Mean Field Control and Mean Field Game Models with Several Populations

TL;DR

This paper explores multi-population mean field control and game models, deriving optimality conditions and highlighting the necessity of Master equations in competitive scenarios, with applications to linear-quadratic cases.

Contribution

It introduces a unified framework for multi-population mean field models, deriving adjoint equations and emphasizing the role of Master equations in competitive settings.

Findings

Derived adjoint equations for multi-population mean field problems

Identified the need for Master equations in competitive multi-population models

Provided linear-quadratic examples with Riccati equations

Abstract

In this paper, we investigate the interaction of two populations with a large number of indistinguishable agents. The problem consists in two levels: the interaction between agents of a same population, and the interaction between the two populations. In the spirit of mean field type control (MFC) problems and mean field games (MFG), each population is approximated by a continuum of infinitesimal agents. We define four different problems in a general context and interpret them in the framework of MFC or MFG. By calculus of variations, we derive formally in each case the adjoint equations for the necessary conditions of optimality. Importantly, we find that in the case of a competition between two coalitions, one needs to rely on a system of Master equations in order to describe the equilibrium. Examples are provided, in particular linear-quadratic models for which we obtain systems of ODEs that can be related to Riccati equations.