereas we deal with games with a finite number of players.