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.”
Fagin, Ronald; Geanakoplos, John; Halpern, Joseph Y.; and Vardi, Moshe Y., "The Hierarchical Approach to Modeling Knowledge and Common Knowledge" (1999). Cowles Foundation Discussion Papers. 1461.