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The hierarchical approach to modeling knowledge and common knowledge

Author

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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".

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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    Cited by:

    1. Fukuda, Satoshi, 2024. "The existence of universal qualitative belief spaces," Journal of Economic Theory, Elsevier, vol. 216(C).
    2. Board, Oliver, 2004. "Dynamic interactive epistemology," Games and Economic Behavior, Elsevier, vol. 49(1), pages 49-80, October.
    3. Satoshi Fukuda, 2025. "The Hierarchical construction of a universal qualitative belief space," International Journal of Game Theory, Springer;Game Theory Society, vol. 54(2), pages 1-37, December.
    4. 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.
    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. Moscati Ivan, 2009. "Interactive and common knowledge in the state-space model," CESMEP Working Papers 200903, University of Turin.
    8. Pivato, Marcus, 2008. "The Discursive Dilemma and Probabilistic Judgement Aggregation," MPRA Paper 8412, University Library of Munich, Germany.
    9. Xiao Luo & Yi-Chun Chen, 2004. "A Unified Approach to Information, Knowledge, and Stability," Econometric Society 2004 Far Eastern Meetings 472, Econometric Society.
    10. Xiao Luo & Xuewen Qian & Chen Qu, 2020. "Iterated elimination procedures," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 70(2), pages 437-465, September.
    11. 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.
    12. Brânzei, R. & Tijs, S.H. & Timmer, J.B., 2000. "Collecting Information to improve Decision-Making," Other publications TiSEM 74fa171d-2799-4747-9c2e-7, Tilburg University, School of Economics and Management.
    13. 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.
    14. Áron Tóbiás, 2021. "Meet meets join: the interaction between pooled and common knowledge," International Journal of Game Theory, Springer;Game Theory Society, vol. 50(4), pages 989-1019, December.

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