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The hierarchical approach to modeling knowledge and common knowledge


  • John Geanakoplos

    () (Cowles Foundation, Yale University, New Haven, CT 06520, USA)

  • (**), Moshe Y. Vardi

    () (Department of Computer Science, Rice University, Houston, TX 77005-1892, USA)

  • Joseph Y. Halpern

    () (Computer Science Department, Cornell University, Ithaca, NY 14853, USA)

  • Ronald Fagin

    () (IBM Almaden Research Center, San Jose, CA 95120, USA)


One approach to representing knowledge or belief of agents, used by economists and computer scientists, involves an infinite hierarchy of beliefs. Such a hierarchy consists of an agent's beliefs about the state of the world, his beliefs about other agents' beliefs about the world, his beliefs about other agents' beliefs about other agents' beliefs about the world, and so on. (Economists have typically modeled belief in terms of a probability distribution on the uncertainty space. In contrast, computer scientists have modeled belief in terms of a set of worlds, intuitively, the ones the agent considers possible.) We consider the question of when a countably infinite hierarchy completely describes the uncertainty of the agents. We provide various necessary and sufficient conditions for this property. It turns out that the probability-based approach can be viewed as satisfying one of these conditions, which explains why a countable hierarchy suffices in this case. These conditions also show that whether a countable hierarchy suffices may depend on the "richness" of the states in the underlying state space. We also consider the question of whether a countable hierarchy suffices for "interesting" sets of events, and show that the answer depends on the definition of "interesting".

Suggested Citation

  • John Geanakoplos & (**), Moshe Y. Vardi & Joseph Y. Halpern & Ronald Fagin, 1999. "The hierarchical approach to modeling knowledge and common knowledge," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(3), pages 331-365.
  • Handle: RePEc:spr:jogath:v:28:y:1999:i:3:p:331-365

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    References listed on IDEAS

    1. Adam Brandenburger & Eddie Dekel, 2014. "Hierarchies of Beliefs and Common Knowledge," World Scientific Book Chapters,in: The Language of Game Theory Putting Epistemics into the Mathematics of Games, chapter 2, pages 31-41 World Scientific Publishing Co. Pte. Ltd..
    2. Harsanyi, John C, 1995. "Games with Incomplete Information," American Economic Review, American Economic Association, vol. 85(3), pages 291-303, June.
    3. Rubinstein, Ariel, 1989. "The Electronic Mail Game: Strategic Behavior under "Almost Common Knowledge."," American Economic Review, American Economic Association, vol. 79(3), pages 385-391, June.
    4. Robert J. Aumann, 1999. "Interactive epistemology I: Knowledge," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(3), pages 263-300.
    5. Werlang, Sérgio Ribeiro da Costa, 1988. "Common knowledge," FGV/EPGE Economics Working Papers (Ensaios Economicos da EPGE) 118, FGV/EPGE - Escola Brasileira de Economia e Finanças, Getulio Vargas Foundation (Brazil).
    6. Heifetz, Aviad & Samet, Dov, 1998. "Knowledge Spaces with Arbitrarily High Rank," Games and Economic Behavior, Elsevier, vol. 22(2), pages 260-273, February.
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    Cited by:

    1. Yi-Chun Chen & Xiao Luo & Chen Qu, 2016. "Rationalizability in general situations," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 61(1), pages 147-167, January.
    2. Board, Oliver, 2004. "Dynamic interactive epistemology," Games and Economic Behavior, Elsevier, vol. 49(1), pages 49-80, October.
    3. Baratgin, Jean, 2009. "Updating our beliefs about inconsistency: The Monty-Hall case," Mathematical Social Sciences, Elsevier, vol. 57(1), pages 67-95, January.
    4. Tohme, Fernando, 2005. "Existence and definability of states of the world," Mathematical Social Sciences, Elsevier, vol. 49(1), pages 81-100, January.
    5. Moscati Ivan, 2009. "Interactive and common knowledge in the state-space model," CESMEP Working Papers 200903, University of Turin.
    6. Pivato, Marcus, 2008. "The Discursive Dilemma and Probabilistic Judgement Aggregation," MPRA Paper 8412, University Library of Munich, Germany.
    7. Xiao Luo & Yi-Chun Chen, 2004. "A Unified Approach to Information, Knowledge, and Stability," Econometric Society 2004 Far Eastern Meetings 472, Econometric Society.
    8. Brânzei, R. & Tijs, S.H. & Timmer, J.B., 2000. "Collecting Information to improve Decision-Making," Discussion Paper 2000-26, Tilburg University, Center for Economic Research.
    9. Mariotti, Thomas & Meier, Martin & Piccione, Michele, 2005. "Hierarchies of beliefs for compact possibility models," Journal of Mathematical Economics, Elsevier, vol. 41(3), pages 303-324, April.


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