Mean Field Games with a Dominating Player

TL;DR

This paper develops a theoretical framework for mean field games involving a dominating player and multiple agents, providing conditions for optimal controls, equilibrium existence, and convergence from finite to mean field games.

Contribution

It introduces a comprehensive theory for mean field games with a dominating player, including necessary and sufficient conditions for equilibrium and convergence analysis.

Findings

Established necessary conditions for optimal controls.

Proved existence and uniqueness of equilibrium controls.

Demonstrated convergence of finite player games to mean field models.

Abstract

In this article, we consider mean field games between a dominating player and a group of representative agents, each of which acts similarly and also interacts with each other through a mean field term being substantially influenced by the dominating player. We first provide the general theory and discuss the necessary condition for the optimal controls and game condition by adopting adjoint equation approach. We then present a special case in the context of linear-quadratic framework, in which a necessary and sufficient condition can be asserted by stochastic maximum principle; we finally establish the sufficient condition that guarantees the unique existence of the equilibrium control. The proof of the convergence result of finite player game to mean field counterpart is provided in Appendix.