Deterministic multi-player Dynkin games

A multi-player Dynkin game is a sequential game in which at every stage one of the players is chosen, and that player can decide whether to continue the game or to stop it, in which case all players receive some terminal payoff. We study a variant of this model, where the order by which players are chosen is deterministic, and the probability that the game terminates once the chosen player decides to stop may be strictly less than 1. We prove that a subgame-perfect epsilon-equilibrium in Markovian strategies exists. If the game is not degenerate this epsilon-equilibrium is actually in pure strategies.