Mean field games: the master equation and the mean field limit

We present here results obtained in the joint work with Delarue, Lasry and Lions [4] on the convergence, as N tends to infinity, of a system of N coupled Hamilton-Jacobi equations, the Nash system. This system arises in differential game theory. The limit problem can be expressed in terms of the “Mean Field Game” system (coupling a Hamilton-Jacobi equation with a Fokker-Planck equation), or, alternatively, in terms of the “master equation” (a kind of second order partial differential equation stated on the space of probability measures). We also discuss the behavior of the optimal trajectories, for which we show a propagation of chaos property. The description of interactions between “rational agents” is often a difficult issue because the agents, being supposed rational, observe each other and react in function of their observation (strategic interaction): this leads to the notion of Nash equilibria, which are often difficult to compute and interpret. Aumann [2] was among the first to notice that this problem simplifies a lot when there are infinitely many “non-atomic” agents. By non-atomic, we mean that the agents have an infinitesimal influence on the global system. For a long time the ideas of nonatomic games have been applied to games in which the action space of the players is relatively simple (one-shot games). The importance of dynamic optimization problems (optimal control) in engineering sciences, economic theory, finance, etc... lead Lasry and Lions [12, 13, 14] and Huang, Caines and Malhamé [8, 9, 10, 11] to develop the counterpart of Aumann’s “non-antomic games” to optimal control. This is the so-called mean field games. It is worth mentioning that similar ideas were discussed in earlier works in the economic literature (heterogeneous agent models). Based on heuristic considerations, these authors derived a system describing interacting, indistinguishable agents. This system, called Mean Field Game (MFG) system, takes the form of a coupling between a backward Hamilton-Jacobi (HJ) equation and a forward Kolmogorov equation: $ & % ́Btu ́∆u`Hpx,Duq “ F px,mptqq in r0, T s ˆ R, Btm ́∆m ́ divpmDpHpx,Duqq “ 0 in r0, T s ˆ R, upT, xq “ Gpx,mpT qq, mp0, ̈q “ mp0q in R, (1) The data are the Hamiltonian H : RdˆRd Ñ R, the horizon T ą 0, the initial measure m0 and the maps F,G which describe the influence of the second equation on the first one. The function u can be thought of as the value function for an average agent seeking to optimize an optimal control problem, while m represents the time-evolving probability distribution of the state of the players when all players play in an optimal way. The terminology Mean Field Games comes from the analogy with statistical physics, which deals with large populations of particules and derives macroscopic laws from microscopic ones. ∗Ceremade, Université Paris-Dauphine, cardaliaguet@ceremade.dauphine.fr Séminaire Laurent-Schwartz — EDP et applications Centre de mathématiques Laurent Schwartz, 2015-2016 Exposé no XVII, 1-10