Wednesday, 7 July 2010

The history of game theory done right?

Over at the Coordination Problem blog Peter Boettke recommends that we read a the new book on the history of game theory by Robert Leonard: Von Neumann, Morgenstern, and The Creation of Game Theory: From Chess to Social Science, 1900–1960 (Cambridge: Cambridge University Press, 2010).

I have read a couple of pages of the book online (pages 16 and 17 to be precise) in which Leonard discusses Ernst Zermelo's contribution to the mathematics of chess. I picked this discussion because I have written on the subject as well. In fact Leonard makes reference to my paper with Ulrich Schwalbe, "Zermelo and the Early History of Game Theory". Games and Economic Behavior, v34 no1, January 2001: 123-37. I have to say that Leonard is one of the few people I have come across that get his discussion of Zermelo right. Most people get Zermelo's Theorem and the proof of it wrong.

In the modern literature on game theory many variant statements of Zermelo's Theorem. Some writers claim that Zermelo showed that Chess is determinate, e.g., Aumann (1989b, p. 1), Eichberger (1993, p. 9), or Hart (1992, p. 30): “In Chess, either White can force a win, or Black can force a win, or both sides can force a draw.” Others state more general propositions under the heading of Zermelo’s theorem, e.g., Mas-Colell et al. (1995, p. 272): “Every finite game of perfect information T_E has a pure strategy Nash equilibrium that can be derived by backward induction. Moreover, if no player has the same payoffs at any two terminal nodes, then there is a unique Nash equilibrium that can be derived in this manner.” Dimand and Dimand (1996, p. 107) claim that Zermelo showed that White cannot lose: “[I]n a finite game, there exists a strategy whereby a first mover ... cannot lose, but it is not clear whether there is a strategy whereby the first mover can win.” These statements are wrong. The questions Zermelo actually addresses are: First, what does it mean for a player to be in a “winning” position and is it possible to define this in an objective mathematical manner; second, if he is in a winning position, can the number of moves needed to force the win be determined?

Leonard gets it right discussing both of Zermelo's questions. If there is one problem Leonard's discussion it is when he writes about Zermelo's proof to do with the second of his questions. Leonard correctly outlines Zermelo's method of proof but does not point out that the proof Zermelo gives is wrong. His theorem is correct and was correctly proved in the 1920s. But this is only a minor point and if the rest of the book is as good as the two pages I have read, albeit a very small sample, then I agree with Boettke that the book would be well worth reading.

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