Are Mean-field Games the Limits of Finite Stochastic Games?

Abstract

Mean-field games model the rational behavior of an infinite number of indistinguishable players in interaction. An important assumption of mean-field games is that, as the number of player is infinite, the decisions of an individual player do not affect the dynamics of the mass. Each player plays against the mass. A mean-field equilibrium corresponds to the case when the optimal decisions of a player coincide with the decisions of the mass. Many authors argue that mean-field games are a good approximation of symmetric stochastic games with a large number of players, the rationale behind this being that the impact of one player becomes negligible when the number of players goes to infinity. In this paper, we question this assertion. We show that, in general, this convergence does not hold. In fact, the “tit for tat” principle allows one to define many equilibria in repeated or stochastic games with N players. However, in mean-field games, the deviation of a single player is not visi- ble by the population and therefore the “tit for tat” principle cannot be applied. The conclusion is that, even if N-player games have many equilibria with a good social cost, this may not be the case for the limit game.