On the value of a time-inconsistent mean-field zero-sum Dynkin game

We study a mean-field zero-sum Dynkin game (MF-ZSDG) with time-inconsistent performance functionals adapted to the Brownian filtration. Despite the time-inconsistency of the MF-ZSDG, we show that it admits a value and that the pair of first times the value process hits the upper and lower obstacles, respectively, is a saddle point for the game. We solve the problem by approximating the associated lower and upper value processes with a sequence of value processes of interacting time-consistent zero-sum Dynkin games for which the saddle point of each of the value processes is the pair of first times each of those value processes hits the associated upper and lower obstacles, respectively. Under mild assumptions, we show that this sequence of saddle points converges in probability to the pair of first hitting times of the value process of the upper and lower obstacles, respectively, and that the limit is a saddle point for the time-inconsistent MF-ZSDG.