This is a linkpost for http://www.scottaaronson.com/blog/?p=2410
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 it common knowledge that n>=k.
Thanks, I did end up figuring out my error.
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.)