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'Knowing Whether', 'Knowing That' and the Cardinality of State Spaces


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  • Sergiu Hart

    (The Hebrew Uni.)

  • Aviad Heifetz

    (Tel Aviv Uni.)

  • Dov Samet

    (kTel Aviv Uni.)


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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Bibliographic Info

Paper provided by EconWPA in its series Game Theory and Information with number 9404002.

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Length: 11 pages
Date of creation: 10 Apr 1994
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Handle: RePEc:wpa:wuwpga:9404002

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  1. Robert J. Aumann, 1999. "Interactive epistemology I: Knowledge," International Journal of Game Theory, Springer, 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. Aviad Heifetz & Martin Meier & Burkhard C. Schipper, 2003. "Interactive Unawareness and Speculative Trade," Bonn Econ Discussion Papers bgse17_2003, University of Bonn, Germany.
  3. Heifetz, Aviad & Meier, Martin & Schipper, Burkhard C., 2006. "Interactive unawareness," Journal of Economic Theory, Elsevier, vol. 130(1), pages 78-94, September.
  4. Ehud Lehrer & Dov Samet, 2003. "Agreeing to agree," Game Theory and Information 0310005, EconWPA.
  5. Gossner, Olivier & Tsakas, Elias, 2007. "Testing Rationality on Primitive Knowledge," Working Papers in Economics 275, University of Gothenburg, Department of Economics.
  6. Moscati Ivan, 2009. "Interactive and common knowledge in the state-space model," CESMEP Working Papers 200903, University of Turin.
  7. Frédéric KOESSLER, 2001. "Persuasion Games with Higher Order Uncertainty," Working Papers of BETA 2001-14, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
  8. Feinberg, Yossi, 2000. "Characterizing Common Priors in the Form of Posteriors," Journal of Economic Theory, Elsevier, vol. 91(2), pages 127-179, April.


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