Long Time Behavior and Stabilization for Displacement Monotone Mean Field Games

TL;DR

This paper investigates the long-term stability and convergence of Nash equilibria in displacement monotone Mean Field Games, demonstrating exponential convergence rates and stability in various metrics for both deterministic and stochastic cases.

Contribution

It establishes the stability and convergence of equilibria and value functions in displacement monotone MFGs over infinite horizons, with quantitative exponential rates.

Findings

Equilibria become close as time horizon increases, independently of initial conditions.

Value functions converge to an infinite horizon limit involving an ergodic constant.

Stability and convergence results hold for both deterministic and stochastic MFGs with displacement monotonicity.

Abstract

This paper is devoted to the study of the long time behavior of Nash equilibria in Mean Field Games within the framework of displacement monotonicity. We first show that any two equilibria defined on the time horizon $[0,T]$ must be close as $T \to \infty$, in a suitable sense, independently of initial/terminal conditions. The way this stability property is made quantitative involves the $L^2$ distance between solutions of the associated Pontryagin system of FBSDEs that characterizes the equilibria. Therefore, this implies in particular the stability in the 2-Wasserstein distance for the two flows of probability measures describing the agent population density and the $L^2$ distance between the co-states of agents, that are related to the optimal feedback controls. We then prove that the value function of a typical agent converges as $T \to \infty$, and we describe this limit via an infinite horizon MFG system, involving an ergodic constant. All of our convergence results hold true in a unified way for deterministic and idiosyncratic noise driven Mean Field Games, in the case of strongly displacement monotone non-separable Hamiltonians. All these are quantitative at exponential rates.