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Department of self-promotion: sequential tests, Blackwell games and the axiom of determinacy.
It takes a particular kind of strength to manage such a productive research career while tolerating the stresses and strains of personal insult, and carrying the aspirations of so many on one’s shoulders. Blackwell was more than a brilliant mathematician, he was also a human being of extraordinary personal fortitude.
I’ll always remember what he told me when I handed him a draft of my thesis. “The best thing about Bayesians is that they’re always right.”
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.
Blackwell and Girshick about the concept of strategy:
Imagine that you are to play the White pieces in a single game of chess, and that you discover you are unable to be present for the occasion. There is available a deputy, who will represent you on the occasion, and who will carry out your instructions exactly, but who is absolutely unable to make any decisions of his own volition. Thus, in order to guarantee that your deputy will be able to conduct the White pieces throughout the game, your instructions to him must envisage every possible circumstance in which he may be required to move, and must specify, for each such circumstance, what his choice is to be. Any such complete set of instructions constitutes what we shall call a strategy.
Now think about an infinite game, like repeated prisoner’s dilemma. If we take the idea about strategy as set of instructions seriously then not every function from past histories to an action is something we would like to call a strategy, because not every function can be described by a set of instructions ! This should be clear even before we formalize what instructions mean, simply because the set of possible `instructions’ is countable, as every such instructions is just a sentence in English.
So what should be the formal definition of a strategy in these games, a definition that will capture the intuition of a complete set of instructions that specify what your choice is to be for each possible circumstances? You know what I think.