A mean field game (MFG) is a stochastic differential game with a continuum of players, describing the limit as n tends to infinity of Nash equilibria of certain n-player games, in which agents interact symmetrically through the empirical measure of their state processes. One way to understand a MFG is through its "master equation," an infinite-dimensional PDE aptly nicknamed the "monster equation." A solution of this equation can be used, for instance, to construct a solution of the original mean field game or prove convergence of n-player equilibria (see Cardaliaguet-Delarue-Lasry-Lions 2015). This talk will not dwell on how to solve the master equation, a difficult issue addressed in only a few papers thus far. Instead, we show how to use a sufficiently smooth solution to answer several open questions about the limit theory for MFGs. In particular, we derive for the first time a central limit theorem and a large deviations principle for the n-player empirical measure (in equilibrium). The proofs use the master equation to quantitatively relate the n-player equilibrium to a McKean-Vlasov system of interacting diffusions for which the limit theory is well understood.

This talk is based on joint work with Francois Delarue and Kavita Ramanan. Speaker(s): Daniel Lacker (Brown)