IDEAS home Printed from https://ideas.repec.org/a/eee/mateco/v84y2019icp73-82.html
   My bibliography  Save this article

Epistemic foundations for set-algebraic representations of knowledge

Author

Listed:
  • Fukuda, Satoshi

Abstract

This paper formalizes an informal idea that an agent’s knowledge is characterized by a collection of sets such as a σ-algebra within the framework of a state space model. The paper fully characterizes why the agent’s knowledge takes (or does not take) such a set algebra as a σ-algebra or a topology, depending on logical and introspective properties of knowledge and on the underlying structure of the state space. The agent’s knowledge is summarized by a collection of events if and only if she can only know what is true, she knows any logical implication of what she knows, and she is introspective about what she knows. In this case, for any event, the collection that represents knowledge has the maximal event included in the original event. When the underlying space is a measurable space, the collection becomes a σ-algebra if and only if the agent is additionally introspective about what she does not know.

Suggested Citation

  • Fukuda, Satoshi, 2019. "Epistemic foundations for set-algebraic representations of knowledge," Journal of Mathematical Economics, Elsevier, vol. 84(C), pages 73-82.
    • Handle: RePEc:eee:mateco:v:84:y:2019:i:c:p:73-82
    DOI: 10.1016/j.jmateco.2019.07.001
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304406819300746
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmateco.2019.07.001?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to look for a different version below or search for a different version of it.

    Other versions of this item:

