Studia Logica

, Volume 84, Issue 2, pp 211–240

An Impossibility Theorem on Beliefs in Games


DOI: 10.1007/s11225-006-9011-z

Cite this article as:
Brandenburger, A. & Keisler, H.J. Stud Logica (2006) 84: 211. doi:10.1007/s11225-006-9011-z


A paradox of self-reference in beliefs in games is identified, which yields a game-theoretic impossibility theorem akin to Russell’s Paradox. An informal version of the paradox is that the following configuration of beliefs is impossible:

Ann believes that Bob assumes that

Ann believes that Bob’s assumption is wrong

This is formalized to show that any belief model of a certain kind must have a ‘hole.’ An interpretation of the result is that if the analyst’s tools are available to the players in a game, then there are statements that the players can think about but cannot assume. Connections are made to some questions in the foundations of game theory.


belief model complete belief model game first order logic modal logic paradox 


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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Stern School of BusinessNew York UniversityNew YorkUSA
  2. 2.Department of MathematicsUniversity of Wisconsin-MadisonMadisonUSA

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