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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Find related papers by JEL classification: C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory D8 - Microeconomics - - Information, Knowledge, and Uncertainty
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HEIFETZ, Aviad & MEIER, Martin & SCHIPPER, Burkhard C., 2004.
"Interactive unawareness,"
CORE Discussion Papers
2004059, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
[Downloadable!]
Other versions:
Aviad Heifetz & Martin Meier & Burkhard C. Schipper, 2005.
"Interactive Unawareness,"
Discussion Papers
52, SFB/TR 15 Governance and the Efficiency of Economic Systems, Free University of Berlin, Humboldt University of Berlin, University of Bonn, University of Mannheim, University of Munich.
[Downloadable!]