The invention of deliberately oversimplified theories is one of the major techniques of science, particularly of the “exact” sciences, which make extensive use of mathematical analysis. If a biophysicist can usefully employ simplified models of the cell and the cosmologist simplified models of the universe then we can reasonably expect that simplified games may prove to be useful models for more complicated conflicts.— JOHN WILLIAMS, The Compleat Strategyst
NASH BECAME AWARE of a new branch of mathematics that was in the air of Fine Hall. It was an attempt, invented by von Neumann in the 1920s, to construct a systematic theory of rational human behavior by focusing on games as simple settings for the exercise of human rationality.
The first edition of The Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern came out in 1944.1 Tucker was running a popular new seminar in Fine on game theory.2 The Navy, which had made use of the theory during the war in antisubmarine warfare, was pouring money into game theory research at Princeton.3 The pure mathematicians around the department and at the Institute were inclined to view the new branch of mathematics, with its social science and military orientation, as “trivial,” “just the latest fad,” and “déclassé,”4 but to many of the students at Princeton at the time it was glamorous, heady stuff, like everything associated with von Neumann.5
Kuhn and Gale were always talking about von Neumann and Morgenstern’s book.6 Nash attended a lecture by von Neumann, one of the first speakers in Tucker’s seminar.7 Nash was intrigued by the apparent wealth of interesting, unsolved problems. He soon became one of the regulars at the seminar that met Thursdays at five o’clock; before long he was identified as a member of “Tucker’s clique.”8
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Mathematicians have always found games intriguing. Just as games of chance led to probability theory, poker and chess began to interest mathematicians around Göttingen, the Princeton of its time, in the 1920s.9 Von Neumann was the first to provide a complete mathematical description of a game and to prove a fundamental result, the min-max theorem.10
Von Neumann’s 1928 paper, Zur Theorie der Gesellschaftspiele, suggests that the theory of games might have applications to economics: “Any event — given the external conditions and the participants in the situation (provided that the latter are acting of their own free will) — may be regarded as a game of strategy if one looks at the effect it has on the participants,” adding, in a footnote, “[this] is the principal problem of classical economics: how is the absolutely selfish ‘homo economicus’ going to act under given external circumstances.”11 But the focal point of the theory — in von Neumann’s lectures and in discussions in mathematical circles during the 1930s — basically remained the exploration of parlor games like chess and poker.12 It was not until von Neumann met Morgenstern, a fellow émigré, in Princeton in 1938 that the link to economics was forged.13
Morgenstern, a tall, imposing expatriate from Vienna who was given to Napoleonic airs, claimed to be the grandson of the Kaiser’s father, Friedrich III of Germany.14 Tall, darkly handsome, “with cool gray eyes and a sensuous mouth,” Morgie cut an elegant figure on horseback, and caused a sensation among his students by abruptly marrying a beautiful redhead named Dorothy, a volunteer for the World Federalists many years his junior.15 Born in Silesia, Germany, in 1902, Morgenstern grew up and was educated in Vienna in a period of great intellectual and artistic ferment.16 After a three-year fellowship abroad financed by the Rockefeller Foundation, he became a professor and, until the Anschluss, was head of an institute for business cycle research. When Hitler marched into Vienna, Morgenstern happened to be visiting Princeton, and he decided it made sense to stay. He joined the university’s economics faculty, but disliked most of his American colleagues. He gravitated to the Institute, where Einstein, von Neumann, and Gödel were working at the time, angling for, but never receiving, an appointment there. “There is a spark missing,” he wrote disdainfully to a friend, referring to the University. “It is too provincial.”17
Morgenstern was, by temperament, a critic. His first book, Wirtschafts-prognose (Economic Prediction), was an attempt to prove that forecasting the ups and downs of the economy was a futile endeavor.18 One reviewer called it as “remarkable for its pessimism as it is for any . . . theoretical innovation.”19 Unlike those in astronomy, economic predictions have the peculiar ability to change outcomes.20 Predict a shortage, and businesses and consumers will react; the result is a glut.
