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Repeated Games with Present-Biased Preferences

By Hector Chade, Pavlo Prokopovych and Lones Smith


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.beta-delta preferences, repeated games, dynamic programming, Strotz-Pollak equilibrium

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