Approximating equilibria for infinite horizon dynamic games.

Abstract

One of the emerging trends in game theory is the increasing interest in dynamic games as the natural extension of the well studied class of repeated games. Early studies yielded specific examples where equilibria are computable in a more or less straightforward manner. However, it has been recently recognized the fact that strategic behaviour in dynamic games is far more complex than in the context of repeated games. We will focus our attention into two of the plausible causes for this complexity; on the one hand, the fact that the classical technique which consists of studying the finite horizon version of the game (which is supposed to be simpler) and taking limits may fail to fully grasp the nature of all equilibria. This difficulty is essentially due to end of horizon effects. Secondly, when game parameters vary in time there may be a substantial change in the equilibrium nature, which is a rather troublesome feature, particularly when there is a great degree of uncertainty on the game parameters. In Chapters 2 and 4, we study the stability of first period equilibrium strategies as the planning horizon diverges to infinity. Interestingly enough, much of the work in stability issues in dynamic games has been concentrated on the convergence of equilibrium state paths to a stationary state. Our work differs in that we are interested in early turnpikes or solution horizons, that is, long but finite horizons such that first period equilibrium outcome is arbitrarily close to an infinite horizon first period equilibrium outcome. We also prove the existence of forecast horizons, a stronger concept that makes rigorous the intuitive belief that play in early periods must be strategically decoupled from changes in game parameters at the tail. The key assumption that underlies these results is the monotonic behaviour of finite horizon equilibrium with respect to parameter changes, a feature that has been widely detected in many applications. Finally, in Chapter 3 we provide a new sequential characterization of infinite horizon equilibria. We do so by defining the notion of a finite horizon constrained equilibria, one in which ending play is clearly prespecified. This allows to overcome the aforementioned end of horizon effects. In view of this result, one need only look at the properties preserved by limits of finite horizon constrained equilibria, to unveil the nature of infinite horizon equilibrium strategies. For instance, one may prove or disprove the sustainability of first best outcome as equilibrium play of the infinite horizon game. This result is also shown to hold in undiscounted games, a subject that has been largely untouched by the existing literature on the subject.