achieving a situation from which White would be sure to win—assuming no dumb moves—no matter what Black does.)

Using principles of set theory (one of von Neumann’s mathematical specialties, by the way), Zermelo proved that proposition. His original proof required some later tweaking by other mathematicians and Zermelo himself. But the main lesson from it all was not so important for strategy in chess as it was to show that math could be used to analyze important features of any such game of strategy.

As it turns out, chess was a good choice because it is a perfect example of a particularly important type of game of strategy, known as a two-person zero-sum game. It’s called “zero-sum” because whatever one player wins, the other loses. The interests of the two competitors are diametrically opposed. (Chess is also a game where the players have “perfect information.” That means the game situation and all the decisions of all the players are known at all times—like playing poker with all the cards always dealt face up.)

Zermelo did not address the question of exactly what the best strategy is to play in chess, or even whether there actually is a surefire best strategy. The first move in that direction came from the brilliant French mathematician Émile Borel. In the early 1920s, Borel showed that there is a demonstrable best strategy in two-person zero-sum games—in some special cases. He doubted that it would be possible to prove the existence of a certain best strategy for such games in general.

But that’s exactly what von Neumann did. In two-person zero-sum games, he determined, there is always a way to find the best strategy possible, the strategy that will maximize your winnings (or minimize your losses) to whatever extent is possible by the rules of the game and your opponent’s choices. That’s the modern minimax7 theorem, which von Neumann first presented in December 1926 to the Göttingen Mathematical Society and then developed fully in his 1928 paper called “Zur Theorie der Gesellshaftsspiele” (Theory of Parlor Games), laying the foundation for von Neumann’s economics revolution.8



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