Abstract
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.
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References
Armbruster, W., and W. Boge, ‘Bayesian Game Theory’, in O. Moeschlin and D. Pallaschke (eds.), Game Theory and Related Topics, North-Holland, Amsterdam, 1979.
Aumann, R., ‘Interactive Epistemology I: Knowledge’, International Journal of Game Theory, 28 (1999), 263–300.
Battigalli, P., and M. Siniscalchi, ‘Hierarchies of Conditional Beliefs and Interactive Epistemology in Dynamic Games’, Journal of Economic Theory, 88 (1999), 188–230.
Battigalli, P., and M. Siniscalchi, ‘Strong Belief and Forward-Induction Reasoning’, Journal of Economic Theory, 106 (2002), 356–391.
Boge, W., and T. Eisele, ‘On Solutions of BayesianGames’, International Journal of Game Theory, 8 (1979), 193–215.
Bonanno, G., ‘A Simple Modal Logic for Belief Revision’, Synthese (Knowledge, Rationality and Action) 147 (2005), 193–228.
Boolos, G., The Logic of Provability, Cambridge University Press, 1993.
Brandenburger, A., ‘On the Existence of a ‘Complete’Possibility Structure’, in M. Basili, N. Dimitri, and I. Gilboa (eds.), Cognitive Processes and Economic Behavior, Routledge, 2003, pp. 30–34.
Brandenburger, A., and E. Dekel, ‘Hierarchies of Beliefs and Common Knowledge’, Journal of Economic Theory, 59 (1993), 189–198.
Brandenburger, A., A. Friedenberg, and H.J. Keisler, ‘Admissibility in Games’, 2006, at http://www.stern.nyu.edu/~abranden.
Epstein, L., and T. Wang, ‘Beliefs about Beliefs Without Probabilities’, Econometrica, 64 (1996), 1343–1373.
Fagin, R., ‘A Quantitative Analysis of Modal Logic’, Journal of Symbolic Logic, 59 (1994), 209–252.
Fagin, R., J. Geanakoplos, J. Halpern, and M. Vardi, ‘The Hierarchical Approach to Modeling Knowledge and Common Knowledge’, International Journal of Game Theory, 28 (1999), 331–365.
Fagin, R., J. Halpern, and M. Vardi, ‘A Model-Theoretic Analysis of Knowledge’, Journal of the Association of Computing Machinery, 38 (1991), 382–428.
Heifetz, A., ‘The Bayesian Formulation of Incomplete Information—The Non-Compact Case’, International Journal of Game Theory, 21 (1993), 329–338.
Heifetz, A., ‘How Canonical is the Canonical Model? A Comment on Aumann's Interactive Epistemology’, International Journal of Game Theory, 28 (1999), 435–442.
Heifetz, A., and D. Samet, ‘Topology-free Typology of Beliefs’, Journal of Economic Theory, 82 (1998a), 324–381.
Heifetz, A., and D. Samet, ‘Knowledge Spaces with Arbitrarily High Rank’, Games and Economic Behavior, 22 (1998b), 260–273.
Heifetz, A., and D. Samet, ‘Coherent Beliefs are Not Always Types’, Journal of Mathematical Economics, 32 (1999), 475–488.
Humberstone, I.L., ‘The Modal Logic of All and Only’, Notre Dame Journalof Formal Logic, 28 (1987), 177–188.
Huynh, H.L., and B. Szentes, ‘Believing the Unbelievable: The Dilemma of Self-Belief’, 1999, at http://home.uchicago.edu/~szentes.
Kechris, A., Classical Descriptive Set Theory, Springer-Verlag, 1995.
Lomuscio, A., Knowledge Sharing Among Ideal Agents, Ph.D. Thesis, University of Birmingham, 1999.
Mariotti, T., M. Meier, and M. Piccione, ‘Hierarchies of Beliefs for Compact Possibility Models’, Journal of Mathematical Economics, 41 (2005), 303–324.
Meier, M., ‘On the Nonexistence of Universal Information Structures’, Journal of Economic Theory, 122 (2005), 132–139.
Meier, M., ‘Finitely Additive Beliefs and Universal Type Spaces’, The Annals of Probability, 34 (2006), 386–422.
Mertens, J-F., and S. Zamir, ‘Formulation of Bayesian Analysis for Games with Incomplete Information’, International Journal of Game Theory, 14 (1985), 1–29.
Pacuit, E., ‘Understanding the Brandenburger-Keisler Paradox’, to appear, Studia Logica, 2006.
Pearce, D., ‘Rational Strategic Behavior and the Problem of Perfection’, Econometrica, 52 (1984), 1029–1050.
Pintér, M., ‘Type Space on a Purely Measurable Parameter Space’, Economic Theory, 26 (2005), 129–139.
Salonen, H., ‘Beliefs, Filters, and Measurability’, 1999, University of Turku.
Thomason, R., ‘A Note on Syntactical Treatments of Modality’, Synthese, 44 (1980), 391–395.
Yanofsky, N., ‘A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points’, The Bulletin of Symbolic Logic, 9 (2003), 362–386.
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Special Issue Ways of Worlds II. On Possible Worlds and Related Notions Edited by Vincent F. Hendricks and Stig Andur Pedersen
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Brandenburger, A., Keisler, H.J. An Impossibility Theorem on Beliefs in Games. Stud Logica 84, 211–240 (2006). https://doi.org/10.1007/s11225-006-9011-z
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DOI: https://doi.org/10.1007/s11225-006-9011-z