IDEAS home Printed from https://ideas.repec.org/a/eee/jetheo/v70y1996i1p249-256.html
   My bibliography  Save this article

"Knowing Whether," "Knowing That," and The Cardinality of State Spaces

Author

Listed:
  • Hart, Sergiu
  • Heifetz, Aviad
  • Samet, Dov

Abstract

We introduce a new operator on information structures which we call `knowing whether' as opposed to the standard knowledge operator which may be called `knowing that'. The difference between these operators is simple. Saying that an agent knows t h a t a certain event occurred implies that this event indeed occurred, while saying that the agent knows w h e t h e r an event occurred does not imply that the event occurred. (Formally, knowing whether X means that either it is known that X occurred or it is known that X did not occur.) We show that iterating `knowing whether' operators of different agents has a remarkable property that iterations of `knowing that' do not have. When we generate a sequence of events, starting with a given event and then applying `knowing that' or `not knowing that' to the previous event, then the events in this sequence may be, somewhat surprisingly, contradictory. In contrast, any sequence of this type, generated with `knowing whether' and `not knowing whether' is never contradictory. We use this property of the `knowing whether' operator to construct a simple and natural state space and information structures for two agents, such that: (1) any two states are distinct relative to some interactive knowledge of a fixed event, (2) the space has the cardinality of the continuum. This result --- originally proved in a complicated manner by Aumann (1989) --- demonstrates the usefulness of the `knowing whether'
(This abstract was borrowed from another version of this item.)

Suggested Citation

  • Hart, Sergiu & Heifetz, Aviad & Samet, Dov, 1996. ""Knowing Whether," "Knowing That," and The Cardinality of State Spaces," Journal of Economic Theory, Elsevier, vol. 70(1), pages 249-256, July.
    • Handle: RePEc:eee:jetheo:v:70:y:1996:i:1:p:249-256
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0022-0531(96)90084-8
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    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. Robert J. Aumann, 1999. "Interactive epistemology I: Knowledge," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(3), pages 263-300.
    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. Simon Grant & John Quiggin, 2005. "Learning and Discovery," Risk & Uncertainty Working Papers WP7R05, Risk and Sustainable Management Group, University of Queensland.
    2. Moscati Ivan, 2009. "Interactive and common knowledge in the state-space model," CESMEP Working Papers 200903, University of Turin.
    3. Heifetz, Aviad & Samet, Dov, 1998. "Knowledge Spaces with Arbitrarily High Rank," Games and Economic Behavior, Elsevier, vol. 22(2), pages 260-273, February.
    4. Heifetz, Aviad & Meier, Martin & Schipper, Burkhard C., 2006. "Interactive unawareness," Journal of Economic Theory, Elsevier, vol. 130(1), pages 78-94, September.
    5. Koessler, Frederic, 2003. "Persuasion games with higher-order uncertainty," Journal of Economic Theory, Elsevier, vol. 110(2), pages 393-399, June.
    6. Konrad Grabiszewski, 2015. "Epistemic Self-Analysis and Epistemic Bounded Rationality," Economics Bulletin, AccessEcon, vol. 35(3), pages 1941-1948.
    7. , & ,, 2011. "Agreeing to agree," Theoretical Economics, Econometric Society, vol. 6(2), May.
    8. Satoshi Fukuda, 2018. "Representing Unawareness on State Spaces," Working Papers 635, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    9. Fukuda, Satoshi, 2021. "Unawareness without AU Introspection," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    10. Feinberg, Yossi, 2000. "Characterizing Common Priors in the Form of Posteriors," Journal of Economic Theory, Elsevier, vol. 91(2), pages 127-179, April.
    11. Gossner, Olivier & Tsakas, Elias, 2007. "Testing Rationality on Primitive Knowledge," Working Papers in Economics 275, University of Gothenburg, Department of Economics.

    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. John Geanakoplos, 1993. "Common Knowledge," Cowles Foundation Discussion Papers 1062, Cowles Foundation for Research in Economics, Yale University.
    2. Heifetz, Aviad & Samet, Dov, 1998. "Topology-Free Typology of Beliefs," Journal of Economic Theory, Elsevier, vol. 82(2), pages 324-341, October.
    3. Katarina Kostelić, 2021. "Game Awareness: A Questionnaire," Games, MDPI, vol. 12(4), pages 1-39, December.
    4. Giacomo Bonanno & Cédric Dégremont, 2013. "Logic and Game Theory," Working Papers 11, University of California, Davis, Department of Economics.
    5. , & ,, 2011. "Agreeing to agree," Theoretical Economics, Econometric Society, vol. 6(2), May.
    6. Dov Samet, 2008. "S5 Knowledge Without Partitions," Levine's Bibliography 122247000000001875, UCLA Department of Economics.
    7. 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.
    8. Giacomo Bonanno & Elias Tsakas, 2017. "Qualitative analysis of common belief of rationality in strategic-form games," Working Papers 181, University of California, Davis, Department of Economics.
    9. Cyril Hédoin, 2016. "Community-Based Reasoning in Games: Salience, Rule-Following, and Counterfactuals," Games, MDPI, vol. 7(4), pages 1-17, November.
    10. Billot, Antoine & Vergopoulos, Vassili, 2018. "Expected utility without parsimony," Mathematical Social Sciences, Elsevier, vol. 93(C), pages 14-21.
    11. 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.
    12. Guilhem Lecouteux, 2018. "Bayesian game theorists and non-Bayesian players," The European Journal of the History of Economic Thought, Taylor & Francis Journals, vol. 25(6), pages 1420-1454, November.
    13. Halpern, Joseph Y., 2001. "Alternative Semantics for Unawareness," Games and Economic Behavior, Elsevier, vol. 37(2), pages 321-339, November.
    14. Giovanna Devetag & Hykel Hosni & Giacomo Sillari, 2012. "You Better Play 7: Mutual versus Common Knowledge of Advice in a Weak-link Experiment," LEM Papers Series 2012/01, Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy.
    15. Stephen Morris & Hyun Song Shin, "undated". "Approximate Common Knowledge and Co-ordination: Recent Lessons from Game Theory," CARESS Working Papres 97-8, University of Pennsylvania Center for Analytic Research and Economics in the Social Sciences.
    16. Giacomoni Gilbert, 2024. "The Need for Products Interchangeability: An Unsolved Problem of Semantic Conflicts No Product Definition System Can Support Perfectly," Journal of the Knowledge Economy, Springer;Portland International Center for Management of Engineering and Technology (PICMET), vol. 15(2), pages 7672-7701, June.
    17. Abigail Devereaux & Roger Koppl & Stuart Kauffman, 2024. "Creative evolution in economics," Journal of Evolutionary Economics, Springer, vol. 34(2), pages 489-514, April.
    18. Spyros Galanis, 2011. "Syntactic foundations for unawareness of theorems," Theory and Decision, Springer, vol. 71(4), pages 593-614, October.
    19. Christian Bach & Jérémie Cabessa, 2012. "Common knowledge and limit knowledge," Theory and Decision, Springer, vol. 73(3), pages 423-440, September.
    20. Holler Manfred J., 2002. "Classical, Modern, and New Game Theory / Klassische, Moderne und Neue Spieltheorie," Journal of Economics and Statistics (Jahrbuecher fuer Nationaloekonomie und Statistik), De Gruyter, vol. 222(5), pages 556-583, October.

    More about this item

    JEL classification:

    • C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
    • D8 - Microeconomics - - Information, Knowledge, and Uncertainty

    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:jetheo:v:70:y:1996:i:1:p:249-256. 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/inca/622869 .

    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.