Aumann's agreement theorem

Aumann's agreement theorem states that Bayesian agents with a common prior cannot have different posterior probabilities for an event when those probabilities are common knowledge.

Common knowledge: A fact is common knowledge when everyone knows it, everyone knows that everyone knows it, and this continues at every level. The theorem’s decisive condition is that the agents’ posterior probabilities are common knowledge.
Common prior assumption: The assumption that Bayesian agents begin with the same probability distribution over possible states. Without a common prior, differing posteriors need not reflect differing information alone.
Bayesian persuasion: The study of how an informed sender can influence a receiver’s beliefs through information disclosure. Disclosure can change posteriors, but common-knowledge posteriors under a common prior cannot remain different.
Harsanyi doctrine: The principle that uncertainty about other agents’ beliefs can be represented through a common prior over states. It helps explain how a shared prior can coexist with private information and initially different beliefs.
Bayesian probability: A framework that updates probabilities by conditioning on observed evidence. Each agent’s posterior is the common prior conditioned on that agent’s information.
Bayesian updating: The revision of probabilities in light of evidence using Bayes’ rule. Agents obtain their posterior probabilities by updating the shared prior on private information.
Information aggregation: The process by which dispersed information is combined into a collective assessment or decision. The theorem constrains persistent disagreement when agents share a prior and know one another’s assessments.
Agreeing to disagree: The persistence of opposing opinions despite discussion or access to shared information. The theorem rules out this situation only when posteriors are common knowledge and the prior is shared.
Conditional expectation: The expectation of a random variable given specified information, represented mathematically by a conditional expectation. Posterior probabilities are conditional expectations of an event’s indicator given each agent’s information.
Knowledge operator: A formal operator describing the states an agent considers possible given their information. Knowledge operators define the nested knowledge conditions behind common knowledge.

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Robert AumannBroader topic: It captures how shared assumptions and public knowledge constrain persistent disagreement.