A systematic process for evaluating structured perfect Bayesian equilibria in dynamic games with asymmetric information

We consider a finite horizon dynamic game with N selfish players who observe their types privately and take actions, which 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. Since each player has a different information set, this forms a dynamic game with asymmetric information and there is no known methodology to find perfect Bayesian equilibria (PBE) for such games in general. In this paper, we provide a two-step backward-forward recursive algorithm to find a class of PBE using a belief state based on common information of the players. We refer to such equilibria as structured Bayesian perfect equilibria (SPBE). 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 defined through a forward recursion. We provide a public goods example to demonstrate the methodology.