Aumann's agreement theorem
Aumann's agreement theorem was stated and proved by Robert Aumann in a paper titled "Agreeing to Disagree", 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, that is have common knowledge of a disagreement over the posterior probability of a given event.
The model used in Aumann to prove the theorem consists of a finite set of states with a prior probability , which is common to all agents. Agent 's knowledge is given by a partition of . The posterior probability of agent , denoted is the conditional probability of given . Fix an event and let be the event that for each , . The theorem claims that if the event that is common knowledge is not empty then all the numbers are the same. The proof follows directly from the definition of common knowledge. The event is a union of elements of for each . Thus, for each , . The claim of the theorem follows since the left hand side is independent of . The theorem was proved for two agents but the proof for any number of agents is similar.
Monderer and Samet relaxed the assumption of common knowledge and assumed instead common -belief of the posteriors of the agents. They gave an upper bound of the distance between the posteriors . This bound approaches 0 when approaches 1.
Ziv Hellman relaxed the assumption of a common prior and assumed instead that the agents have priors that are -close in a well defined metric. He showed that common knowledge of the posteriors in this case implies that they are -close. When goes to zero, Aumann's original theorem is recapitulated.
Knowledge which is defined in terms of partitions has the property of negative introspection. That is, agents know that they do not know what they do not know. However, it is possible to show that it is impossible to agree to disagree even when knowledge does not have this property. , 
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.
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 , one for each agent , such the . The bet is favorable to agent in a state if the expected value of at is positive. 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. , 
A question arises whether such an agreement can be reached in a reasonable time and, from a mathematical perspective, whether this can be done efficiently. Scott Aaronson has shown that this is indeed the case. Of course, the assumption of common priors is a rather strong one and may not hold in practice. However, Robin Hanson has presented an argument that Bayesians who agree about the processes that gave rise to their priors (e.g., genetic and environmental influences) should, if they adhere to a certain pre-rationality condition, have common priors.
The non-probabilistic case
- Aumann, Robert J. (1976). "Agreeing to Disagree" (PDF). The Annals of Statistics. 4 (6): 1236–1239. doi:10.1214/aos/1176343654. ISSN 0090-5364. JSTOR 2958591.
- Monderer, dov; Dov Samet (1989). "Approximating common knowledge with common beliefs". Games and Economic Behavior. 1 (2): 170–190.
- Hellman, Ziv (2013). "Almost Common Priors". International Journal of Game Theory. 42 (2): 399–410. doi:10.1007/s00182-012-0347-5.
- Bacharach, Michael (1985). "Some extensions of a claim of Aumann in an axiomatic model of knowledge". Journal of Economic Theory. 37 (1): 167–190. doi:10.1016/0022-0531(85)90035-3.
- Samet, Dov (1990). "Ignoring ignorance and agreeing to disagree". Journal of Economic Theory. 52 (1): 190–207. doi:10.1016/0022-0531(90)90074-T.
- Halpern, Joseph; Willemien Kets (2013-10-28). "Ambiguous Language and Consensus" (PDF). Retrieved 2014-01-13.
- Feinberg, Yossi (2000). "Characterizing Common Priors in the Form of Posteriors". Journal of Economic Theory. 91: 127–179. doi:10.1006/jeth.1999.2592.
- Samet, Dov (1998). "Common Priors and Separation of Convex Sets". Games and Economic Behavior. 91: 172–174. doi:10.1006/game.1997.0615.
- Aaronson, Scott (2005). The complexity of agreement (PDF). Proceedings of ACM STOC. pp. 634–643. doi:10.1145/1060590.1060686. ISBN 978-1-58113-960-0. Retrieved 2010-08-09.
- Hanson, Robin (2006). "Uncommon Priors Require Origin Disputes". Theory and Decision. 61 (4): 319–328. CiteSeerX 10.1.1.63.4669. doi:10.1007/s11238-006-9004-4.