The multi-player nonzero-sum Dynkin game in discrete time

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We study the infinite horizon discrete time N-player nonzero-sum Dynkin game ( $$N \ge 2$$ ) with stopping times as strategies (or pure strategies). The payoff depends on the set of players that stop at the termination stage (where the termination stage is the minimal stage in which at least one player stops). We prove existence of a Nash equilibrium point for the game provided that, for each player $$\pi _i$$ and each nonempty subset $$S$$ of players that does not contain $$\pi _i$$ , the payoff if $$S$$ stops at a given time is at least the payoff if $$S$$ and $$\pi _i$$ stop at that time.