This year’s Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel has been awarded to two game theorists: Robert Aumann of Hebrew University, and Thomas Schelling of the University of Maryland.
In 1976, Aumann wrote a famous paper called “Agreeing to Disagree,” which proved the following fact. Suppose you and your friend are perfectly rational Bayesians, who’d both form the same opinions if given the same information. Then provided your opinions are “common knowledge” (meaning you both know them, you both know you both know them, etc.), those opinions must be equal — even if neither of you knows the evidence on which the other’s opinion is based! Loosely speaking, then, you can never “agree to disagree.”
As an example, suppose Alice offers to sell you some stock. Then the mere fact that she’s trying to sell it gives you useful information — namely, that something must’ve convinced her the stock is headed south. So even if you have no idea what that something is, her offer should cause you to decrease your own valuation of the stock. Similarly, if you agree to buy the stock, that should cause Alice to increase her valuation. As observed by Milgrom and Stokey, the end result of all this second-guessing is that you might as well never trade at all! This is assuming three conditions: (i) that you and Alice would believe the same things if given the same information, (ii) that you’re both trying to maximize expected wealth, and (iii) that you both have the same liquidity needs (i.e. neither desperately needs to pay off a mortgage). If you’re still confused, read this delightful survey by Cowen and Hanson.
A year ago I proved a complexity-theoretic analogue of Aumann’s theorem: that not only will two Bayesians agree “in the limit” of common knowledge, but they’ll also (probably, approximately) agree after a really short conversation. I sent my paper to Aumann just for fun, not expecting any response from the great man. To my surprise, Aumann promptly wrote back with a thoughtful critique — telling me to cut out my philosophical musings and let the math speak for itself. I hated this advice at the time, but eventually came to the grudging realization that it was right.
Interestingly, besides being a world expert on rationality, Aumann is also an Orthodox Jew, who’s written several papers applying game theory to the Talmud. He was born in Germany in 1930 and escaped to the US in 1938.
Congratulations to Aumann and Schelling!