A Beautiful Mind Part 3

JOHN VON N NEUMANN was the very brightest star in Princeton's mathematical firmament and the apostle of the new mathematical era. At forty-five, he was universally considered the most cosmopolitan, multifaceted, and was the very brightest star in Princeton's mathematical firmament and the apostle of the new mathematical era. At forty-five, he was universally considered the most cosmopolitan, multifaceted, and intelligent intelligent mathematician the twentieth century had produced. mathematician the twentieth century had produced.1 No one was more responsible for the newly found importance of mathematics in America's intellectual elite. Less of a celebrity than Oppenheimer, not as remote as Einstein, as one biographer put it, von Neumann was the role model for Nash's generation. No one was more responsible for the newly found importance of mathematics in America's intellectual elite. Less of a celebrity than Oppenheimer, not as remote as Einstein, as one biographer put it, von Neumann was the role model for Nash's generation.2 He held a dozen consultancies, but his presence in Princeton was much felt. He held a dozen consultancies, but his presence in Princeton was much felt.3 "We were all drawn by von Neumann," Harold Kuhn recalled. "We were all drawn by von Neumann," Harold Kuhn recalled.4 Nash was to come under his spell. Nash was to come under his spell.5 Possibly the last true polymath, von Neumann made a brilliant career - half a dozen brilliant careers - by plunging fearlessly and frequently into any area where highly abstract mathematical thought could provide fresh insights. His ideas ranged from the first rigorous proof of the erG.o.dic theorem to ways of controlling the weather, from the implosion device for the A-bomb to the theory of games, from a new algebra [of rings of operators] for studying quantum physics to the notion of outfitting computers with stored programs.6 A giant among pure mathematicians by the time he was thirty years old, he had become in turn physicist, economist, weapons expert, and computer visionary. Of his 150 published papers, 60 are in pure mathematics, 20 in physics, and 60 in applied mathematics, including statistics and game theory. A giant among pure mathematicians by the time he was thirty years old, he had become in turn physicist, economist, weapons expert, and computer visionary. Of his 150 published papers, 60 are in pure mathematics, 20 in physics, and 60 in applied mathematics, including statistics and game theory.7 When he died in 1957 of cancer at fifty-three, he was developing a theory of the structure of the human brain. When he died in 1957 of cancer at fifty-three, he was developing a theory of the structure of the human brain.8 Unlike the austere and otherworldly G. H. Hardy, the Cambridge number theorist idolized by the previous generation of American mathematicians, von Neumann was worldly and engaged. Hardy abhorred politics, considered applied mathematics repellent, and saw pure mathematics as an esthetic pursuit best practiced for its own sake, like poetry or music.9 Von Neumann saw no contradiction between the purest mathematics and the grittiest engineering problems or between the role of the detached thinker and the political activist. Von Neumann saw no contradiction between the purest mathematics and the grittiest engineering problems or between the role of the detached thinker and the political activist.

He was one of the first of those academic consultants who were always on a train or plane bound for New York, Washington, or Los Angeles, and whose names frequently appeared in the news. He gave up teaching when he went to the Inst.i.tute in 1933 and gave up full-time research in 1955 to become a powerful member of the Atomic Energy Commission. in 1933 and gave up full-time research in 1955 to become a powerful member of the Atomic Energy Commission.10 He was one of the people who told Americans how to think about the bomb and the Russians, as well as how to think about the peaceful uses of atomic energy. He was one of the people who told Americans how to think about the bomb and the Russians, as well as how to think about the peaceful uses of atomic energy.11 An alleged model for Dr. Strangelove in the 1963 Stanley Kubrick film, An alleged model for Dr. Strangelove in the 1963 Stanley Kubrick film,12 he was a pa.s.sionate Cold Warrior, advocating a first strike against Russia he was a pa.s.sionate Cold Warrior, advocating a first strike against Russia13 and defending nuclear testing. and defending nuclear testing.14 Twice married and wealthy, he loved expensive clothes, hard liquor, fast cars, and dirty jokes. Twice married and wealthy, he loved expensive clothes, hard liquor, fast cars, and dirty jokes.15 He was a worka-holic, blunt and even cold at times. He was a worka-holic, blunt and even cold at times.16 Ultimately he was hard to know; the standing joke around Princeton was that von Neumann was really an extraterrestrial who had learned how to imitate a human perfectly. Ultimately he was hard to know; the standing joke around Princeton was that von Neumann was really an extraterrestrial who had learned how to imitate a human perfectly.17 In public, though, von Neumann exuded Hungarian charm and wit. The parties he gave in his brick mansion on Princeton's fashionable Library Place were "frequent and famous and long," according to Paul Halmos, a mathematician who knew von Neumann. In public, though, von Neumann exuded Hungarian charm and wit. The parties he gave in his brick mansion on Princeton's fashionable Library Place were "frequent and famous and long," according to Paul Halmos, a mathematician who knew von Neumann.18 His rapid-fire repartee in any of four languages was packed with references to history, politics, and the stock market. His rapid-fire repartee in any of four languages was packed with references to history, politics, and the stock market.19 His memory was astounding and so was the speed with which his mind worked. He could instantly memorize a column of phone numbers and virtually anything else. Stories of von Neumann's beating computers in mammoth feats of calculation abound. Paul Halmos tells the story in an obituary of the first test of von Neumann's electronic computer. Someone suggested a question like "What is the smallest power of 2 with the property that its decimal digit fourth from the right is 7?" As Halmos recounts, "The machine and Johnny started at the same time, and johnny finished first."20 Another time somebody asked him to solve the famous fly puzzle:21 Two bicyclists start twenty miles apart and head toward each other, each going at a steady rate of 10 m.p.h. At the same time, a fly that travels at a steady 15 m.p.h. starts from the front wheel of the southbound bicycle and flies to the front wheel of the northbound one, then turns around and flies to the front wheel of the southbound one again, and continues in this manner till he is crushed between the two front wheels. Question: what total distance did the fly cover?

