Structured Perfect Bayesian Equilibrium in Infinite Horizon Dynamic Games with Asymmetric Information

TL;DR

This paper introduces Structured Perfect Bayesian Equilibrium (SPBE) for infinite horizon dynamic games with asymmetric information, simplifying the computation by using public beliefs as a sufficient statistic, and provides a recursive solution method.

Contribution

It defines a new subset of PBE called SPBE that leverages public beliefs to decouple strategies and beliefs over time, enabling a single fixed-point equation solution.

Findings

SPBE can be computed via a fixed-point equation.

The public belief dimension remains constant over time.

Method demonstrated with a public goods example.

Abstract

In dynamic games with asymmetric information structure, the widely used concept of equilibrium is perfect Bayesian equilibrium (PBE). This is expressed as a strategy and belief pair that simultaneously satisfy sequential rationality and belief consistency. Unlike symmetric information dynamic games, where subgame perfect equilibrium (SPE) is the natural equilibrium concept, to date there does not exist a universal algorithm that decouples the interdependence of strategies and beliefs over time in calculating PBE. In this paper we find a subset of PBE for an infinite horizon discounted reward asymmetric information dynamic game. We refer to it as Structured PBE or SPBE; in SPBE, any agents' strategy depends on the public history only through a common public belief and on private history only through the respective agents' latest private information (his private type). The public belief acts as a summary of all the relevant past information and it's dimension does not increase with time. The motivation for this comes the common information approach proposed in Nayyar et al. (2013) for solving decentralized team (non-strategic) resource allocation problems with asymmetric information. We calculate SPBE by solving a single-shot fixed-point equation and a corresponding forward recursive algorithm. We demonstrate our methodology by means of a public goods example.