The hierarchical approach to modeling knowledge and common knowledge |
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Authors: | Ronald Fagin John Geanakoplos Joseph Y Halpern Moshe Y Vardi |
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Institution: | (1) IBM Almaden Research Center, San Jose, CA 95120, USA (e-mail: fagin@almaden.ibm.com), US;(2) Cowles Foundation, Yale University, New Haven, CT 06520, USA (e-mail: john.geanakoplos@yale.edu), US;(3) Computer Science Department, Cornell University, Ithaca, NY 14853, USA (e-mail: halpern@cs.cornell.edu), US;(4) Department of Computer Science, Rice University, Houston, TX 77005-1892, USA (e-mail: vardi@cs.rice.edu), US |
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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”. |
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Keywords: | : Common knowledge belief/knowledge hierarchies |
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