There are two ways to answer the problem. One is to calculate the distance the fly covers on each leg of its trips between the two bicycles and finally sum the infinite series so obtained. The quick way is to observe that the bicycles meet exactly an hour after they start so that the fly had just an hour for his travels; the answer must therefore be 15 miles. When the question was put to von Neumann, he solved it in an instant, and thereby disappointed the questioner: "Oh, you must have heard the trick before!" "What trick," asked von Neumann, "all I did was sum the infinite series."

This seems astounding until one learns that at six, von Neumann could divide two eight-digit numbers in his head.22

Born in Budapest to a family of Jewish bankers, von Neumann was undeniably precocious. 23 23 At age eight, he had mastered calculus. At age twelve, he was reading works aimed at professional mathematicians, such as Emile Borel's At age eight, he had mastered calculus. At age twelve, he was reading works aimed at professional mathematicians, such as Emile Borel's Theorie des Fonctions. Theorie des Fonctions. But he also loved to invent mechanical toys and became a child expert on Byzantine history, the Civil War, and the trial of Joan of Arc. When it was time to go off to university, he agreed to study chemical engineering as a compromise with his father, who feared that his son couldn't make a living as a mathematician. Von Neumann kept his bargain by enrolling at the University of Budapest and promptly leaving for Berlin, where he spent his time doing mathematics, including visiting lectures by Einstein, and returning to Budapest at the end of every semester to take examinations. He published his second mathematics paper, in which he gave the modern definition of ordinal numbers which superseded Cantor's, at age nineteen. But he also loved to invent mechanical toys and became a child expert on Byzantine history, the Civil War, and the trial of Joan of Arc. When it was time to go off to university, he agreed to study chemical engineering as a compromise with his father, who feared that his son couldn't make a living as a mathematician. Von Neumann kept his bargain by enrolling at the University of Budapest and promptly leaving for Berlin, where he spent his time doing mathematics, including visiting lectures by Einstein, and returning to Budapest at the end of every semester to take examinations. He published his second mathematics paper, in which he gave the modern definition of ordinal numbers which superseded Cantor's, at age nineteen.24 By age twenty-five he had published ten major papers; by age thirty, nearly three dozen. By age twenty-five he had published ten major papers; by age thirty, nearly three dozen.25 As a student in Berlin, von Neumann frequently took the three-hour train trip to Gottingen, where he got to know Hilbert. The relationship led to von Neumann's famous 1928 paper on the axiomatization of set theory. Later he found the first mathematically rigorous proof of the erG.o.dic theorem, solved Hilbert's so-called Fifth Problem for compact groups, invented a new algebra and a new field called "continuous geometry," which is the geometry of dimensions that vary continuously (instead of a fourth dimension, one could now speak of three and three-quarters dimension). He was also a leader in the drive among mathematicians to colonize other disciplines by inventing new approaches.26 Von Neumann was still in his twenties when he wrote his famous paper on the theory of parlor games and his groundbreaking book on the mathematics of the new quantum physics, Von Neumann was still in his twenties when he wrote his famous paper on the theory of parlor games and his groundbreaking book on the mathematics of the new quantum physics, Mathematische Grundlagen der Quantenmechanik - the Mathematische Grundlagen der Quantenmechanik - the one Nash studied in the original German at Carnegie. one Nash studied in the original German at Carnegie.27 Von Neumann was a privatdozent privatdozent first at Berlin and then at Hamburg. He became a half-time professor at Princeton in 1931 and joined the Inst.i.tute for Advanced Study in 1933 at age thirty. When the war came, his interests shifted once again. Halmos says that "till then he was a top-flight pure mathematician who understood physics; after that he was an applied mathematician who remembered his pure work." first at Berlin and then at Hamburg. He became a half-time professor at Princeton in 1931 and joined the Inst.i.tute for Advanced Study in 1933 at age thirty. When the war came, his interests shifted once again. Halmos says that "till then he was a top-flight pure mathematician who understood physics; after that he was an applied mathematician who remembered his pure work."28 During the war, he collaborated with