I will have more to say about the Stony Brook conference, but first a word about David Blackwell, who passed away last week. We game theorists know Blackwell for several seminal contributions. Blackwell’s approachability theorem is at the heart of Aumann and Maschler’s result about repeated games with incomplete information which Eilon mentions below, and also of the calibration results which I mentioned in my presentation in Stony Brook (Alas, I was too nervous and forgot to mention Blackwell as I intended too). Blackwell’s theory of comparison of experiments has been influential in the game-theoretic study of value of information, and Olivier presented a two-person game analogue for Blackwell’s theorem in his talk. Another seminal contribution of Blackwell, together with Lester Dubins, is the theorem about merging of opinions, which is the major tool in the Ehuds’ theory of Bayesian learning in repeated games. And then there are his contributions to the theory of infinite games with Borel payoffs (now known as Blackwell games) and Blackwell and Fergurson’s solution to the Big Match game.
One conspicuous aspect of many of Blackwell’s awesome papers is that they are extremely short — often a couple of pages long. He had an amazing ability to prove theorems in the right way, and he wrote with eloquence and clarity. He is the only writer I know who uses the `as the reader can verify’ trick productively, exactly at those occasions when the reader will indeed find it easier to convince himself in the validity of an assertion than to read a formal proof of it. It is very rare that I succeed in reading proofs in papers that were written dozens of years ago: Notations and perspectives change, and important results are usually reproduced in clearer way over the years. But Blackwell’s papers are still the best place to read the proofs of his theorems.
Blackwell’s game theoretic papers have also been influential outside game theory. Let me mention here the two-page paper `infinite games and analytic sets’, which is probably my favorite math paper, in which Blackwell gives a game theoretic proof for the co-reduction principle for analytic sets, that for every pair of analytic subsets of a standard Borel space there exist analytic supersets such that and . Such direct applications of game theory in math are rare gems, but the general approach of applying strategic/game theoretic perspective is now prevalent in statistics and probability.
I hope I am not forcing my own agenda on Blackwell’s research when I say that for him game and decision theory were a tool to study conceptual questions about the meaning of probability and information. At any rate, he was clearly interested in these questions. Here is a nice passage from an interview with Morris Degroot in which Blackwell recounts an incident that took place before his meeting with Savage and his `conversion’ to Bayesianism. An economist who was consulting the Ministry of Defense in their budget allocation asked Blackwell about the probability of a major war in the next five years:
[The economist:] “I am not going to ask you to tell me a number, but if you could give me any guide as to how I could go about finding such a number I would be grateful.” Oh, I said to him, that question just doesn’t make sense. Probability applies to a long sequence of repeatable events, and this is clearly a unique situation. The probability is either 0 or 1, but we won’t know for five years, I pontificated…. that conversation bothered me
The full interview here. I hope the game theory society will find a way to celebrate Blackwell’s contribution to our community.