Siegfried, Tom. "3 Nash’s Equilibrium--Game theory’s foundation." A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature. Washington, DC: The National Academies Press, 2006.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature
problems. Game theory worked just fine if Robinson Crusoe was playing games with Friday, but the math for Gilligan’s Island wasn’t as rigorous.
Von Neumann’s approach to multiple-player games was to assume that coalitions would form. If Gilligan, the Skipper, and Mary Ann teamed up against the Professor, the Howells, and Ginger, you could go back to the simple two-person game rules. Many players might be involved, but if they formed two teams, the teams could take the place of individual players in the mathematical analysis.
But as later commentators noted, von Neumann had led himself into an inconsistency, threatening his theory’s internal integrity. A key part of two-person zero-sum games was choosing a strategy that was the best you could do against a smart opponent. Your best bet was to play your optimal (probably mixed) strategy no matter what anybody else did. But if coalitions formed among players in many-person games, as von Neumann believed they would, that meant your strategy would in fact depend on coordinating it with at least some of the other players. In any event, game theory describing many players interacting in non-zero-sum situations— that is, game theory applicable to real life—needed something more than the original Theory of Games had to offer. And that’s what John Nash provided.
The book A Beautiful Mind offers limited insight into Nash’s math, particularly in regard to all the many areas of science where that math has lately become prominent.8 But the book reveals a lot about Nash’s personal troubles. Sylvia Nasar’s portrait of Nash is not very flattering, though. He is depicted as immature, self-centered, arrogant, uncaring, and oblivious. But brilliant.
Nash was born in West Virginia, in the coal-mining town of Bluefield, in 1928. While showing some interest in math in high school (he even took some advanced courses at a local college), he planned to become an electrical engineer, like his father. But by the time he enrolled at Carnegie Tech (the Carnegie Institute of Tech-