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

TY - JOUR

T1 - Mean-field games of optimal stopping

T2 - a relaxed solution approach

AU - Bouveret, Géraldine

AU - Dumitrescu, Roxana

AU - Tankov, Peter

PY - 2020/7/6

Y1 - 2020/7/6

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85088965171&partnerID=8YFLogxK

U2 - 10.1137/18M1233480

DO - 10.1137/18M1233480

M3 - Article

SN - 0363-0129

VL - 58

SP - 1795

EP - 1821

JO - SIAM JOURNAL ON CONTROL AND OPTIMIZATION

JF - SIAM JOURNAL ON CONTROL AND OPTIMIZATION

IS - 4

ER -