'Knowing Whether', 'Knowing That' and the Cardinality of State Spaces
AbstractWe 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'
Download InfoIf you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
Bibliographic InfoPaper provided by EconWPA in its series Game Theory and Information with number 9404002.
Length: 11 pages
Date of creation: 10 Apr 1994
Date of revision:
Note: Plain TeX, 11 pages.
Contact details of provider:
Web page: http://22.214.171.124
Other versions of this item:
- Hart, Sergiu & Heifetz, Aviad & Samet, Dov, 1996. ""Knowing Whether," "Knowing That," and The Cardinality of State Spaces," Journal of Economic Theory, Elsevier, vol. 70(1), pages 249-256, July.
- C7 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory
- D8 - Microeconomics - - Information, Knowledge, and Uncertainty
Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Robert J. Aumann, 1999. "Interactive epistemology I: Knowledge," International Journal of Game Theory, Springer, vol. 28(3), pages 263-300.
- HEIFETZ, Aviad & MEIER, Martin & SCHIPPER, Burkhard C., 2004.
CORE Discussion Papers, UniversitÃ© catholique de Louvain, Center for Operations Research and Econometrics (CORE)
2004059, UniversitÃ© catholique de Louvain, Center for Operations Research and Econometrics (CORE).
- Heifetz, Aviad & Meier, Martin & Schipper, Burkhard C., 2005. "Interactive Unawareness," Discussion Paper Series of SFB/TR 15 Governance and the Efficiency of Economic Systems 52, Free University of Berlin, Humboldt University of Berlin, University of Bonn, University of Mannheim, University of Munich.
- Frédéric KOESSLER, 2001.
"Persuasion Games with Higher Order Uncertainty,"
Working Papers of BETA
2001-14, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.
- Lehrer, Ehud & Samet, Dov, 2011.
"Agreeing to agree,"
Econometric Society, vol. 6(2), May.
- Moscati Ivan, 2009. "Interactive and common knowledge in the state-space model," CESMEP Working Papers 200903, University of Turin.
- Feinberg, Yossi, 2000. "Characterizing Common Priors in the Form of Posteriors," Journal of Economic Theory, Elsevier, vol. 91(2), pages 127-179, April.
- Simon Grant & John Quiggin, 2005.
"Learning and Discovery,"
Risk & Uncertainty Working Papers
WP7R05, Risk and Sustainable Management Group, University of Queensland.
- Grant, Simon & Quiggin, John, 2006. "Learning and Discovery," Risk and Sustainable Management Group Working Papers, University of Queensland, School of Economics 151174, University of Queensland, School of Economics.
- Aviad Heifetz & Martin Meier & Burkhard C. Schipper, 2003. "Interactive Unawareness and Speculative Trade," Bonn Econ Discussion Papers, University of Bonn, Germany bgse17_2003, University of Bonn, Germany.
- Gossner, Olivier & Tsakas, Elias, 2007. "Testing Rationality on Primitive Knowledge," Working Papers in Economics 275, University of Gothenburg, Department of Economics.
For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (EconWPA).
If references are entirely missing, you can add them using this form.