Aumann's agreement theorem

Aumann's agreement theorem states that two Bayesian agents with the same prior beliefs cannot "agree to disagree" about the probability of an event if their individual beliefs are common knowledge. In other words, if it is commonly known what each agent believes about some event, and both agents are rational and update their beliefs using Bayes' rule, then their updated (posterior) beliefs must be the same.

Informally, the theorem implies that rational individuals who start from the same assumptions and share all relevant information—even just by knowing each other’s opinions—must eventually come to the same conclusions. If their differing beliefs about something are common knowledge, they must in fact agree.

The theorem was proved by Robert Aumann in his 1976 paper "Agreeing to Disagree", which also introduced the formal, set-theoretic definition of common knowledge.