We study inﬁnitely repeated games with observable actions, where players have present-biased (so-called beta-delta) preferences. We give a two-step procedure to characterize Strotz–Pollak equilibrium payoﬀs: compute the continuation payoﬀ set using recursive techniques, and then use this set to characterize the equilibrium payoﬀ set U(beta,delta). While Strotz–Pollak equilibrium and subgame perfection diﬀer here, the generated paths and payoﬀs nonetheless coincide. We then explore the cost of the present-time bias. Fixing the total present value of 1 util flow, lower beta or higher delta shrinks the payoﬀ set. Surprisingly, unless the minimax outcome is a Nash equilibrium of the stage game, the equilibrium payoﬀ set U(beta,delta) is not separately monotonic in beta or delta. While U(beta,delta) is contained in payoﬀ set of a standard repeated game with smaller discount factor, the present-time bias precludes any lower bound on U(beta,delta) that would easily generalize the beta = 1 folk-theorem.
Chade, Hector; Prokopovych, Pavlo; and Smith, Lones, "Repeated Games with Present-Biased Preferences" (2006). Cowles Foundation Discussion Papers. 1843.