One result that’s related to Aumann’s Agreement Theorem is that if you and I alternate saying our posterior probabilities of some event, we converge on the same probability if we have common priors. You might therefore wonder why we ever do anything else. The answer is that describing evidence is strictly more informative than stating one’s posterior. For instance, imagine that we’ve both secretly flipped coins, and want to know whether both coins landed on the same side. If we just state our posteriors, we’ll immediately converge to 50%, without actually learning the answer, which we could have learned pretty trivially by just saying how our coins landed. This is related to the original proof of the Aumann agreement theorem in a way that I can’t describe shortly.
One result that’s related to Aumann’s Agreement Theorem is that if you and I alternate saying our posterior probabilities of some event, we converge on the same probability if we have common priors. You might therefore wonder why we ever do anything else. The answer is that describing evidence is strictly more informative than stating one’s posterior. For instance, imagine that we’ve both secretly flipped coins, and want to know whether both coins landed on the same side. If we just state our posteriors, we’ll immediately converge to 50%, without actually learning the answer, which we could have learned pretty trivially by just saying how our coins landed. This is related to the original proof of the Aumann agreement theorem in a way that I can’t describe shortly.