Morgenstern on a twelve-hundred-page ma.n.u.script that became During the war, he collaborated with Morgenstern on a twelve-hundred-page ma.n.u.script that became The Theory of Games and Economic Behavior. The Theory of Games and Economic Behavior. He was also the top mathematician in Oppenheimer's Manhattan Project from 1943 onward. His contribution to the A-bomb was his proposal for an implosion method for triggering an explosion with nuclear fuel, an idea credited with shortening the time needed to develop the bomb by as much as a year. He was also the top mathematician in Oppenheimer's Manhattan Project from 1943 onward. His contribution to the A-bomb was his proposal for an implosion method for triggering an explosion with nuclear fuel, an idea credited with shortening the time needed to develop the bomb by as much as a year.29 In 1948, he was back at the Inst.i.tute and very much a presence in Princeton. He did not teach any courses, but he edited and held court at the IAS.30 He dropped in at Fine Hall teas from time to time. He and Oppenheimer were already deep into their great debate over whether the H-bomb, or the Super, as it was known, could and should be built. He dropped in at Fine Hall teas from time to time. He and Oppenheimer were already deep into their great debate over whether the H-bomb, or the Super, as it was known, could and should be built.31 He was fascinated by meteorological prediction and control, suggesting once that the north and south poles be dyed blue in order to raise the earth's temperature. He not only showed the physicists, economists, He was fascinated by meteorological prediction and control, suggesting once that the north and south poles be dyed blue in order to raise the earth's temperature. He not only showed the physicists, economists, and electrical engineers that formal mathematics could yield fresh breakthroughs in their fields but made the enterprise of applying mathematics to real-world disciplines seem glamorous to the purest of young mathematicians. and electrical engineers that formal mathematics could yield fresh breakthroughs in their fields but made the enterprise of applying mathematics to real-world disciplines seem glamorous to the purest of young mathematicians.

By the end of the war, von Neumann's real pa.s.sion had become computers, though he called his interest in them "obscene."32 While he did not build the first computer, his ideas about computer architecture were accepted, and he invented mathematical techniques needed for computers. He and his collaborators, who included the future scientific director of IBM, Hermann Goldstine, invented stored rather than hardwired programs, a prototype digital computer, and a system for weather prediction. The theoretically oriented Inst.i.tute had no interest in building a computer, so von Neumann sold the idea to the Navy, arguing that the Normandy invasion had almost failed because of poor weather predictions. He promoted the MANIAC, as the machine was eventually named, as a device for improving meteorological prediction. More than anything, though, von Neumann was the one who saw the potential of these "thinking machines" most clearly, arguing in a speech in Montreal in 1945 that "many branches of both pure and applied mathematics are in great need of computing instruments to break the present stalemate created by the failure of the purely a.n.a.lytical approach to nonlinear problems." While he did not build the first computer, his ideas about computer architecture were accepted, and he invented mathematical techniques needed for computers. He and his collaborators, who included the future scientific director of IBM, Hermann Goldstine, invented stored rather than hardwired programs, a prototype digital computer, and a system for weather prediction. The theoretically oriented Inst.i.tute had no interest in building a computer, so von Neumann sold the idea to the Navy, arguing that the Normandy invasion had almost failed because of poor weather predictions. He promoted the MANIAC, as the machine was eventually named, as a device for improving meteorological prediction. More than anything, though, von Neumann was the one who saw the potential of these "thinking machines" most clearly, arguing in a speech in Montreal in 1945 that "many branches of both pure and applied mathematics are in great need of computing instruments to break the present stalemate created by the failure of the purely a.n.a.lytical approach to nonlinear problems."33 Everything von Neumann touched was imbued with his glamour. By wading fearlessly into fields far beyond mathematics, he inspired other young geniuses, Nash among them, to do the same. His success in applying similar approaches to dissimilar problems was a green light for younger men who were problem solvers rather than specialists.

