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

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  • 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'
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  • 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
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    References listed on IDEAS

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    1. Robert J. Aumann, 1999. "Interactive epistemology I: Knowledge," International Journal of Game Theory, Springer;Game Theory Society, vol. 28(3), pages 263-300.
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    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. Heifetz, Aviad & Samet, Dov, 1998. "Knowledge Spaces with Arbitrarily High Rank," Games and Economic Behavior, Elsevier, vol. 22(2), pages 260-273, February.
    3. Konrad Grabiszewski, 2015. "Epistemic Self-Analysis and Epistemic Bounded Rationality," Economics Bulletin, AccessEcon, vol. 35(3), pages 1941-1948.
    4. , & ,, 2011. "Agreeing to agree," Theoretical Economics, Econometric Society, vol. 6(2), May.
    5. Fukuda, Satoshi, 2021. "Unawareness without AU Introspection," Journal of Mathematical Economics, Elsevier, vol. 94(C).
    6. Moscati Ivan, 2009. "Interactive and common knowledge in the state-space model," CESMEP Working Papers 200903, University of Turin.
    7. Heifetz, Aviad & Meier, Martin & Schipper, Burkhard C., 2006. "Interactive unawareness," Journal of Economic Theory, Elsevier, vol. 130(1), pages 78-94, September.
    8. Koessler, Frederic, 2003. "Persuasion games with higher-order uncertainty," Journal of Economic Theory, Elsevier, vol. 110(2), pages 393-399, June.
    9. Satoshi Fukuda, 2018. "Representing Unawareness on State Spaces," Working Papers 635, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University.
    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.

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    More about this item

    JEL classification:

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

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