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

  • 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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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
Date of revision:
Handle: RePEc:wpa:wuwpga:9404002
Note: Plain TeX, 11 pages.
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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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