    References listed on IDEAS

    as
    1. Brandenburger, Adam & Dekel, Eddie & Geanakoplos, John, 1992. "Correlated equilibrium with generalized information structures," Games and Economic Behavior, Elsevier, vol. 4(2), pages 182-201, April.
    2. João Correia-da-Silva & Carlos Hervés-Beloso, 2007. "Private Information: Similarity as Compatibility," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 30(3), pages 395-407, March.
    3. Robert J. Aumann, 1999. "Interactive epistemology II: Probability," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(3), pages 301-314.
    4. Samet, Dov, 1990. "Ignoring ignorance and agreeing to disagree," Journal of Economic Theory, Elsevier, vol. 52(1), pages 190-207, October.
    5. Kreps,David M. & Wallis,Kenneth F. (ed.), 1997. "Advances in Economics and Econometrics: Theory and Applications," Cambridge Books, Cambridge University Press, number 9780521589819, November.
    6. Nielsen, Lars Tyge, 1984. "Common knowledge, communication, and convergence of beliefs," Mathematical Social Sciences, Elsevier, vol. 8(1), pages 1-14, August.
    7. Cotter, Kevin D., 1986. "Similarity of information and behavior with a pointwise convergence topology," Journal of Mathematical Economics, Elsevier, vol. 15(1), pages 25-38, February.
    8. Milgrom, Paul, 1981. "An Axiomatic Characterization of Common Knowledge," Econometrica, Econometric Society, vol. 49(1), pages 219-222, January.
    9. Yannelis, Nicholas C, 1991. "The Core of an Economy with Differential Information," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 1(2), pages 183-197, April.
    10. 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.
    11. Meier, Martin, 2008. "Universal knowledge-belief structures," Games and Economic Behavior, Elsevier, vol. 62(1), pages 53-66, January.
    12. Shin Hyun Song, 1993. "Logical Structure of Common Knowledge," Journal of Economic Theory, Elsevier, vol. 60(1), pages 1-13, June.
    13. Morris, Stephen, 1996. "The Logic of Belief and Belief Change: A Decision Theoretic Approach," Journal of Economic Theory, Elsevier, vol. 69(1), pages 1-23, April.
    14. Jong Jae Lee, 2018. "Formalization of information: knowledge and belief," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 66(4), pages 1007-1022, December.
    15. Modica, Salvatore & Rustichini, Aldo, 1999. "Unawareness and Partitional Information Structures," Games and Economic Behavior, Elsevier, vol. 27(2), pages 265-298, May.
    16. Bacharach, Michael, 1985. "Some extensions of a claim of Aumann in an axiomatic model of knowledge," Journal of Economic Theory, Elsevier, vol. 37(1), pages 167-190, October.
    17. Kreps,David M. & Wallis,Kenneth F. (ed.), 1997. "Advances in Economics and Econometrics: Theory and Applications," Cambridge Books, Cambridge University Press, number 9780521589833, November.
    18. Robert Wilson, 2005. "Information, efficiency, and the core of an economy," Studies in Economic Theory, in: Dionysius Glycopantis & Nicholas C. Yannelis (ed.), Differential Information Economies, pages 55-64, Springer.
    19. Dubra, Juan & Echenique, Federico, 2004. "Information is not about measurability," Mathematical Social Sciences, Elsevier, vol. 47(2), pages 177-185, March.
    20. Noguchi, Mitsunori, 2018. "Alpha cores of games with nonatomic asymmetric information," Journal of Mathematical Economics, Elsevier, vol. 75(C), pages 1-12.
    21. Kajii, Atsushi & Morris, Stephen, 1997. "Commonp-Belief: The General Case," Games and Economic Behavior, Elsevier, vol. 18(1), pages 73-82, January.
    22. Lipman Barton L., 1994. "A Note on the Implications of Common Knowledge of Rationality," Games and Economic Behavior, Elsevier, vol. 6(1), pages 114-129, January.
    23. Satoshi Fukuda, 2018. "Representing Unawareness on State Spaces," Working Papers 635, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    24. Brandenburger, Adam & Dekel, Eddie, 1987. "Common knowledge with probability 1," Journal of Mathematical Economics, Elsevier, vol. 16(3), pages 237-245, June.
    25. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June.
    26. Eddie Dekel & Barton L. Lipman & Aldo Rustichini, 1998. "Standard State-Space Models Preclude Unawareness," Econometrica, Econometric Society, vol. 66(1), pages 159-174, January.
    27. Dov Monderer & Dov Samet, 1996. "Proximity of Information in Games with Incomplete Information," Mathematics of Operations Research, INFORMS, vol. 21(3), pages 707-725, August.
    28. Stinchcombe, Maxwell B., 1990. "Bayesian information topologies," Journal of Mathematical Economics, Elsevier, vol. 19(3), pages 233-253.
    29. Kreps,David M. & Wallis,Kenneth F. (ed.), 1997. "Advances in Economics and Econometrics: Theory and Applications," Cambridge Books, Cambridge University Press, number 9780521589826, November.
    30. Allen, Beth, 1983. "Neighboring information and distributions of agents' characteristics under uncertainty," Journal of Mathematical Economics, Elsevier, vol. 12(1), pages 63-101, September.
    31. John Geanakoplos, 1989. "Game Theory Without Partitions, and Applications to Speculation and Consensus," Cowles Foundation Discussion Papers 914, Cowles Foundation for Research in Economics, Yale University.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Fukuda, Satoshi, 2024. "The existence of universal qualitative belief spaces," Journal of Economic Theory, Elsevier, vol. 216(C).
    2. Satoshi Fukuda, 2024. "On the consistency among prior, posteriors, and information sets," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 78(2), pages 521-565, September.
    3. Fukuda, Satoshi, 2021. "Unawareness without AU Introspection," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    4. Áron Tóbiás, 2021. "A unified epistemological theory of information processing," Theory and Decision, Springer, vol. 90(1), pages 63-83, February.
    5. Fukuda, Satoshi, 2020. "Formalizing common belief with no underlying assumption on individual beliefs," Games and Economic Behavior, Elsevier, vol. 121(C), pages 169-189.
    6. Satoshi Fukuda, 2018. "Representing Unawareness on State Spaces," Working Papers 635, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    7. Áron Tóbiás, 2023. "Cognitive limits and preferences for information," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 46(1), pages 221-253, June.
    8. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Áron Tóbiás, 2023. "Cognitive limits and preferences for information," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 46(1), pages 221-253, June.
    2. Fukuda, Satoshi, 2021. "Unawareness without AU Introspection," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    3. Satoshi Fukuda, 2024. "On the consistency among prior, posteriors, and information sets," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 78(2), pages 521-565, September.
    4. 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.
    5. Fukuda, Satoshi, 2020. "Formalizing common belief with no underlying assumption on individual beliefs," Games and Economic Behavior, Elsevier, vol. 121(C), pages 169-189.
    6. Fukuda, Satoshi, 2024. "The existence of universal qualitative belief spaces," Journal of Economic Theory, Elsevier, vol. 216(C).
    7. Gossner, Olivier & Tsakas, Elias, 2007. "Testing Rationality on Primitive Knowledge," Working Papers in Economics 275, University of Gothenburg, Department of Economics.
    8. Moscati Ivan, 2009. "Interactive and common knowledge in the state-space model," CESMEP Working Papers 200903, University of Turin.
    9. Tsakas, Elias & Voorneveld, Mark, 2007. "Efficient communication, common knowledge, and consensus," Working Papers in Economics 255, University of Gothenburg, Department of Economics.
    10. Satoshi Fukuda, 2018. "Representing Unawareness on State Spaces," Working Papers 635, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    11. John Geanakoplos, 1993. "Common Knowledge," Cowles Foundation Discussion Papers 1062, Cowles Foundation for Research in Economics, Yale University.
    12. Bernard Walliser, 1991. "Logique épistémique et théorie des jeux," Revue Économique, Programme National Persée, vol. 42(5), pages 801-832.
    13. Áron Tóbiás, 2021. "A unified epistemological theory of information processing," Theory and Decision, Springer, vol. 90(1), pages 63-83, February.
    14. Meier, Martin, 2005. "On the nonexistence of universal information structures," Journal of Economic Theory, Elsevier, vol. 122(1), pages 132-139, May.
    15. Bonanno, G. & Nehring, K., 1995. "Intersubjective Consistency of Beliefs and the Logic of Common Belief," Department of Economics 95-08, California Davis - Department of Economics.
    16. Noguchi, Mitsunori, 2018. "Alpha cores of games with nonatomic asymmetric information," Journal of Mathematical Economics, Elsevier, vol. 75(C), pages 1-12.
    17. Khan, M. Ali & Sun, Yeneng & Tourky, Rabee & Zhang, Zhixiang, 2008. "Similarity of differential information with subjective prior beliefs," Journal of Mathematical Economics, Elsevier, vol. 44(9-10), pages 1024-1039, September.
    18. Khrennikov, Andrei, 2015. "Quantum version of Aumann’s approach to common knowledge: Sufficient conditions of impossibility to agree on disagree," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 89-104.
    19. Ezra Einy & Ori Haimanko & Diego Moreno & Benyamin Shitovitz, 2005. "On the continuity of equilibrium and core correspondences in economies with differential information," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 26(4), pages 793-812, November.
    20. Konrad Grabiszewski, 2015. "Epistemic Self-Analysis and Epistemic Bounded Rationality," Economics Bulletin, AccessEcon, vol. 35(3), pages 1941-1948.

    More about this item

    Keywords

    Knowledge; Information; Set algebra; σ-algebra; Introspection;
    All these keywords.

    JEL classification:

    • C70 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - General
    • D83 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Search; Learning; Information and Knowledge; Communication; Belief; Unawareness

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:mateco:v:84:y:2019:i:c:p:73-82. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jmateco .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.