The hierarchical approach to modeling knowledge and common knowledge

One approach to representing knowledge or belief of agents, used by economi sts and computer scientists, involves an infinite hierarchy of beliefs. Suc h 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 o n. (Economists have typically modeled belief in terms of a probability dist ribution on the uncertainty space. In contrast, computer scientists have mo deled belief in terms of a set of worlds, intuitively, the ones the agent c onsiders possible.) We consider the question of when a countably infinite h ierarchy completely describes the uncertainty of the agents. We provide var ious necessary and sufficient conditions for this property. It turns out th at 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 a lso consider the question of whether a countable hierarchy suffices for "in teresting" sets of events, and show that the answer depends on the definiti on of "interesting".