# Mean-field games of optimal stopping: a relaxed solution approach

## TL;DR

This paper introduces a relaxed solution framework for mean-field games involving optimal stopping, establishing existence, uniqueness, and a numerical method for equilibrium computation.

## Contribution

It develops a relaxed optimal stopping approach for mean-field games, proving equilibrium existence, uniqueness, and connecting relaxed solutions to pure strategies.

## Key findings

- Proved existence of relaxed Nash equilibrium.
- Established conditions for pure strategy optimality.
- Presented a convergent numerical method for potential games.

## Abstract

We consider the mean-field game where each agent determines the optimal time to exit the game by solving an optimal stopping problem with reward function depending on the density of the state processes of agents still present in the game. We place ourselves in the framework of relaxed optimal stopping, which amounts to looking for the optimal occupation measure of the stopper rather than the optimal stopping time. This framework allows us to prove the existence of the relaxed Nash equilibrium and the uniqueness of the associated value of the representative agent under mild assumptions. Further, we prove a rigorous relation between relaxed Nash equilibria and the notion of mixed solutions introduced in earlier works on the subject, and provide a criterion, under which the optimal strategies are pure strategies, that is, behave in a similar way to stopping times. Finally, we present a numerical method for computing the equilibrium in the case of potential games and show its convergence.