Probabilistic Approach to Mean Field Games and Mean Field Type Control Problems with Multiple Populations

TL;DR

This paper develops a probabilistic framework for analyzing mean field games and control problems involving multiple populations, addressing cooperative and non-cooperative scenarios and establishing equilibrium existence and approximation results.

Contribution

It introduces a coupled stochastic differential equation approach to multi-population mean field problems and provides conditions for equilibrium existence and finite-agent approximation.

Findings

Established existence of mean field equilibria for various cooperation scenarios.

Proved that mean field solutions approximate Nash equilibria in large finite-agent games.

Analyzed multiple population interactions with shared and differing dynamics.

Abstract

In this work, we systematically investigate mean field games and mean field type control problems with multiple populations using a coupled system of forward-backward stochastic differential equations of McKean-Vlasov type stemming from Pontryagin's stochastic maximum principle. Although the same cost functions as well as the coefficient functions of the state dynamics are shared among the agents within each population, they can be different population by population. We study the mean field limit for the three different situations; (i) every agent is non-cooperative; (ii) the agents within each population are cooperative; and (iii) the agents in some populations are cooperative but those in the other populations are not. We provide several sets of sufficient conditions for the existence of a mean field equilibrium for each of these cases. Furthermore, under appropriate conditions, we show that the mean field solution to each of these problems actually provides an approximate Nash equilibrium for the corresponding game with a large but finite number of agents.