CHAPTER 8

The Theory of Games

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 a.n.a.lysis. 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 W WILLIAMS, The Compleat Strategyst 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. 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 The Theory of Games and Economic Behavior by von Neumann and Oskar Morgenstern came out in 1944. by von Neumann and Oskar Morgenstern came out in 1944.1 Tucker was running a popular new seminar in Fine on game theory. 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. 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 Inst.i.tute were inclined to view the new branch of mathematics, with its social science and military orientation, as "trivial," "just the latest fad," and "decla.s.se," The pure mathematicians around the department and at the Inst.i.tute were inclined to view the new branch of mathematics, with its social science and military orientation, as "trivial," "just the latest fad," and "decla.s.se,"4 but to many of the students at Princeton at the time it was glamorous, heady stuff, like everything a.s.sociated with von Neumann. but to many of the students at Princeton at the time it was glamorous, heady stuff, like everything a.s.sociated 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. 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." 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 Mathematicians have always found games intriguing. Just as games of chance led to probability theory, poker and chess began to interest mathematicians around Gottingen, 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. 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, Zur Theorie der Gesellschaftspiele, suggests that the theory of games might have applications to economics: "Any event - given the external conditions and the partic.i.p.ants 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 partic.i.p.ants," adding, in a footnote, "[this] is the princ.i.p.al problem of cla.s.sical economics: how is the absolutely selfish 'h.o.m.o economicus' going to act under given external circ.u.mstances." suggests that the theory of games might have applications to economics: "Any event - given the external conditions and the partic.i.p.ants 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 partic.i.p.ants," adding, in a footnote, "[this] is the princ.i.p.al problem of cla.s.sical economics: how is the absolutely selfish 'h.o.m.o economicus' going to act under given external circ.u.mstances."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. 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 emigre, in Princeton in 1938 that the link to economics was forged. It was not until von Neumann met Morgenstern, a fellow emigre, 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. 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. 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 inst.i.tute 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 Inst.i.tute, where Einstein, von Neumann, and G.o.del 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." After a three-year fellowship abroad financed by the Rockefeller Foundation, he became a professor and, until the Anschluss, was head of an inst.i.tute 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 Inst.i.tute, where Einstein, von Neumann, and G.o.del 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, Wirtschaftsprognose (Economic Prediction), Wirtschaftsprognose (Economic Prediction), was an attempt to prove that forecasting the ups and downs of the economy was a futile endeavor. 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." 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. 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. 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." 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." 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 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 compet.i.tion. It was in a discussion of attempts to schematize mathematically such processes that the present shape of this theory began to take form." monopoly, oligopoly and free compet.i.tion. 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. 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. 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 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 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 Principia and the effort to put economics on a rigorous mathematical footing to Newton's mathematization, using his invention of the calculus, of physics. 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 a.s.sured." One reviewer, Leo Hurwicz, wrote, "Ten more such books and the future of economics is a.s.sured."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. 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. 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 Rather, they claimed, "Economic problems are not formulated clearly and are often stated in such vague terms as to make mathematical treatment a priori a priori appear hopeless because it is quite uncertain what the problems really are." 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." 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." 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 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. 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 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. - 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. 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 inst.i.tutional 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 Foundations of Economic Theory - making economic theory more rigorous through the use of calculus and advanced statistical methods. - 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. 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? 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 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. 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 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. 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. 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. 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. 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. He couldn't prove that a solution existed for all such games.49 The last eighty pages of The last eighty pages of The Theory of Games and Economic Behavior 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 fict.i.tious player who consumes the excess or makes up the deficit. dealt with non-zero-sum games, but von Neumann's theory reduced such games formally to zero-sum games by introducing a fict.i.tious 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." 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 the most important test of the new theory. most important test of the new theory.

CHAPTER 9

The Bargaining Problem Princeton, Spring 1949 Princeton, Spring 1949

We hope however to obtain a real understanding of the problem of exchange by studying it from an altogether different angle; that is, from the perspective of a "game of strategy."

