An Extension of Major-Minor Mean Field Game Theory

TL;DR

This paper extends mean field game theory with a dominating player by incorporating joint state-control distributions and the direct impact of the dominating player's controls, leading to a more comprehensive modeling framework.

Contribution

It introduces a generalized framework for mean field games with a dominating player, accounting for joint state-control distributions and the influence of the dominating player's controls.

Findings

Formulation of the problem using conditional distributions.

Derivation of necessary optimality conditions via stochastic maximum principles.

Development of a coupled SHJB--FP system characterizing equilibrium.

Abstract

This work extends the theory presented in Mean Field Games with a Dominating Player by Bensoussan, Chau and Yam on mean field games with a dominating player, to the case in which the utility and cost functions depend not only on the law of the states, but on the joint state--control law. We incorporate the conditional distribution of the state--control pair of the representative agent given the common noise of the dominating player. In addition, we generalize the role of the dominating player to include the direct impact of its controls $u_0$ on the dynamics and functionals of the system. The optimization problems are reformulated in terms of the conditional distribution of the state--control pair, the necessary optimality conditions are established via stochastic maximum principles, and a coupled SHJB--FP system of equations is obtained that synthesizes the equilibrium conditions. This framework provides a significant extension of the existing literature on MFG with a dominating player.