The Bayesian Solution and Hierarchies of Beliefs

Tang, Qianfeng (2010): The Bayesian Solution and Hierarchies of Beliefs.

Abstract

The Bayesian solution is a notion of correlated equilibrium proposed by Forges (1993), and hierarchies of beliefs over conditional beliefs are introduced by Ely and Pęski (2006) in their study of interim rationalizability. We study the connection between the two concepts. We say that two type spaces are equivalent if they represent the same set of hierarchies of beliefs over conditional beliefs. We show that the correlation embedded in equivalent type spaces can be characterized by partially correlating devices, which send correlated signals to players in a belief invariant way. Since such correlating devices also implement the Bayesian solution, we establish that the Bayesian solution is invariant across equivalent type spaces.

• Item Type: MPRA Paper
• Language: English
• Keywords: Games with incomplete information, Correlated equilibrium, The Bayesian solution, Common knowledge, Hierarchies of beliefs.
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C70 - General; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games

References

[1] Aumann, Robert (1998), �Common priors: A reply to Gul.�Econometrica, 66, 929-938.

[2] Brandenburger, Adam and Eddie Dekel (1993), �Hierarchies of beliefs and common knowledge.�Journal of Economic Theory, 59, 189-198.

[3] Dekel, Eddie, Drew Fudenberg and Stephen Morris (2007), �Interim correlated rationalizability.�Theoretical Economics, 2, 15-40.

[4] Ely, Jeff and Marcin Peski (2007), �Hierarchies of belief and interim rationalizability.� Theoretical Economics, 1, 19-65.

[5] Forges, Françoise (1993), �Five legitimate de�nitions of correlated equilibrium in games with incomplete information.�Theory and decision, 35, 277-310.

[6] Forges, Françoise (2006), �Correlated equilibrium in games with incomplete information revisited.�Theory and decision, 61, 329-344.

[7] Harsanyi, John (1967-1968), "Games with incomplete information played by �Bayesian� players, Parts I, II, III." Management Science, 14, 159-182, 320-334, 486-502.

[8] Liu, Qingmin (2009), �On redundant types and Bayesian formulation of incomplete information.�Journal of Economic Theory, 144, 2115-2145.

[9] Liu, Qingmin (2005), �Representation of belief hierarchies in incomplete information games.�working paper.

[10] Mertens, Jean-François and Shmuel Zamir (1985), �Formulation of Bayesian analysis for games with incomplete information.� International Journal of Game Theory, 14, 1-29.

[11] Tang, Qianfeng (2010), "Interim partially correlated rationalizability." working paper.