How common are common priors?

Abstract

To answer the question in the title we vary agents' beliefs against the background of a fixed knowledge space, that is, a state space with a partition for each agent. Beliefs are the posterior probabilities of agents, which we call type profiles. We then ask what is the topological size of the set of consistent type profiles, those that are derived from a common prior (or a common improper prior in the case of an infinite state space). The answer depends on what we term the tightness of the partition profile. A partition profile is tight if in some state it is common knowledge that any increase of any single agent's knowledge results in an increase in common knowledge. We show that for partition profiles that are tight the set of consistent type profiles is topologically large, while for partition profiles that are not tight this set is topologically small.

From Posteriors to Priors Via Cycles: An Addendum

“…* For helpful discussions and comments, I thank Sophie Bade, Christoph Engel, Yossi Feinberg, Alia Gizatulina, and Christian Hellwig. 1 For a similar result, see Proposition 4 in Hellman and Samet [2].…”

The cycles approach

“…A mathematically equivalent result was established independently by Hellman and Samet (2012) (in their paper, this is in Proposition 2 and again in Theorem 1). They defined a property of knowledge models named ''tightness'', which links individual and common learning.…”

“…In independent work, Hellman and Samet (2012) introduce the concept of 'tightness', a property that is mathematically equivalent to acyclicity. They also reproduce our formula (1) and study some topological aspects that we do not cover, such as the conditions under which profiles of consistent types are dense in the set of all type profiles, or are of first category.…”

Incomplete-Information Models of Large Economies with Anonymity: Existence and Uniqueness of Common Priors

“…Technically, any element in the information partition intersects every element in the information partition of the other player. 6 In the large economy, these conditions, i.e., Properties 1 and 2, do not concern the information that one participant may form about another participants'beliefs, but, rather, a limit on the information about the aggregate state that individuals cna infer from the observation of their types. In Bierbrauer and Hellwig [2], this limitation is characterized by saying that the belief system is "relatively uninformative", i.e., the information that a person has does not allow this person rule out any aggregate state that was considered possible ex ante.…”