Deterministic limit of mean field games associated with nonlinear Markov processes

TL;DR

This paper investigates the deterministic limit of mean field games with nonlocal coupling, where the dynamics are modeled by nonlinear Markov processes, and establishes convergence to a minimax solution framework.

Contribution

It introduces a framework for analyzing the deterministic limit of mean field games with nonlinear Markov dynamics using minimax solutions.

Findings

Proves convergence of mean field game solutions to minimax solutions

Extends mean field game theory to nonlinear Markov processes

Connects stochastic and deterministic mean field game models

Abstract

The paper is concerned with the deterministic limit of mean field games with the nonlocal coupling. It is assumed that the dynamics of mean field games are given by nonlinear Markov processes. This type of games includes stochastic mean field games as well as mean field games with finite state space. We consider the limiting deterministic mean field game within the framework of minimax approach. The concept of minimax solutions is close to the probabilistic formulation. In this case the Hamilton--Jacobi equation is considered in the minimax/viscosity sense, whereas the flow of probabilities is determined by the probability on the set of solutions of the differential inclusion associated with the Hamilton-Jacobi equation such that those solutions are viable in the graph of the minimax solution. The main result of the paper is the convergence (up to subsequence) of the solutions of the mean field games to the minimax solution of deterministic mean field game in the case when the underlying dynamics converge to the deterministic evolution.