Essence of Aumanns Agreeing to Disagree

Consider agents who have common beliefs about something, say some series of events (perhaps I see , you see , etc.) for which beliefs can be Bayesian updated, and then have some posterior beliefs about that something.

If the posterior beliefs are common knowledge, meaning I know yours, you know I know yours, I know you know I know yours, etc., then the posterior beliefs must be equivalent.

For instance, if there is a biased coin, and we have some common belief about the probability of “heads.” You then witness coin flips, and then I go into another room and witness flips, with and not known to the other party.

We then state our posterior belief about the probability of “heads.”

If the beliefs are different, I will update my belief based on knowing your belief, and you will do the same, onward and onward, until our posterior beliefs are precisely the same.

The proof of this completely non-obvious result turns out to be trivial once “common knowledge” is properly written with mathematics!