Stochastic Games in Continuous Time: Persistent Actions in Long-Run Relationships, Second Version

This paper studies a class of continuous-time stochastic games in which the actions of a long-run player have a persistent effect on payoffs. For example, the quality of a firm's product depends on past as well as current effort, or the level of a policy instrument depends on a government's past choices. The long-run player faces a population of small players, and its actions are imperfectly observed. I establish the existence of Markov equilibria, characterize the Perfect Public Equilibria (PPE) pay-offset as the convex hull of the Markov Equilibria payoff set, and identify conditions for the uniqueness of a Markov equilibrium in the class of all PPE. The existence proof is constructive: it characterizes the explicit form of Markov equilibria payoffs and actions, for any discount rate. Action persistence creates a crucial new channel to generate intertemporal incentives in a setting where traditional channels fail, and can provide permanent non-trivial incentives in many settings. These results offer a novel framework for thinking about reputational dynamics of firms, governments, and other long-run agents.