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Interactive epistemology II: Probability

Author

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  • Robert J. Aumann

    () (Center for Rationality and Interactive Decision Theory, The Hebrew University, Feldman Building, Givat Ram, 91904 Jerusalem, Israel)

Abstract

Formal Interactive Epistemology deals with the logic of knowledge and belief when there is more than one agent or "player." One is interested not only in each person's knowledge and beliefs about substantive matters, but also in his knowledge and beliefs about the others' knowledge and beliefs. This paper examines two parallel approaches to the subject. The first is the semantic, in which knowledge and beliefs are represented by a space of states of the world, and for each player i, partitions ℐi of and probability distributions i(·; ) on for each state of the world. The atom of ℐi containing a given state represents i's knowledge at that state - the set of those other states that i cannot distinguish from ; the probability distributions i(·; ) represents i's beliefs at the state . The second is the syntactic approach, in which beliefs are embodied in sentences constructed according to certain syntactic rules. This paper examines the relation between the two approaches, and shows that they are in a sense equivalent. In game theory and economics, the semantic approach has heretofore been most prevalent. A question that often arises in this connection is whether, in what sense, and why the space , the partitions ℐi, and the probability distributions i(·; ) can be taken as given and commonly known by the players. An answer to this question is provided by the syntactic approach.

Suggested Citation

  • Robert J. Aumann, 1999. "Interactive epistemology II: Probability," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(3), pages 301-314.
  • Handle: RePEc:spr:jogath:v:28:y:1999:i:3:p:301-314
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    Citations

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    Cited by:

    1. Cubitt, Robin P. & Sugden, Robert, 2014. "Common Reasoning In Games: A Lewisian Analysis Of Common Knowledge Of Rationality," Economics and Philosophy, Cambridge University Press, vol. 30(03), pages 285-329, November.
    2. Robin P. Cubitt & Robert Sugden, 2008. "Common reasoning in games," Discussion Papers 2008-01, The Centre for Decision Research and Experimental Economics, School of Economics, University of Nottingham.
    3. repec:eee:gamebe:v:104:y:2017:i:c:p:146-164 is not listed on IDEAS
    4. Heifetz, Aviad & Mongin, Philippe, 2001. "Probability Logic for Type Spaces," Games and Economic Behavior, Elsevier, vol. 35(1-2), pages 31-53, April.
    5. Dmitry Levando & Maxim Sakharov, 2018. "Natural Instability of Equilibrium Prices," Working Papers 2018:01, Department of Economics, University of Venice "Ca' Foscari".
    6. Samet, Dov, 2010. "Agreeing to disagree: The non-probabilistic case," Games and Economic Behavior, Elsevier, vol. 69(1), pages 169-174, May.
    7. Samet, Dov, 2000. "Quantified Beliefs and Believed Quantities," Journal of Economic Theory, Elsevier, vol. 95(2), pages 169-185, December.
    8. Meier, Martin, 2005. "On the nonexistence of universal information structures," Journal of Economic Theory, Elsevier, vol. 122(1), pages 132-139, May.
    9. Robin Cubitt & Robert Sugden, 2005. "Common reasoning in games: a resolution of the paradoxes of ‘common knowledge of rationality’," Discussion Papers 2005-17, The Centre for Decision Research and Experimental Economics, School of Economics, University of Nottingham.
    10. Asheim, Geir B. & Sovik, Ylva, 2005. "Preference-based belief operators," Mathematical Social Sciences, Elsevier, vol. 50(1), pages 61-82, July.
    11. Christian Bach & Jérémie Cabessa, 2012. "Common knowledge and limit knowledge," Theory and Decision, Springer, vol. 73(3), pages 423-440, September.
    12. Giovanna Devetag & Hykel Hosni & Giacomo Sillari, 2012. "You Better Play 7: Mutual versus Common Knowledge of Advice in a Weak-link Experiment," CEEL Working Papers 1201, Cognitive and Experimental Economics Laboratory, Department of Economics, University of Trento, Italia.
    13. Pintér, Miklós, 2010. "The non-existence of a universal topological type space," Journal of Mathematical Economics, Elsevier, vol. 46(2), pages 223-229, March.
    14. Meier, Martin, 2008. "Universal knowledge-belief structures," Games and Economic Behavior, Elsevier, vol. 62(1), pages 53-66, January.
    15. Tsakas, Elias, 2014. "Rational belief hierarchies," Journal of Mathematical Economics, Elsevier, vol. 51(C), pages 121-127.
    16. Yildiz, Muhamet, 2015. "Invariance to representation of information," Games and Economic Behavior, Elsevier, vol. 94(C), pages 142-156.
    17. Slikker, M. & Norde, H.W. & Tijs, S.H., 2000. "Information Sharing Games," Discussion Paper 2000-100, Tilburg University, Center for Economic Research.
    18. Feinberg, Yossi, 2000. "Characterizing Common Priors in the Form of Posteriors," Journal of Economic Theory, Elsevier, vol. 91(2), pages 127-179, April.
    19. Pier Luigi Porta & Gianni Viaggi, 2002. "Employment, Technology and Institutions in the Process of Structural Change. A History of Economic Thought Perspective," Working Papers 51, University of Milano-Bicocca, Department of Economics, revised Jul 2002.
    20. repec:hal:journl:halshs-00344461 is not listed on IDEAS
    21. Shmuel Zamir, 2008. "Bayesian games: Games with incomplete information," Discussion Paper Series dp486, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    22. MEIER, Martin, 2001. "An infinitary probability logic for type spaces," CORE Discussion Papers 2001061, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    23. Feinberg, Yossi, 2005. "Subjective reasoning--dynamic games," Games and Economic Behavior, Elsevier, vol. 52(1), pages 54-93, July.
    24. Schmidt, Christian, 2003. "Que reste-t-il du Treatise on Probability de Keynes?," L'Actualité Economique, Société Canadienne de Science Economique, vol. 79(1), pages 37-55, Mars-Juin.
    25. Dekel, Eddie & Siniscalchi, Marciano, 2015. "Epistemic Game Theory," Handbook of Game Theory with Economic Applications, Elsevier.

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