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

Sergiu Hart, Aviad Heifetz and Dov Samet
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Aviad Heifetz: Tel Aviv Uni.

Game Theory and Information from University Library of Munich, Germany

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'

JEL-codes: C7 D8 (search for similar items in EconPapers)
Pages: 11 pages
Date: 1994-04-10
Note: Plain TeX, 11 pages.
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Citations: View citations in EconPapers (5)

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Journal Article: "Knowing Whether," "Knowing That," and The Cardinality of State Spaces (1996)
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Persistent link: https://EconPapers.repec.org/RePEc:wpa:wuwpga:9404002

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