Markov Stationary Equilibria in Stochastic Supermodular Games with Imperfect Private and Public Information

Lukasz Balbus,Kevin L Reffett,Lukasz Patryk Wozny

doi:10.2139/ssrn.2029918

Abstract

We study a class of discounted, infinite horizon stochastic games with public and private signals and strategic complementarities. Using monotone operators defined on the function space of values and strategies (equipped with a product order), we prove existence of a Stationary Markov Nash Equilibrium via constructive methods. In addition, we provide monotone comparative statics results for ordered perturbations of our space of stochastic games. We present examples from industrial organization literature and discuss possible extensions of our techniques for studying principal-agent models.

Highlights

Introduction and Related LiteratureSince the class of discounted infinite horizon stochastic games was first introduced by Shapley [50], the question of existence and characterization of equilibrium has been the object of extensive study in game theory.1 In addition, more recently, stochastic games have become a fundamental tool for studying dynamic equilibrium in economic models where there is repeated strategic interaction among agents with limited commitment
This paper proposes a new set of monotone methods for a class of discounted, infinite horizon stochastic games with both public and private signals, as well as strategic complementarities
Our analysis shares some of the properties of the belief-free equilibria studied in [23], as we assume players have Markovian beliefs that depend only on public and individual signals, and we do not need to model beliefs off the equilibrium path as in their work

Summary

Introduction and Related Literature

Since the class of discounted infinite horizon stochastic games was first introduced by Shapley [50], the question of existence and characterization of equilibrium has been the object of extensive study in game theory. In addition, more recently, stochastic games have become a fundamental tool for studying dynamic equilibrium in economic models where there is repeated strategic interaction among agents with limited commitment. This approach arises in, for example, calibration approaches to characterizing MSNE (as in macroeconomics), or estimation/simulation methods (as in industrial organization) For such questions, one needs to unify the theory of existence of equilibrium with a theory to numerical implementation, which requires one to present (i) constructive arguments to verify existence, and (ii) sharp characterizations of the set of equilibria being computed, and (iii) methods of relating error analysis to particular approximation schemes at hand. One needs to unify the theory of existence of equilibrium with a theory to numerical implementation, which requires one to present (i) constructive arguments to verify existence, and (ii) sharp characterizations of the set of equilibria being computed, and (iii) methods of relating error analysis to particular approximation schemes at hand Our paper proposes such a framework for the class of stochastic games we study. Appendix states the auxiliary theorem we use in our proofs, while Sect. 5 concludes with a discussion of related methods

The Class of Games

Main Results

Examples

Dynamic Price Competition with Private Information

Dynamic Cournot Competition with Learning-by-Doing and Incomplete Information

Conclusions and Related Techniques