Major-Minor Mean Field Game of Stopping: An Entropy Regularization Approach

TL;DR

This paper introduces an entropy regularization approach to establish the existence of relaxed equilibria in a discrete-time major-minor mean field game of stopping, addressing complexities due to the major player's strategic choices.

Contribution

It develops a novel entropy regularization method to prove the existence of equilibria in complex mean field games involving stopping and control decisions.

Findings

Existence of regularized equilibria via fixed point theorems

Convergence of regularized equilibria to original problem solutions

Extension of method to mean field games with control choices

Abstract

This paper studies a discrete-time major-minor mean field game of stopping where the major player can choose either an optimal control or stopping time. We look for the relaxed equilibrium as a randomized stopping policy, which is formulated as a fixed point of a set-valued mapping, whose existence is challenging by direct arguments. To overcome the difficulties caused by the presence of a major player, we propose to study an auxiliary problem by considering entropy regularization in the major player's problem while formulating the minor players' optimal stopping problems as linear programming over occupation measures. We first show the existence of regularized equilibria as fixed points of some simplified set-valued operator using the Kakutani-Fan-Glicksberg fixed-point theorem. Next, we prove that the regularized equilibrium converges as the regularization parameter $\lambda$ tends to 0, and the limit corresponds to a fixed point of the original operator, thereby confirming the existence of a relaxed equilibrium in the original mean field game problem. We also extend this entropy regularization method to the mean-field game problem where the minor players choose optimal controls.