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 to disagree, that is have common knowledge of a disagreement over the posterior probability of a given event.

The Theorem

The model used in Aumann to prove the theorem consists of a finite set of states $S$ with a prior probability $p$, which is common to all agents. Agent $a$'s knowledge is given by a partition $\Pi_a$ of $S$. The posterior probability of agent $a$, denoted $p_a$ is the conditional probability of $p$ given $\Pi_a$.
Fix an event $E$ and let $X$ be the event that for each $a$, $p_a(E)=x_a$. The theorem claims that if the event $C(X)$ that $X$ is common knowledge is not empty then all the numbers $x_a$ are the same. The proof follows directly from the definition of common knowledge. The event $C(X)$ is a union of elements of $\Pi_a$ for each $a$. Thus, for each $a$, $p(E , C(x))=x_a$. The claim of the theorem follows since the left hand side is independent of $a$. The theorem was proved for two agents but the proof for any number of agents is similar.

Extensions

Monderer and Samet
relaxed the assumption of common knowledge and assumed instead common $p$-belief of the posteriors of the agents. They gave an upper bound of the distance between the posteriors $x_a$. This bound approaches 0 when $p$ approaches 1.

Ziv Hellman relaxed the assumption of a common prior and assumed instead that the agents have priors that are $\varepsilon$-close in a well defined metric. He showed that common knowledge of the posteriors in this case implies that they are $\varepsilon$-close. When $\varepsilon$ goes to zero, Aumann's original theorem is recapitulated.

Nielsen extended the theorem to non-discrete models in which knowledge is described by $\sigma$-algebras rather than partitions.

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 $f_a$, one for each agent $a$, such the $\sum_a f_a=0$. The bet is ''favorable'' to agent $a$ in a state $s$ if the expected value of $f_a$ at $s$ 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.
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Dynamics

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 $E$. Upon gaining this new information, each is updating her posterior of $E$.
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. Geanakoplos and Polemarchakis
proved it for dialogues in finite state spaces. Polemarchakis showed that any pair of finite sequences of the same length that end with the same number can be obtained as a dialogue. In contrast, Di Tillio ''et al'' showed that infinite dialogues must satisfy certain restrictions on their variation.
Scott Aaronson studied the complexity and rate of convergence of various types of dialogues with more than two agents.

References

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Bayesian statistics
Economics theorems
Game theory
Probability theorems
Rational choice theory
Theorems in statistics