On Mean-Field Games

Stochastic differential games with a large number of players are notoriously challenging, both theoretically and numerically, typically when it comes to computing Nash equilibria. Yet, when many players interact somehow symmetrically by responding only to the average behavior of the others, the game can surprisingly become more tractable by taking the limit of an infinite number of players. This is in direct analogy with the so-called « mean-field theory » which simplifies the analysis of large systems of interacting particles in statistical physics. Introduced independently about two decades ago by Lasry and Lions (mathematics) and Caines, Huang and Malahamé (engineering), the theory of Mean-Field Games has since been greatly developed with various applications in engineering, economical, social and biological sciences. The goal of this short lecture is to introduce the key concepts, particularly the deep connections between game theory, Partial Differential Equations and stochastic analysis, and to showcase a few striking recent applications.