The excellent Scott Aaronson has posted on his blog a version of a talk he recently gave at SPARC, about Aumann’s agreement theorem and related topics. I think a substantial fraction of LW readers would enjoy it. As well as stating Aumann’s theorem and explaining why it’s true, the article discusses other instances where the idea of “common knowledge” (the assumption that does a lot of the work in the AAT) is important, and offers some interesting thoughts on the practical applicability (if any) of the AAT.

(Possibly relevant: an earlier LW discussion of AAT.)

It includes a great test of whether a given discussion is Aumann-rational:

as opposed to

Maybe I’m confused, in the ‘muddy children puzzle’ it seems it would be common knowledge from the start that at least 98 children have muddy foreheads. Each child sees 99 muddy foreheads. Each child could reason that every other child must see at least 98 muddy foreheads. 100 minus their own forehead which they cannot see minus the other child’s forehead which the other child cannot see equals 98.

What am I missing?

Common knowledge means I know, and I know that you know, and I know that you know that he knows, and she knows that I know that you know that he knows, and so on—any number of iterations.

Each child sees 99 muddy foreheads and therefore knows n >= 99. Each child can tell that each other child knows n >= 98. But, e.g., it isn’t true that A knows B knows C knows that n >= 98; only that A knows B knows C knows that n>=

97: each link in the chain reduces the number by 1. So for no k>0 is itcommon knowledgethat n>=k.Thanks, I did end up figuring out my error.