NancyLebovitz comments on Navigating disagreement: How to keep your eye on the evidence - Less Wrong
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Re: Problem 4: Roughly speaking: yes.
Ordinary disagreements persist after hearing others' estimates. A and B may start out asserting "50" and "10", and then argue their way to "25" and "12", then "23" and "17". But if you want each estimate to be as accurate as possible, this is silly behavior; if A can predict that his estimate will go down over time (as he integrates more of B's evidence), he can also predict that his current estimate is too high -- and so he can improve his accuracy by lowering his estimate right now. The two parties should be as likely to overshoot as to undershoot in their disagreements, e.g.: A: 50; B: 10 A: 18; B: 22 A: 21; B: 21.
So next time you're in a dispute, try applying Principle 3: ask what an outside observer would say about the situation. If Alfred and Betty both apply this principle, they'll each ask: "What would an outside observer guess about Lake L, given that Betty has studied geography and said "10", while Alfred said "50"?" And, thus viewing the situation from the (same) outside, Betty and Alfred will both weigh Betty's evidence about equally. Alfred may underweight Betty's impression (e.g., because he doesn't realize she wrote her thesis on Lake L) -- but he may equally overweight Betty's opinion (e.g., because he doesn't realize that she's never heard of Lake L either). If he could predict that he was (over/under) weighting her opinion, he'd quit doing it.
More precisely: if you and your interlocutor can predict your direction of disagreement, at least one of you is forming needlessly inaccurate estimates.
Before I read your reply, I assume that Alfred will lower his estimate a lot, and Betty might raise her estimate a little. I expect Betty's estimate to still be lower than Alfred's, though the size of these effects would be dependent on how much more geography Betty knows than Alfred.
After reading your reply, I think you're right about convergence, and definitely right about driving your answer towards what you think is correct as fast as possible rather than holding back for fear of seeming to give in.