Repeated Games with Stochastic Discounting

Abstract

This paper provides folk theorems for infinitely repeated games where the discount factor is stochastic. When discount factors are independently distributed and the current discount fac- tor is unobservable prior to current actions, standard trigger strategies support a 'full' folk theorem when the infimum of the mean of the sequence of discount factors is sufficiently close to one. When players choose actions in each period after having observed the current discount factor, this result breaks down; payoffs on the boundary of the set of individually rational payoffs are unobtainable as Nash equilibrium average payoffs to the supergame. In order to highlight the impact of stochastic discounting on the analysis of supergames, we provide the stochastic discounting analogue to Friedman's perfect folk theorem. This paper examines the implications of stochastic discount factors for Nash equilibria in repeated games. Repeated games have broad applicability in olig- opolistic markets, where optimizing owners make decisions based on the expected present value of alternative actions. In this setting, firms can invest one dollar today at the present rate of interest to obtain principal and interest tomorrow; thus it is natural to interpret the discount factor as 1/(1 + r,), where rt is the real rate of interest between periods t and t + 1. The paper is motivated by recent evidence (Kinal and Lahiri 1988, among others) that the real rate of interest varies stochastically over time, and the fact that the existing literature on repeated games assumes the discount factor to be deterministic. Our results are also relevant for finitely repeated games where the probability that the game ends (owing to bankruptcy, say) varies stochastically over time. Our Theorem 2 shows that, when discount factors are independently dis- tributed and the current discount factor is unobservable prior to current actions, standard trigger strategies support a 'full' folk theorem when the infi- mum of the mean of the sequence of discount factors is sufficiently close to one. Theorem 3 demonstrates that, when players choose actions in each period after having observed the current discount factor this result breaks down; pay- offs on the boundary of the set of individually rational payoffs are unobtain- able as Nash equilibrium average payoffs to the supergame. The reason is that a sufficiently low realization of the discount factor induces players to behave myopically, since in such periods the expected discounted value of sustaining one-shot cooperative actions is arbitrarily low. In order to highlight the impact of stochastic discounting on the analysis of supergames, we provide in Theorem 4 the stochastic discounting analogue to Friedman's perfect folk theorem. The supergame strategies we employ never break down, and involve actions correlated with current realizations of the discount rate. Players revert to a one-shot Nash equilibrium for low realiza- tions of the discount factor, and return to one-shot collusive behaviour for high realizations.