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 its 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 timeinvariant 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.

Zero-sum Stochastic Games with Asymmetric Information

“…4. Common information based perfect Bayesian equilibria in stochastic games of asymmetric information: Authors in [30] and [39] consider a stochastic game model (can be non-zero sum) in which the system state can be decomposed into a public state that is commonly observed by all players and a private state that is privately observed by each player. In this model, all the players' past actions are commonly observed and additionally, an imperfect version of players' private state may be disclosed to all the players at each time.…”

“…In this paper, we only consider two-player zerosum games. However, our general model captures many other models with system dynamics and information structures that may not conform to the model considered in [30,39]. For instance, unlike in [30,39], players' actions may not be fully observed in our model.…”

“…For instance, unlike in [30,39], players' actions may not be fully observed in our model. Further, the solutions in [30,39] rely on strong existence assumptions that may not be true in general.…”

Linear Equilibria for Dynamic LQG Games with Asymmetric Information and Dependent Types

“…We are interested in PBEs with strategies, g i t (·|h i t ) = ψ i t (·|S(h i t )), that are functions of h i t only through the summaries S(h i t ). These PBEs are called structured PBEs [22]. In contrast to H i t , the set of summaries does not grow in time and therefore, finding such structured PBEs is less complicated than a general PBE.…”

“…In contrast to H i t , the set of summaries does not grow in time and therefore, finding such structured PBEs is less complicated than a general PBE. According to [22], we can show that players can guarantee the same rewards by playing structured strategies compared to the general non-structured ones. In the dynamic games with asymmetric information, the summaries are usually the belief of players over the unknown variables of the game.…”