An Introduction to Mean Field Game Theory

Abstract

These notes are an introduction to Mean Field Game (MFG) theory, which models differential games involving infinitely many interacting players. We focus here on the Partial Differential Equations (PDEs) approach to MFGs. The two main parts of the text correspond to the two emblematic equations in MFG theory: the first part is dedicated to the MFG system, while the second part is devoted to the master equation. The MFG system describes Nash equilibrium configurations in the mean field approach to differential games with infinitely many players. It consists in the coupling between a backward Hamilton-Jacobi equation (for the value function of a single player) and a forward Fokker-Planck equation (for the distribution law of the individual states). We discuss the existence and the uniqueness of the solution to the MFG system in several frameworks, depending on the presence or not of a diffusion term and on the nature of the interactions between the players (local or nonlocal coupling). We also explain how these different frameworks are related to each other. As an application, we show how to use the MFG system to find approximate Nash equilibria in games with a finite number of players and we discuss the asymptotic behavior of the MFG system. The master equation is a PDE in infinite space dimension: more precisely it is a kind of transport equation in the space of measures. The interest of this equation is that it allows to handle more complex MFG problems as, for instance, MFG problems involving a randomness affecting all the players. To analyse this equation, we first discuss the notion of derivative of maps defined on the space of measures; then we present the master equation in several frameworks (classical form, case of finite state space and case with common noise); finally we explain how to use the master equation to prove the convergence of Nash equilibria of games with finitely many players as the number of players tends to infinity. As the works on MFGs are largely inspired by P.L. Lions’ courses held at the Collège de France in the years 2007–2012, we complete the text with an appendix describing the organization of these courses.