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.

[1]  Eitan Altman,et al.  Stochastic games with one step delay sharing information pattern with application to power control , 2009, 2009 International Conference on Game Theory for Networks.

[2]  A. Banerjee,et al.  A Simple Model of Herd Behavior , 1992 .

[3]  A. Pakes,et al.  Markov-Perfect Industry Dynamics: A Framework for Empirical Work , 1995 .

[4]  M. Spence.  Job Market Signaling , 1973 .

[5]  Demosthenis Teneketzis,et al.  On the design of globally optimal communication strategies for real-time noisy communication systems with noisy feedback , 2008, IEEE Journal on Selected Areas in Communications.

[6]  Ashutosh Nayyar,et al.  On globally optimal real-time encoding and decoding strategies in multi-terminal communication systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[7]  David M. Kreps,et al.  S I G N a L L I N G , 2022 .

[8]  Ashutosh Nayyar,et al.  Optimal Control Strategies in Delayed Sharing Information Structures , 2010, IEEE Transactions on Automatic Control.

[9]  Joseph E. Stiglitz,et al.  Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information , 1976 .

[10]  Eric van Damme,et al.  Non-Cooperative Games , 2000 .

[11]  David M. Kreps,et al.  Signaling Games and Stable Equilibria , 1987 .

[12]  Lones Smith,et al.  Pathological Outcomes of Observational Learning , 2000 .

[13]  Yu-Chi Ho.  Team decision theory and information structures , 1980, Proceedings of the IEEE.

[14]  Sanford J. Grossman.  The Informational Role of Warranties and Private Disclosure about Product Quality , 1981, The Journal of Law and Economics.

[15]  Achilleas Anastasopoulos,et al.  Stochastic Control of Relay Channels With Cooperative and Strategic Users , 2014, IEEE Transactions on Communications.

[16]  Harold L. Cole,et al.  Dynamic Games with Hidden Actions and Hidden States , 2001, J. Econ. Theory.

[17]  L. Shapley,et al.  Stochastic Games* , 1953, Proceedings of the National Academy of Sciences.

[18]  Ariel Pakes,et al.  A Framework for Applied Dynamic Analysis in I.O , 2000 .

[19]  Yuval Peres,et al.  Perfect Bayesian Equilibria in Repeated Sales , 2014, SODA.

[20]  Achilleas Anastasopoulos,et al.  Signaling equilibria for dynamic LQG games with asymmetric information , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[21]  Tamer Basar,et al.  Common Information Based Markov Perfect Equilibria for Stochastic Games With Asymmetric Information: Finite Games , 2014, IEEE Transactions on Automatic Control.

[22]  Achilleas Anastasopoulos,et al.  Decentralized Bayesian learning in dynamic games , 2016, 2016 54th Annual Allerton Conference on Communication, Control, and Computing (Allerton).

[23]  Eric Maskin,et al.  Markov Perfect Equilibrium: I. Observable Actions , 2001, J. Econ. Theory.

[24]  D. Bergemann,et al.  Learning and Strategic Pricing , 1996 .

[25]  A. Zahavi.  Mate selection-a selection for a handicap. , 1975, Journal of theoretical biology.

[26]  J. Filar,et al.  Competitive Markov Decision Processes , 1996 .

[27]  Charles A. Wilson,et al.  A model of insurance markets with incomplete information , 1977 .

[28]  Tamer Basar,et al.  Common Information based Markov Perfect Equilibria for Linear-Gaussian Games with Asymmetric Information , 2014, SIAM J. Control. Optim..

[29]  Yi Ouyang,et al.  Dynamic oligopoly games with private Markovian dynamics , 2015, 2015 54th IEEE Conference on Decision and Control (CDC).

[30]  Aditya Mahajan,et al.  Optimal Decentralized Control of Coupled Subsystems With Control Sharing , 2011, IEEE Transactions on Automatic Control.

[31]  J. Sobel,et al.  Equilibrium Selection in Signaling Games , 1987 .

[32]  S. Bikhchandani,et al.  You have printed the following article : A Theory of Fads , Fashion , Custom , and Cultural Change as Informational Cascades , 2007 .

[33]  T. Başar,et al.  Dynamic Noncooperative Game Theory , 1982 .

[34]  G. Mailath,et al.  Repeated Games and Reputations: Long-Run Relationships , 2006 .

[35]  T. Başar,et al.  Dynamic Noncooperative Game Theory, 2nd Edition , 1998 .

[36]  Daron Acemoglu,et al.  A Theory of Political Transitions , 1999 .

[37]  Jeff S. Shamma,et al.  LP formulation of asymmetric zero-sum stochastic games , 2014, 53rd IEEE Conference on Decision and Control.

[38]  Ashutosh Nayyar,et al.  Decentralized Stochastic Control with Partial History Sharing: A Common Information Approach , 2012, IEEE Transactions on Automatic Control.