Repeated Games with Probabilistic Horizon
by
Ivan Arribas
Department of Economics, University of Valencia
Coauthors: Amparo Urbano (Department of Economics, University of Valencia)

Assumptions about the length of players' horizons often have profound implications for behavior. As it is well know equilibria in infinitely repeated games are characterized by folk theorems. However, in instances where the stage game gives rise to a unique equilibrium, a large but finite horizon does not allow players to sustain anything other than the repetition of the stage game equilibrium. Games with infinite horizon and constant discounting imply that the end of the game never gets any closer (in a probability sense). Yet, the expected horizons of agents do not remain constant, in general, over time. Repeated games with probabilistic horizon are defined as those games where players have a common structure of probability over the length of their repetition. In particular, they assign a probability p(t) to the event that "the game ends in period t". In this framework we analyze a Generalized Prisoners' Dilemma game in both finite stage and differentiable stage games. For finite stage games we completely characterize the existence of sub-game perfect cooperative equilibrium strategies by the speed at which p(t) converges to zero when t tends to infinity or the (first order) convergence speed (the limit of the ratio between the ending probabilities of two consecutive periods).Our approach to modeling the uncertainty over the length of the game allows us to solve analytically a wider family of problems than those analyzed by Jones(1999, 1998) and Bernheim and Dasgupta(1995). "Leptokurtic distributions" are defined as those distributions for which the (first order) convergence speed is zero and they preclude cooperation in finite stage games with probabilistic horizon and we relate them with Jones' quasifinite continuation probabilities and with Berheim and Dasgupta's asymptotically finite continuation probabilities. Moreover, finite stage games with probabilistic horizon unify the analysis for finitely repeated games, infinitely repeated games with discount factor and infinitely repeated games with limit average payoffs. Thus, common and wide families of probability distributions as the positive Poisson, the geometric, the harmonic and the negative binomial distributions, among others are analyzed. For differentiable games we reinterpret the Bernheim and Dasgupta's results and we classify the games according to both the (first order) convergence speed of the original probability distribution and the second order convergence speed of its logarithm. Only five possibilities are available and we analyze in which of them cooperation is attainable. Also, just a subset of the leptokurtic distributions precludes cooperation in differentiable stage games with probabilistic horizon.

JEL C.S: C72; UNESCO: 120706

Date received: June 12, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafi-55.