Major-Minor Mean Field Game of Stopping: An Entropy Regularization Approach
Xiang Yu, Jiacheng Zhang, Keyu Zhang, Zhou Zhou
TL;DR
This work addresses a discrete-time major-minor mean-field game where the major player can either exert an optimal control or stop, and each minor player faces a stopping problem influenced by the major’s state. By introducing entropy regularization on the major player’s objective, the authors obtain a unique Gibbs-type BR and recast the minor players’ problem as a linear program over occupation measures, enabling a fixed-point analysis via Kakutani–Fan–Glicksberg. They prove the existence of regularized equilibria for the auxiliary problem and show that, as the regularization parameter $\lambda$ tends to zero, these equilibria converge to a relaxed equilibrium for the original MFG; the methodology is extended to the setting where minor players solve optimal control problems as well. The results establish well-posedness for a broad class of major-minor MFGs with stopping and, more generally, with minor-control interactions, using entropy regularization to overcome nonconvexity and nonuniqueness obstacles. This contributes a robust existence framework with potential applications in economics and finance where a dominant agent interacts with a large population of strategically stopping agents.
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 $λ$ 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.