Mean-field approximation for large-population beauty-contest games

We study a class of Keynesian beauty contest games where a large number of heterogeneous players attempt to estimate a common parameter based on their own observations. The players are rewarded for producing an estimate close to a certain multiplicative factor of the average decision, this factor being specific to each player. This model is motivated by scenarios arising in commodity or financial markets, where investment decisions are sometimes partly based on following a trend. We provide a method to compute Nash equilibria within the class of affine strategies. We then develop a mean-field approximation, in the limit of an infinite number of players, which has the advantage that computing the best-response strategies only requires the knowledge of the parameter distribution of the players, rather than their actual parameters. We show that the mean-field strategies lead to an ε-Nash equilibrium for a system with a finite number of players. We conclude by analyzing the impact on individual behavior of changes in aggregate population behavior.