So “at no point in a conversation can Bayesians have common knowledge that they will disagree,” means “‘Common knowledge’ is a far stronger condition than it sounds,” and nothing more and nothing less?
No, for a couple of reasons.
First, I misunderstood the context of that quote. I thought that it was from Wei Dai’s post (because he was the last-named source that you’d quoted). Under this misapprehension, I took him to be pointing out that common knowledge of anything is a fantastically strong condition, and so, in particular, common knowledge of disagreement is practically impossible. It’s theoretically possible for two Bayesians to have common knowledge of disagreement (though, by the theorem, they must have had different priors). But can’t happen in the real world, such as in Luke’s conversations with Anna.
But I now see that this whole line of thought was based on a silly misunderstanding on my part.
In the context of the LW wiki entry, I think that the quote is just supposed to be a restatement of Aumann’s result. In that context, Bayesian reasoners are assumed to have the the same prior (though this could be made clearer). Then I unpack the quote just as you do:
“At no point can two ideal reasoners both know true fact X, where true fact X is that they will disagree on posteriors, and that each knows that they will disagree on posteriors, etc.”
As you point out, by Aumann’s theorem, they won’t disagree on posteriors, so they will never have common knowledge of disagreement, just as the quote says. Conversely, if they have common knowledge of posteriors, but, per the quote, they can’t have common knowledge of disagreement, then those posteriors must agree, which is Aumann’s theorem. In this sense, the quote is equivalent to Aumann’s result.
Apparently the author doesn’t use the word “knowledge” in such a way that to say “A can’t have knowledge of X” is to imply that X is true. (Nor do I, FWIW.)
No, for a couple of reasons.
First, I misunderstood the context of that quote. I thought that it was from Wei Dai’s post (because he was the last-named source that you’d quoted). Under this misapprehension, I took him to be pointing out that common knowledge of anything is a fantastically strong condition, and so, in particular, common knowledge of disagreement is practically impossible. It’s theoretically possible for two Bayesians to have common knowledge of disagreement (though, by the theorem, they must have had different priors). But can’t happen in the real world, such as in Luke’s conversations with Anna.
But I now see that this whole line of thought was based on a silly misunderstanding on my part.
In the context of the LW wiki entry, I think that the quote is just supposed to be a restatement of Aumann’s result. In that context, Bayesian reasoners are assumed to have the the same prior (though this could be made clearer). Then I unpack the quote just as you do:
As you point out, by Aumann’s theorem, they won’t disagree on posteriors, so they will never have common knowledge of disagreement, just as the quote says. Conversely, if they have common knowledge of posteriors, but, per the quote, they can’t have common knowledge of disagreement, then those posteriors must agree, which is Aumann’s theorem. In this sense, the quote is equivalent to Aumann’s result.
Apparently the author doesn’t use the word “knowledge” in such a way that to say “A can’t have knowledge of X” is to imply that X is true. (Nor do I, FWIW.)