Long time behavior of the master equation in mean-field game theory

TL;DR

This paper investigates the long-term behavior of mean field game systems and their master equations, establishing convergence results as the horizon grows or the discount factor vanishes, with implications for ergodic limits.

Contribution

It introduces new estimates for convergence rates and proves the existence of an ergodic master equation linking long-term behaviors of MFG systems.

Findings

Convergence of the time-dependent master equation to an ergodic solution.

Convergence of the discounted master equation as the discount factor tends to zero.

Existence of solutions to the ergodic master equation.

Abstract

Mean Field Game (MFG) systems describe equilibrium configurations in games with infinitely many interacting controllers. We are interested in the behavior of this system as the horizon becomes large, or as the discount factor tends to $0$. We show that, in the two cases, the asymptotic behavior of the Mean Field Game system is strongly related with the long time behavior of the so-called master equation and with the vanishing discount limit of the discounted master equation, respectively. Both equations are nonlinear transport equations in the space of measures. We prove the existence of a solution to an ergodic master equation, towards which the time-dependent master equation converges as the horizon becomes large, and towards which the discounted master equation converges as the discount factor tends to $0$. The whole analysis is based on the obtention of new estimates for the exponential rates of convergence of the time-dependent MFG system and the discounted MFG system.