Here is a concrete example. Two referees independently read the same paper, giving it a mark in the range [0,1], where 0 is a definite reject and 1 is a definite accept. They then meet to decide on a joint verdict.

Alice rates the paper at 0.9. Bob rates the paper at 0.1. Assuming their perfect rationality:

1. How should they proceed to reach an Aumann-style agreement?
2. How accurate is the resulting common estimate likely to be?

Assume that they both have the same prior over the merits of the papers they receive to review: their true worths are uniformly distributed over [0,1]. They have read the same paper and have honestly attempted to judge it according to the same criteria. They may have other, differing information available to them, but the Aumann agreement process does not involve sharing such information.

I've been trying to analyse this in terms of Aumann's original paper and Scott Aaronson's more detailed treatment but I am not getting very far. In Aaronson's framework, if we require a 90% chance of Alice and Bob agreeing to within 0.2, then this can be achieved (Theorem 5) with at most 1/(0.1*0.2^2) = 250 messages in which one referee tells the other their current estimate of the paper. It is as yet unclear to me what calculations Alice and Bob must perform to update their estimates, or what values they might converge on.

In practice, disagreements like this are resolved by sharing not posteriors, but evidence. In this example, Bob might know something that Alice does not, viz. that the authors already published almost the same work in another venue a year ago and that the present paper contains almost nothing new. Or, on the other hand, Bob might simply have missed the point of the paper due to lacking some background knowledge that Alice has.