Why should ideal Bayesian rationalists alter their estimates to something that is not more likely to be true according to the available evidence? The theorem states that they reach agreement because it is the most likely way to be correct.
The parties do update according to their available evidence. However, neither has access to all the evidence. Also, evidence can be misleading—and subsets of the evidence are more likely to mislead.
Parties can become less accurate after updating, I think.
For another example, say A privately sees 5 heads, and A’s identical twin, B privately sees 7 tails—and then they Auman agree on the issue of whether the coin is fair. A will come out with more confidence in thinking that the coin is biased. If the coin is actually fair, A will have become more wrong.
If A and B had shared all their evidence—instead of going through an Auman agreement exchange—A would have realised that the coin was probably fair—thereby becoming less wrong.
Sometimes following the best available answer will lead you to an answer that is incorrect, but from your own perspective it is always the way to maximize your chance of being right.
I don’t think it is true that both parties necessarily wind up with more accurate estimates after updating and agreeing, or even an estimate closer to what they would have obtained by sharing all their data.
The scenario in the grandparent provides an example of an individual’s estimate becoming worse after Aumann agreeing—and also an example of their estimate getting further away from what they would have believed if both parties had shared all their evidence.
I am unable to see where we have any disagreement. If you think we disagree, perhaps this will help you to pinpoint where.
Perhaps I was reading an implication into your comment that you didn’t intend, but I took it that you were saying that Aumann’s Agreement Theorem leads to agreement between the parties, but not necessarily as a result of their each attempting to revise their estimates to what is most likely given the data they have..
Imagine I think there are 200 balls in the urn, but Robin Hanson thinks there are 300 balls in the urn. Once Robin tells me his estimate, and I tell him mine, we should converge upon a common opinion. In essence his opinion serves as a “sufficient statistic” for all of his evidence.
My comments were intended to suggest that the results of going through an Aumann agreement exchange could quite be different from what you would get if the parties shared all their relevant evidence.
The main similarity is that the parties end up agreeing with each other in both cases.
Why should ideal Bayesian rationalists alter their estimates to something that is not more likely to be true according to the available evidence? The theorem states that they reach agreement because it is the most likely way to be correct.
The parties do update according to their available evidence. However, neither has access to all the evidence. Also, evidence can be misleading—and subsets of the evidence are more likely to mislead.
Parties can become less accurate after updating, I think.
For example, consider A in this example.
For another example, say A privately sees 5 heads, and A’s identical twin, B privately sees 7 tails—and then they Auman agree on the issue of whether the coin is fair. A will come out with more confidence in thinking that the coin is biased. If the coin is actually fair, A will have become more wrong.
If A and B had shared all their evidence—instead of going through an Auman agreement exchange—A would have realised that the coin was probably fair—thereby becoming less wrong.
Sometimes following the best available answer will lead you to an answer that is incorrect, but from your own perspective it is always the way to maximize your chance of being right.
To recap, what I originally said here was:
The scenario in the grandparent provides an example of an individual’s estimate becoming worse after Aumann agreeing—and also an example of their estimate getting further away from what they would have believed if both parties had shared all their evidence.
I am unable to see where we have any disagreement. If you think we disagree, perhaps this will help you to pinpoint where.
Perhaps I was reading an implication into your comment that you didn’t intend, but I took it that you were saying that Aumann’s Agreement Theorem leads to agreement between the parties, but not necessarily as a result of their each attempting to revise their estimates to what is most likely given the data they have..
That wasn’t intended. Earlier, I cited this.
http://www.marginalrevolution.com/marginalrevolution/2005/10/robert_aumann_n.html
My comments were intended to suggest that the results of going through an Aumann agreement exchange could quite be different from what you would get if the parties shared all their relevant evidence.
The main similarity is that the parties end up agreeing with each other in both cases.