Characterizing the limit set of perfect and public equilibrium payoffs with unequal discounting

Abstract

We study repeated games with imperfect public monitoring and unequal dis- counting. We characterize the limit set of perfect and public equilibrium payoffs as discount factors converge to 1 with the relative patience between players fixed. We show that the pairwise and individual full rank conditions are sufficient for the folk theorem. In this paper, we characterize the equilibrium payoffs in repeated games with imperfect public monitoring and unequal discounting as discount factors converge to 1 with rel- ative patience fixed. In particular, we show that the pairwise and individual full rank conditions are sufficient for the folk theorem. Lehrer and Pauzner (1999) (henceforth LP) analyze two-player repeated games with perfect monitoring and unequal discounting. They define the set of feasible and sequen- tially individually rational (henceforth SIR) payoffs and show that in two-player games with perfect monitoring, the limit set of subgame perfect equilibrium payoffs coincides with that of SIR payoffs as discount factors converges to 1 with the relative patience fixed (the folk theorem). Recently, Chen and Takahashi (2012) extend the result to n-player games with perfect monitoring. This paper extends their results to imperfect public monitoring. While the proofs of both Lehrer and Pauzner (1999 )a ndChen and Takahashi (2012) are constructive, we em- ploy a nonconstructive approach using the recursive structure of the perfect and public equilibrium (henceforth PPE). Specifically, we attain a characterization of the set of PPE payoffs as discount factors converge to1. In addition, we characterize SIR payoffs. Given these characterizations, we show that if the pairwise and individual full rank conditions are satisfied, these two sets coincide, that is, the folk theorem holds.