This paper analyses a stochastic Bayesian prisoners' dilemma where players' discount factor is private information. We consider an infinitely repeated game where two states of the world may occur. According to her own discount factor, a player chooses a cooperative behaviour in both states (patient), in none of the states (impatient) or in one state only (mildly patient). The presence of different states of the world affects the strategic role of beliefs. A mildly patient player has an incentive in "faking patience", which increases with the competitor's belief that the player is patient. Interestingly, this effect prevents cooperative equilibria to occur when the belief in patience is strong.
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