The ideal would be two people, comfortable with bayesian reasoning, who have a demonstrated track record of updating their positions in response to new information from 3rd parties.

I think Anders Sandberg clearly would be one good choice. I’ll have to think more about the second. I would nominate TGGP as a smart person with a demonstrated capacity to update his beliefs as a result of new information from third parties, but I’m unaware of his facility with the bayesian aspect.

Let’s suppose we throw a die such that only I can see the outcome. We are going to estimate the value that comes up. I know the value, but for you any of the 6 alternatives is equally likely, so your weighted estimate is 3.5. Now, suppose I tell you, “My estimate of the die value is less than what I think yours will be after you hear this.” (I think that is an accurate paraphrase of the paper’s “P=1″.) What are you going to do? What is your new estimate?

You might think, my estimate was 3.5, he knows what it is and says that is lower than what mine will be, so it was probably a 1, 2, or 3, in which case my estimate would be 2, but he says it is lower than what I will think, so it would have had to have been a 1, but then my estimate is 1, and his can’t be lower. You reach a contradiction. There is no estimate you can pick that makes my statement true, because whatever number you pick, I’m saying that the actual die value is lower than that, so you would have to go lower.

Because of this contradiction, then, I guess the conclusion is that I shouldn’t have said that, even if the die really was a 1. I’m tempted to say that because of its self-contradictory nature, my statement gave you no information, so you would leave your estimate at 3.5, in which case my statement was actually true! But then we would have to take another turn on the merry go round and it would become contradictory again.

The paper seems to suggest that I can say, “My estimate of the die value is less or equal to than what I think yours will be after you hear this.” (That would be “N=0″ in the paper.) But I don’t see how I can always say this. What is your estimate going to be? What if I rolled a 6? It doesn’t seem like it works much better.

In the end I’m not sure this result tells us much about the nature of disagreement, rather it seems to be more of a logical paradox similar to the Unexpected Hanging.

Caledonian, it is usually extremely impractical to share all your data.

J, the first link in the post above is to a previous post discussing that very research.

Hal, I’m not sure what you have in mind about a complex path to convergence. I prefer to focus on this result, saying we can’t foresee to disagree.

Then we need to consider, what would be the most informative (or entertaining!?) format for the argument. Tradition would call for them to explain their reasons to each other, periodically indicate when they have changed their minds somewhat, either leading in the end to convergence or “agreeing to disagree”. The Aumann theorem implies that agreement should be reached even with the much more restrictive communication channel modeled here by Robin, where the two simply take turns reciting their estimates regarding the disputed issue. Robin has a paper showing that the path to convergence will often be rather more complicated in this scenario than one might have expected. It would be exciting to come up with a case study which demonstrates this effect.

(In case the link doesn’t work, look for “When Rational Disagreement is Impossible” by Keith Lehrer, in Nous (1976)).

Given that, they should agree every time.

1) I’ve only ever met one Bayesian yet, and we didn’t end up disagreeing on anything…
2) The assumption that all of the Xi are independently and normally distributed with zero mean and a known variance Vi, seems unlikely. When attempting to gather evidence, one takes all possible observations accessible to one, and they often, I would think, have internal correlations.
3) Every time I’ve ever had a serious discussion where I’ve tried to get to the root of a disagreement with someone logical (even if not Bayesian), we’ve traced it to a set of different priors. In practice, I’ve found, it can be extremely difficult to figure out the set of priors we agree on, and which we differ on, and how they interrelate (because real people don’t have independent priors either).