“Agreeing to Disagree,” R. Aumann (1976)

Recently I was discussing with a fellow student mathematical ideas in social science which are 1) nonobvious, 2) trivial to proof mathematically, and 3) important. I led with the Revelation Principle: consider a group of people with individual states s (say, willingness to pay for a good in an auction). Give a mechanism for aggregating claims about those states (possibly false) and spitting out an allocation (for instance, “you may bid in a first-price sealed-bid auction”). Then any allocation which can be reached in some mechanism can also be reached in a truthful mechanism, a mechanism where each person has no incentive to lie about their state. There is a 2 line mathematical proof of this result, and it led to a number of Nobel prizes!

A friend, however, countered with Aumann’s result in this short paper. Consider agents who have common beliefs about something, see some series of events (perhaps I see A, you see B, 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, imagine there is a biased coin, and we have some common belief about the probability of “heads.” You then witness N coin flips, and then I go into another room and witness M flips, with M and N 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!

These results are great counterpunches to those who criticize the mathematization of economic theory (and certainly there are many more fantastic, non-obvious results that are *nontrivial* to prove!). Do you know of any other great proofs along these lines?