Stochastic discounting in repeated games: awaiting the almost inevitable

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We study repeated games with pure strategies and stochastic discounting under perfect information, with the requirement that the stage game has at least one pure Nash action profile. Players discount future payoffs with a common, but stochastic, discount factor where associated stochastic discounting processes are required to satisfy Markov property, martingale property, having bounded increments, and possessing state spaces with rich ergodic subsets. We, additionally, demand that there are states resulting in discount factors arbitrarily close to 0, and that they are reachable with positive (yet, possibly arbitrarily small) probability in the long run. In this setting, we prove both the perfect Folk Theorem and our main result: The occurrence of any finite number of consecutive repetitions of the period Nash action profile, must almost surely happen within a finite time window no matter which subgame perfect equilibrium strategy is considered and no matter how high the initial discount factor is.