  We introduce a zero-sum game problem of mean-field type as an extension of
the classical zero-sum Dynkin game problem to the case where the payoff
processes might depend on the value of the game and its probability law. We
establish sufficient conditions under which such a game admits a value and a
saddle point. Furthermore, we provide a characterization of the value of the
game in terms of a specific class of doubly reflected backward stochastic
differential equations (BSDEs) of mean-field type, for which we derive an
existence and uniqueness result. We then introduce a corresponding system of
weakly interacting zero-sum Dynkin games and show its well-posedness. Finally,
we provide a propagation of chaos result for the value of the zero-sum
mean-field Dynkin game.
