## “We Can’t Disagree Forever,” J. Geanakoplos & H. Polemarchakis (1982)

Aumann famously proved, as previously discussed on this site, that under common priors, common knowledge, and knowledge of each other’s information partitions, agents must agree on posteriors even if they witness different events. The common knowledge restriction turns out to be pretty strong – it essentially says that, if Pi(A) is the elements of agent i’s partition containing A, then each agent must share probabilities of every element of their partition contained in Pi(A) conditional on A. In general, this is not the case. Consider information partitions p1={1,2},{3,4} and p2={1,2,3},{4} with both agents having prior .25 on each number. Consider the event {3,4}. By Bayes’ formula, 1 has posterior 1; if {3,4} happens, he knows it. 2 has posterior .5. The agents clearly disagree, and given the information partitions, they will not realize they disagree.

It turns out communication can simplify this proof and make it more general. Assume agents can state their posterior, that there is no common knowledge assumption, and that each agent knows the other’s information partition. Then let each agent update their beliefs and state the posterior probability again. It turns out that a number of steps equal to the sum of elements of each agent’s information partition is sufficient to get agents to converge on their posterior belief. Incredibly, it can happen that for n steps, with n arbitrary, we each state exactly the same posteriors (say I state .5 and you state .4) but then in the nth stage we each state a probability we agree on!