We Can't Disagree Forever
Under the assumption of common priors, if the information partitions of two agents are ﬁnite, then simply by communicating back and forth and revising their posteriors the two agents will converge to a common equilibrium posterior, even though they may base their posteriors on quite diﬀerent information. Furthermore, given any integer, n, one can construct an example in which the revision process not only takes n steps to converge, but no evident revision occurs — for (n – 1) steps both agents repeat the same conflicting posteriors — until the last step when the two agents decide to agree. Common knowledge of each other’s posterior does not necessarily lead agents to the posterior they would have agreed upon had information been directly exchanged. On the other hand, the examples that are characterized by a discrepancy between the direct and indirect communication equilibrium are rare: with probability 1, the revision process constructed here leads the two agents in one step to the direct communication equilibrium.
Geanakoplos, John and Polemarchakis, Heracles M., "We Can't Disagree Forever" (1982). Cowles Foundation Discussion Papers. 874.