Aumann's agreement theorem
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-(logic)|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.
The theorem The model of Aumann They gave an upper bound of the distance between the posteriors <math>x_a</math>. This bound approaches 0 when <math>p</math> approaches 1.
Ziv Hellman relaxed the assumption of a common prior and assumed instead that the agents have priors that are <math>\varepsilon</math>-close in a well defined metric. He showed that common knowledge of the posteriors in this case implies that they are <math>\varepsilon</math>-close. When <math>\varepsilon</math> goes to zero, Aumann's original theorem is recapitulated.
Nielsen extended the theorem to non-discrete models in which knowledge is described by <math>\sigma</math>-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 <math>f_a</math>, one for each agent <math>a</math>, such that <math>\sum_a f_a=0</math> (the idea being that no money is created or destroyed, only transferred, in these bets). The bet is favorable to agent <math>a</math> in a state <math>s</math> if the expected value of <math>f_a</math> at <math>s</math> 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.