Agreeing to agree

Abstract

Aumann (1976) shows that agents who have a common prior cannot have common knowledge of their posteriors for event E if these posteriors do not coincide. But given an event E, can the agents have posteriors with a common prior such that it is common knowledge that the posteriors for E do coincide? We show that a necessary and sufficient condition for this is the existence of a nonempty finite event F with the following two properties. First, it is common knowledge at F that the agents cannot tell whether E occurred. Second, this still holds true at F, when F itself becomes common knowledge.

Agreeing to agree and Dutch books

“…Due to ignorance we can choose the types such that t π i (E) = 1/2 for i = 1, 2. Thus agreeing to agree is possible for E. For details, see Lehrer and Samet (2011).…”

“…For instance, if we take the uniform distribution on the four states as the common prior, then both firms have the same posterior for E namely, 1/2. However, the characterization of Lehrer and Samet (2011) does not hold when the number of agents exceeds two. Specifically, ignorance is no longer sufficient for agreeing to agree with more than two agents, as demonstrated by the following example.…”

“…For such state spaces with two agents, Lehrer and Samet (2011) showed that agreeing to agree is possible if and only if there exists a nonempty finite event F at which it is common knowledge that the agents cannot tell whether or not E occurred and this still holds true after F becomes common knowledge. This characterization also implies that whenever agreeing to agree is possible for two agents, it is possible with a prior that has a finite support.…”

“…We say in this case that the agent is ignorant of E. Lehrer and Samet (2011) observed that for two agents in a finite state space, ignorance of all agents in all states is also a sufficient condition for agreeing to agree. Consider the following example from Lehrer and Samet (2011). The eight states in Example 2 are depicted as black dots.…”

“…That is, under what conditions there are posteriors of the agents that are derived from a common prior, such that the agents have common knowledge (agreement) that their posteriors of E are the same (agree). Lehrer and Samet (2011) obtained a characterization of the possibility of agreeing to agree for two agents. In this paper, we study the possibility of agreeing to agree for an arbitrary finite set of agents.…”