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

Listed author(s):
  • Hart, Sergiu
  • Heifetz, Aviad
  • Samet, Dov

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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Article provided by Elsevier in its journal Journal of Economic Theory.

Volume (Year): 70 (1996)
Issue (Month): 1 (July)
Pages: 249-256

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Handle: RePEc:eee:jetheo:v:70:y:1996:i:1:p:249-256
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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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