Mean-Field Games of Optimal Stopping: A Relaxed Solution Approach

Geraldine Bouveret, Roxana Dumitrescu, Peter Tankov

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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 a 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.
Original languageEnglish
Pages (from-to)1795-1821
Number of pages27
JournalSIAM JOURNAL ON CONTROL AND OPTIMIZATION
Volume58
Issue number4
Early online date6 Jul 2020
DOIs
Publication statusE-pub ahead of print - 6 Jul 2020

Keywords

  • Infinite-dimensional linear programming
  • Mean-field games
  • Optimal stopping
  • Relaxed solutions

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