Equilibrium paths in discounted supergames

Abstract

This paper examines the subgame-perfect pure-strategy equilibria in discounted supergames with perfect monitoring. It is shown that all the equilibrium paths are composed of fragments called elementary subpaths. This characterization result makes it possible to compute and analyze the equilibrium paths and payoffs by using a collection of elementary subpaths. It is also shown that all the equilibrium paths can be compactly represented by a directed graph when there are finitely many elementary subpaths. In general, there may be infinitely many elementary subpaths, but it is always possible to construct finite approximations. When the subpaths are allowed to be approximatively incentive compatible, it is possible to compute in a finite number of steps a graph that represents all the equilibrium paths. The directed graphs can be used in analyzing the complexity of equilibrium outcomes. In particular, it is shown that the size and the density of the equilibrium set can be measured by the asymptotic growth rate of equilibrium paths and the Hausdorff dimension of the payoff set. JEL Classification: C72, C73players' available moves at different positions, we use graphs to describe the variety of equilibrium behavior. Moreover, we emphasize that this characterization concerns all the equilibrium paths simultaneously rather than gives a condition for individual paths as in [1,2].We propose two complexity measures to analyze the equilibrium paths and payoffs based on the graph presentation. The first one is the asymptotic growth rate of the equilibrium paths. This measure tells us the rate at which the number of finitely long equilibrium paths grows when their length is increased. Because the paths of action profiles are given by strategies, the growth rate reflects the size of the equilibrium set and the increase of strategies producing the finitely long equilibrium paths.The second complexity measure is the Hausdorff dimension of the payoff set. This measure reflects the density of the equilibrium payoff set, which is a fractal in general [10]. The phenomenon that the payoff set behaves in a rather complex manner, as fractals do, is not completely new, see [31] and [34]. We offer a more comprehensive view to the structure of equilibria: when the discount factors vary, the elementary subpaths change, which affects the graph that generates the payoffs. The proposed complexity measures make it possible, e.g., to compare different repeated games in terms of equilibrium behavior.Our approach to the complexity of repeated game equilibria is new, and it differs from the previous literature on strategic complexity [15,28] and computational complexity [16,21]. We analyze the complexity of all the equilibrium outcomes without relying on the complexity of individual strategies nor their computation. It has been shown that computing even an approximate equilibrium in a stage game is difficult [20], and the task is not any easier in repeated games [13,33]. However, there are efficient algorithms that work for a class of repeated gam...

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