Equilibrium paths in discounted supergames

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.