The Folk Theorem for Repeated Games With Time-Dependent Discounting

This paper defines a general framework to study infinitely repeated games with time-dependent discounting, in which we distinguish and discuss both time-consistent and time-inconsistent preferences. To study the long-term properties of repeated games, we introduce an asymptotic condition to characterize the fact that players become more and more patient, that is, the discount factors at all stages uniformly converge to $1$. Two types of folk theorem's are proven under perfect observations of past actions and without the public randomization assumption: the asymptotic one, i.e. the equilibrium payoff set converges to the individual rational set as players become patient, and the uniform one, i.e. any payoff in the individual rational set is sustained by a single strategy profile which is an approximate subgame perfect Nash equilibrium in all games with sufficiently patient discount factors. As corollaries, our results of time-inconsistency imply the corresponding folk theorem's with the quasi-hyperbolic discounting.