The expressive power of the hierarchical approach to modeling knowledge and common knowledge

• Authors: Ronald Fagin;John Geanakoplos;Joseph Y. Halpern;Moshe Y. Vardi
• Venue: TARK '92 Proceedings of the 4th conference on Theoretical aspects of reasoning about knowledge
• Year: 1992

One approach to representing knowledge or belief of agents, which has been explored independently by economists (Böge and Eisele; Mertens and Zamir; Brandenburger and Dekel; Tan and Werlang) and by computer scientists (Fagin, Halpern, and Vardi) 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 worlds, his beliefs about other agents' beliefs about other agents' beliefs about the worlds, etc. Economists and computer scientists differ, however, in the way they model beliefs. Economists prefer a probability-based framework, where belief is modeled as a probability distribution on the uncertainty space. In contrast, computer scientists prefer an information-based framework, where belief is modeled as a subset of the underlying space. The idea is that whatever is in the subset is believed to be possible, and whatever is not in the subset is believed to be impossible. We consider the question of when such an 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 the infinite hierarchy always completely describes the uncertainty of the agents in the probability-based approach. An interesting consequence of our conditions is that adequacy of an infinite hierarchy may depend on the "richness" of the states in the underlying state space. We also consider the question of whether an infinite hierarchy completely describes the uncertainty of the agents with respect to "interesting" sets of events and show that the answers depends on the definition of "interesting".