Aumann's agreement theorem was stated and proved by Robert Aumann in a paper titled "Agreeing to Disagree",[1] which introduced the set theoretic description of common knowledge.
The theorem concerns agents who share a common prior and update their probabilistic beliefs by Bayes' rule.
It states that if the probabilistic beliefs of such agents, regarding a fixed event, are common knowledge then these probabilities must coincide.
Thus, agents cannot agree to disagree, that is have common knowledge of a disagreement over the posterior probability of a given event.
The model used in Aumann[1] to prove the theorem consists of a finite set of states
The posterior probability of agent
is common knowledge is not empty then all the numbers
The proof follows directly from the definition of common knowledge.
The claim of the theorem follows since the left hand side is independent of
Monderer and Samet relaxed the assumption of common knowledge and assumed instead common
[2] They gave an upper bound of the distance between the posteriors
Ziv Hellman relaxed the assumption of a common prior and assumed instead that the agents have priors that are
[3] He showed that common knowledge of the posteriors in this case implies that they are
goes to zero, Aumann's original theorem is recapitulated.
Nielsen extended the theorem to non-discrete models in which knowledge is described by
[4] Knowledge which is defined in terms of partitions has the property of negative introspection.
However, it is possible to show that it is impossible to agree to disagree even when knowledge does not have this property.
[5] Halpern and Kets argued that players can agree to disagree in the presence of ambiguity, even if there is a common prior.
However, allowing for ambiguity is more restrictive than assuming heterogeneous priors.
[6] The impossibility of agreeing to disagree, in Aumann's theorem, is a necessary condition for the existence of a common prior.
A stronger condition can be formulated in terms of bets.
A bet is a set of random variables
The impossibility of agreeing on the profitability of a bet is a stronger condition than the impossibility of agreeing to disagree, and moreover, it is a necessary and sufficient condition for the existence of a common prior.
[7][8] A dialogue between two agents is a dynamic process in which, in each stage, the agents tell each other their posteriors of a given event
Upon gaining this new information, each is updating their posterior of
Aumann suggested that such a process leads the agents to commonly know their posteriors, and hence, by the agreement theorem, the posteriors at the end of the process coincide.
[1] Geanakoplos and Polemarchakis proved it for dialogues in finite state spaces.
[9] Polemarchakis showed that any pair of finite sequences of the same length that end with the same number can be obtained as a dialogue.
[10] In contrast, Di Tillio and co-authors showed that infinite dialogues must satisfy certain restrictions on their variation.
[11] Scott Aaronson studied the complexity and rate of convergence of various types of dialogues with more than two agents.