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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. Moscati Ivan, 2009. "Interactive and common knowledge in the state-space model," CESMEP Working Papers 200903, University of Turin.
    3. Feinberg, Yossi, 2000. "Characterizing Common Priors in the Form of Posteriors," Journal of Economic Theory, Elsevier, vol. 91(2), pages 127-179, April.
    4. Heifetz, Aviad & Meier, Martin & Schipper, Burkhard C., 2006. "Interactive unawareness," Journal of Economic Theory, Elsevier, vol. 130(1), pages 78-94, September.
    5. Heifetz, Aviad & Samet, Dov, 1998. "Knowledge Spaces with Arbitrarily High Rank," Games and Economic Behavior, Elsevier, vol. 22(2), pages 260-273, February.
    6. Koessler, Frederic, 2003. "Persuasion games with higher-order uncertainty," Journal of Economic Theory, Elsevier, vol. 110(2), pages 393-399, June.
    7. Gossner, Olivier & Tsakas, Elias, 2007. "Testing Rationality on Primitive Knowledge," Working Papers in Economics 275, University of Gothenburg, Department of Economics.
    8. Lehrer, Ehud & Samet, Dov, 2011. "Agreeing to agree," Theoretical Economics, Econometric Society, vol. 6(2), May.
    9. Aviad Heifetz & Martin Meier & Burkhard C. Schipper, 2003. "Interactive Unawareness and Speculative Trade," Bonn Econ Discussion Papers bgse17_2003, University of Bonn, Germany.

    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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