A Systematic Process for Evaluating Structured Perfect Bayesian Equilibria in Dynamic Games With Asymmetric Information

Abstract

We consider both finite-horizon and infinite-horizon versions of a dynamic game with $N$ selfish players who observe their types privately and take actions that are publicly observed. Players’ types evolve as conditionally independent Markov processes, conditioned on their current actions. Their actions and types jointly determine their instantaneous rewards. In dynamic games with asymmetric information, a widely used concept of equilibrium is perfect Bayesian equilibrium (PBE) which consists of a strategy and belief pair that simultaneously satisfy sequential rationality and belief consistency. In general, there does not exist a universal algorithm that decouples the interdependence of strategies and beliefs over time in calculating PBE. In this paper, for the finite-horizon game with independent types, we develop a two-step backward–forward recursive algorithm that sequentially decomposes the problem (w.r.t. time) to obtain a subset of PBEs, which we refer to as structured Bayesian perfect equilibria (SPBE). In such equilibria, a player's strategy depends on his/her history only through a common public belief and its current private type. The backward recursive part of this algorithm defines an equilibrium generating function. Each period in the backward recursion involves solving a fixed-point equation on the space of probability simplexes for every possible belief on types. Using this function, equilibrium strategies and beliefs are generated through a forward recursion. We then extend this methodology to the infinite-horizon model, where we propose a time-invariant single-shot fixed-point equation, which in conjunction with a forward recursive step, generates the SPBE. Sufficient conditions for the existence of SPBE are provided. With our proposed method, we find equilibria that exhibit signaling behavior. This is illustrated with the help of a concrete public goods example.