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The hierarchical approach to modeling knowledge and common knowledge

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  • 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)

Abstract

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".

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Bibliographic Info

Article provided by Springer in its journal International Journal of Game Theory.

Volume (Year): 28 (1999)
Issue (Month): 3 ()
Pages: 331-365

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Handle: RePEc:spr:jogath:v:28:y:1999:i:3:p:331-365

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Related research

Keywords: Common knowledge · belief/knowledge hierarchies;

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References

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  1. Brandenburger Adam & Dekel Eddie, 1993. "Hierarchies of Beliefs and Common Knowledge," Journal of Economic Theory, Elsevier, vol. 59(1), pages 189-198, February.
  2. Harsanyi, John C., 1994. "Games with Incomplete Information," Nobel Prize in Economics documents 1994-1, Nobel Prize Committee.
  3. Robert J. Aumann, 1999. "Interactive epistemology I: Knowledge," International Journal of Game Theory, Springer, vol. 28(3), pages 263-300.
  4. Heifetz, Aviad & Samet, Dov, 1998. "Knowledge Spaces with Arbitrarily High Rank," Games and Economic Behavior, Elsevier, vol. 22(2), pages 260-273, February.
  5. Werlang, Sérgio Ribeiro da Costa, 1988. "Common knowledge," Economics Working Papers (Ensaios Economicos da EPGE) 118, FGV/EPGE Escola Brasileira de Economia e Finanças, Getulio Vargas Foundation (Brazil).
  6. Rubinstein, Ariel, 1989. "The Electronic Mail Game: Strategic Behavior under "Almost Common Knowledge."," American Economic Review, American Economic Association, vol. 79(3), pages 385-91, June.
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Citations

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Cited by:
  1. 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.
  2. Oliver Board, 2002. "Dynamic Interactive Epistemology," Economics Series Working Papers 125, University of Oxford, Department of Economics.
  3. 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.
  4. Moscati Ivan, 2009. "Interactive and common knowledge in the state-space model," CESMEP Working Papers 200903, University of Turin.
  5. Baratgin, Jean, 2009. "Updating our beliefs about inconsistency: The Monty-Hall case," Mathematical Social Sciences, Elsevier, vol. 57(1), pages 67-95, January.
  6. Tohme, Fernando, 2005. "Existence and definability of states of the world," Mathematical Social Sciences, Elsevier, vol. 49(1), pages 81-100, January.
  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. Pivato, Marcus, 2008. "The Discursive Dilemma and Probabilistic Judgement Aggregation," MPRA Paper 8412, University Library of Munich, Germany.

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