“Repeated Games with Present-biased Preferences“, Chade, Prokopovych, and Smith (Journal of Economic Theory, 2008)
We study infinitely repeated games with perfect monitoring, where players have beta-delta preferences. We compute the continuation payoff set using recursive techniques and then characterize equilibrium payoffs. We then explore the cost of the present-time bias, producing comparative statics. Unless the minimax outcome is a Nash equilibrium of the stage game, the equilibrium payoff set is not monotonic in or . Finally, we show how the equilibrium payoff set is contained in that of a repeated game with smaller discount factor.