Mean Field Games and Nonlinear Markov Processes

TL;DR

This paper studies mean field games involving multiple agent classes with dynamics modeled by nonlinear Markov processes, demonstrating the solvability of associated kinetic equations and their relation to Nash equilibria.

Contribution

It introduces a framework for mean field games with multiple agent classes and nonlinear Markov dynamics, extending existing models to more general integro-differential generators.

Findings

Kinetic equations for large N agent systems are solvable.

Solutions correspond to 1/N-Nash equilibria.

Framework accommodates general Levy-Khintchine type generators.

Abstract

In this paper, we investigate the mean field games with $K$ classes of agents who are weakly coupled via the empirical measure. The underlying dynamics of the representative agents is assumed to be a controlled nonlinear Markov process associated with rather general integro-differential generators of L\'evy-Khintchine type (with variable coefficients). We show that nonlinear measure-valued kinetic equations describing the dynamic law of large numbers limit for system with large number N of agents are solvable and that their solutions represent 1/N-Nash equilibria for approximating systems of N agents.