COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1555

Repeated Games with Present-Biased Preferences

Hector Chade
Department of Economics, Arizona State University
Pavlo Prokopovych
EERC, Ukraine
Lones Smith
Economics Department, University of Michigan

January 2006

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.

Keywords: Beta-delta preferences, Repeated games, Dynamic programming, Strotz–Pollak equilibrium

JEL Classification: C73