The Hierarchical Approach to Modeling Knowledge and Common Knowledge

Posted: 26 Oct 1999

See all articles by Ronald Fagin

Ronald Fagin

IBM Research - Almaden Research Center

John Geanakoplos

Yale University; Santa Fe Institute

Joseph Y. Halpern

Cornell University - Department of Computer Science

Moshe Y. Vardi

Rice University - Department of Computer Science

Abstract

One approach to representing knowledge or belief of agents, used by economists and computer scientists, involves an infinite hierarchy of beliefs. Such a hierarchy consists of an agent's beliefs about the state of the world, his beliefs about other agents' beliefs about the world, his beliefs about other agents' beliefs about other agents' beliefs about the world, and so on. (Economists have typically modeled belief in terms of a probability distribution on the uncertainty space. In contrast, computer scientists have modeled belief in terms of a set of worlds, intuitively, the ones the agent considers possible.) We consider the question of when a countably infinite hierarchy completely describes the uncertainty of the agents. We provide various necessary and sufficient conditions for this property. It turns out that the probability-based approach can be viewed as satisfying one of these conditions, which explains why a countable hierarchy suffices in this case. These conditions also show that whether a countable hierarchy suffices may depend on the "richness" of the states in the underlying state space. We also consider the question of whether a countable hierarchy suffices for "interesting" sets of events, and show that the answer depends on the definition of "interesting".

JEL Classification: C70

Suggested Citation

Fagin, Ronald and Geanakoplos, John D and Halpern, Joseph Yehuda and Vardi, Moshe Y., The Hierarchical Approach to Modeling Knowledge and Common Knowledge. Available at SSRN: https://ssrn.com/abstract=176569

Ronald Fagin (Contact Author)

IBM Research - Almaden Research Center ( email )

San Jose, CA 95120
United States

John D Geanakoplos

Yale University ( email )

30 Hillhouse Avenue
New Haven, CT 06511
United States
203-432-3397 (Phone)

HOME PAGE: http://https://economics.yale.edu/people/faculty/john-geanakoplos

Santa Fe Institute ( email )

1399 Hyde Park Road
Santa Fe, NM 87501
United States

Joseph Yehuda Halpern

Cornell University - Department of Computer Science ( email )

4130 Upson Hall
Ithaca, NY 14853
United States
607-255-9562 (Phone)
607-255-4428 (Fax)

Moshe Y. Vardi

Rice University - Department of Computer Science ( email )

Houston, TX 77005-1892
United States

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