Doing it now, later, or never

We study centipede games played by an infinite sequence of players. Following the literature on time-inconsistent preferences, we distinguish two types of decision makers, naive and sophisticated, and the corresponding solution concepts, naive ϵ-equilibrium and sophisticated ϵ-equilibrium. We show the existence of both naive and sophisticated ϵ-equilibria for each positive ϵ. Under the assumption that the payoff functions are upper semicontinuous, we furthermore show that there exist both naive and sophisticated 0-equilibria in pure strategies. We also compare the probability to stop of a naive versus a sophisticated decision maker and show that a sophisticated decision maker stops earlier.