His larger theme was the failure of economic theory to take proper account of interdependence among economic actors. He saw interdependence as the salient feature of all economic decisions, and he was always criticizing other economists for ignoring it.21 Robert Leonard, the historian, writes: “To some extent, his increasingly harsh views of economic theory were the product of mathematicians’ critical stance on the subject.”22 Von Neumann, he found, “focused on the black hole in the middle of economic theory.”23 According to one of von Neumann’s biographers, Morgenstern “interested him in aspects of economic situations, specifically in problems of exchange of goods between two or more persons, in problems of monopoly, oligopoly and free competition. It was in a discussion of attempts to schematize mathematically such processes that the present shape of this theory began to take form.”24
Morgenstern yearned to do “something in the truly scientific spirit.”25 He convinced von Neumann to write a treatise with him arguing that the theory of games was the correct foundation for all economic theory. Morgenstern, who had studied philosophy, not mathematics, could not contribute to the elaboration of the theory, but played muse and producer.26 Von Neumann wrote almost the whole twelve-hundred-page treatise, but it was Morgenstern who crafted the book’s provocative introduction and framed the issues in such a way that the book captured the attention of the mathematical and economic community.27
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The Theory of Games and Economic Behavior was in every way a revolutionary book. In line with Morgenstern’s agenda, the book was “a blistering attack” on the prevailing paradigm in economics and the Olympian Keynesian perspective, in which individual incentives and individual behavior were often subsumed, as well as an attempt to ground the theory in individual psychology. It was also an effort to reform social theory by applying mathematics as the language of scientific logic, in particular set theory and combinatorial methods. The authors wrapped the new theory in the mantle of past scientific revolutions, implicitly comparing their treatise to Newton’s Principia and the effort to put economics on a rigorous mathematical footing to Newton’s mathematization, using his invention of the calculus, of physics.28 One reviewer, Leo Hurwicz, wrote, “Ten more such books and the future of economics is assured.”29
The essence of von Neumann and Morgenstern’s message was that economics was a hopelessly unscientific discipline whose leading members were busily peddling solutions to pressing problems of the day — such as stabilizing employment — without the benefit of any scientific basis for their proposals.30 The fact that much of economic theory had been dressed up in the language of calculus struck them as “exaggerated” and a failure.31 This was not, they said, because of the “human element” or because of poor measurement of economic variables.32 Rather, they claimed, “Economic problems are not formulated clearly and are often stated in such vague terms as to make mathematical treatment a priori appear hopeless because it is quite uncertain what the problems really are.”33
Instead of pretending that they had the expertise to solve urgent social problems, economists should devote themselves to “the gradual development of a theory.”34 The authors argued that a new theory of games was “the proper instrument with which to develop a theory of economic behavior.”35 The authors claimed that “the typical problems of economic behavior become strictly identical with the mathematical notions of suitable games of strategy.”36 Under the heading “necessary limitations of the objectives,” von Neumann and Morgenstern admitted that their efforts to apply the new theory to economic problems had led them to “results that are already fairly well known,” but defended themselves by contending that exact proofs for many well-known economic propositions had been lacking.37
Before they have been given the respective proofs, theory simply does not exist as a scientific theory. The movements of the planets were known long before their courses had been calculated and explained by Newton’s theory. . . .
We believe that it is necessary to know as much as possible about the behavior of the individual and about the simplest forms of exchange. This standpoint was actually adopted with remarkable success by the founders of the marginal utility school, but nevertheless it is not generally accepted. Economists frequently point to much larger, more burning questions and brush everything aside which prevents them from making statements about them. The experience of more advanced sciences, for example, physics, indicates this impatience merely delays progress, including the treatment of the burning questions.
When the book appeared in 1944, von Neumann’s reputation was at its peak. It got the kind of public attention — including a breathless front-page story in The New York Times — that no other densely mathematical work had ever received, with the exception of Einstein’s papers on the special and general theories of relativity.38 Within two or three years, a dozen reviews appeared by top mathematicians and economists.39
The timing, as Morgenstern had sensed, was perfect. The war had unleashed a search for systematic attacks on all sorts of problems in a wide variety of fields, especially economics, previously thought to be institutional and historical in character. Quite apart from the new theory of games, a major transformation was under way — led by Samuelson’s Foundations of Economic Theory— making economic theory more rigorous through the use of calculus and advanced statistical methods.40 Von Neumann was critical of these efforts, but they surely prepared the ground for the reception of game theory.41
Economists were actually somewhat standoffish, at least compared to mathematicians, but Morgenstern’s antagonism to the economics profession no doubt contributed to that reaction. Samuelson later complained to Leonard, the historian, that although Morgenstern made “great claims, he himself lacked the mathematical wherewithal to substantiate them. Moreover [Morgenstern] had the irksome habit of always invoking the authority of some physical scientist or another.”42 In Princeton, Jacob Viner, the chairman of the economics department, heaped scorn on the unpopular Morgenstern by saying that if game theory couldn’t even solve a game like chess, what good was it, since economics was far more complicated than chess?43
It must have become obvious to Nash fairly early on that “the bible,” as The Theory of Games and Economic Behavior was known to students, though mathematically innovative, contained no fundamental new theorems beyond von Neumann’s stunning min-max theorem.44 He reasoned that von Neumann had succeeded neither in solving a major outstanding problem in economics using the new theory nor in making any major advance in the theory itself.45 Not a single one of its applications to economics did more than restate problems that economists had already grappled with.46 More important, the best-developed part of the theory — which took up one-third of the book — concerned zero-sum two-person games, which, because they are games of total conflict, appeared to have little applicability in social science.47 Von Neumann’s theory of games of more than two players, another large chunk of the book, was incomplete.48 He couldn’t prove that a solution existed for all such games.49 The last eighty pages of The Theory of Games and Economic Behavior dealt with non-zero-sum games, but von Neumann’s theory reduced such games formally to zero-sum games by introducing a fictitious player who consumes the excess or makes up the deficit.50 As one commentator was later to write, “This artifice helped but did not suffice for a completely adequate treatment of the non-zero-sum case. This is unfortunate because such games are the most likely to be found useful in practice.”51
To an ambitious young mathematician like Nash, the gaps and flaws in von Neumann’s theory were as alluring as the puzzling absence of ether through which light waves were supposed to travel was to the young Einstein. Nash immediately began thinking about the problem that von Neumann and Morgenstern described as the most important test of the new theory.