The master equation and the convergence problem in mean field games

TL;DR

This paper establishes the well-posedness of the master equation in mean field games and proves the convergence of the Nash system of coupled Hamilton-Jacobi equations to this limit, including a propagation of chaos result.

Contribution

It introduces a rigorous framework for the master equation in mean field games and proves convergence and well-posedness results that were previously unestablished.

Findings

Well-posedness of the master equation established

Convergence of Nash system solutions proven

Propagation of chaos for optimal trajectories demonstrated

Abstract

The paper studies 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. We describe the limit problem in terms of the so-called "master equation", a kind of second order partial differential equation stated on the space of probability measures. Our first main result is the well-posedness of the master equation. To do so, we first show the existence and uniqueness of a solution to the "mean field game system with common noise", which consists in a coupled system made of a backward stochastic Hamilton-Jacobi equation and a forward stochastic Kolmogorov equation and which plays the role of characteristics for the master equation. Our second main result is the convergence, in average, of the solution of the Nash system and a propagation of chaos property for the associated "optimal trajectories".