- VON N NEUMANN AND M MORGENSTERN, The Theory of Games and Economic Behavior, The Theory of Games and Economic Behavior, second edition, 1947 second edition, 1947

NASH WROTE HIS FIRST PAPER, one of the great cla.s.sics of modern economics, during his second term at Princeton. one of the great cla.s.sics of modern economics, during his second term at Princeton.1 "The Bargaining Problem" is a remarkably down-to-earth work for a mathematician, especially a young mathematician. Yet no one but a brilliant mathematician could have conceived the idea. In the paper, Nash, whose economics training consisted of a single undergraduate course taken at Carnegie, adopted "an altogether different angle" on one of the oldest problems in economics and proposed a completely surprising solution. "The Bargaining Problem" is a remarkably down-to-earth work for a mathematician, especially a young mathematician. Yet no one but a brilliant mathematician could have conceived the idea. In the paper, Nash, whose economics training consisted of a single undergraduate course taken at Carnegie, adopted "an altogether different angle" on one of the oldest problems in economics and proposed a completely surprising solution.2 By so doing, he showed that behavior that economists had long considered part of human psychology, and therefore beyond the reach of economic reasoning, was, in fact, amenable to systematic a.n.a.lysis. By so doing, he showed that behavior that economists had long considered part of human psychology, and therefore beyond the reach of economic reasoning, was, in fact, amenable to systematic a.n.a.lysis.

The idea of exchange, the basis of economics, is nearly as old as man, and deal-making has been the stuff of legend since the Levantine kings and the pharaohs traded gold and chariots for weapons and slaves.3 Despite the rise of the great impersonal capitalist marketplace, with its millions of buyers and sellers who never meet face-to-face, the one-on-one bargain - involving wealthy individuals, powerful governments, labor unions, or giant corporations - dominates the headlines. But two centuries after the publication of Adam Smith's Despite the rise of the great impersonal capitalist marketplace, with its millions of buyers and sellers who never meet face-to-face, the one-on-one bargain - involving wealthy individuals, powerful governments, labor unions, or giant corporations - dominates the headlines. But two centuries after the publication of Adam Smith's The Wealth of Nations, The Wealth of Nations, there were still no principles of economics that could tell one how the parties to a potential bargain would interact, or how they would split up the pie. there were still no principles of economics that could tell one how the parties to a potential bargain would interact, or how they would split up the pie.4 The economist who first posed the problem of the bargain was a reclusive Oxford don, Francis Ysidro Edgeworth, in 1881.5 Edgeworth and several of his Victorian contemporaries were the first to abandon the historical and philosophical tradition of Smith, Ricardo, and Marx and to attempt to replace it with the mathematical Edgeworth and several of his Victorian contemporaries were the first to abandon the historical and philosophical tradition of Smith, Ricardo, and Marx and to attempt to replace it with the mathematical tradition of physics, writes Robert Heilbroner in tradition of physics, writes Robert Heilbroner in The Worldly Philosophers. The Worldly Philosophers.6 Edgeworth was not fascinated with economics because it justified or explained or condemned the world, or because it opened new vistas, bright or gloomy, into the future. This odd soul was fascinated by economics because economics dealt with quant.i.ties quant.i.ties and because anything that dealt with quant.i.ties could be translated into and because anything that dealt with quant.i.ties could be translated into mathematics. mathematics.7

Edgeworth thought of people as so many profit-and-loss calculators and recognized that the world of perfect compet.i.tion had "certain properties peculiarly favorable to mathematical calculation; namely a certain indefinite multiplicity and dividedness, a.n.a.logous to that infinity and infinitesimality which facilitate so large a portion of Mathematical Physics ... (consider the theory of Atoms, and all applications of the Differential Calculus)."8 The weak link in his creation, as Edgeworth was uncomfortably aware, was that people simply did not behave in a purely compet.i.tive fashion. Rather, they did not behave this way all the time. True, they acted on their own. But, equally often, they collaborated, cooperated, struck deals, evidently also out of self-interest. They joined trade unions, they formed governments, they established large enterprises and cartels. His mathematical models captured the results of compet.i.tion, but the consequences of cooperation proved elusive.9 Is it peace or war? asks the lover of "Maud" of economic compet.i.tion. It is both, pax or pact between contractors during contract, war, when some of the contractors without consent of others contract.

The first principle of Economics is that every agent is actuated only by self-interest. The workings of this principle may be viewed under two aspects, according as the agent acts without, or with, the consent of others affected by his actions. In a wide sense, the first species of action may be called war; the second contract.

Obviously, parties to a bargain were acting on the expectation that cooperation would yield more than acting alone. Somehow, the parties reached an agreement to share the pie. How they would split it depended on bargaining power, but on that score economic theory had nothing to say and there was no way of finding one solution in the haystack of possible solutions that met this rather broad criterion. Edgeworth admitted defeat: "The general answer is -(a) Contract without compet.i.tion is indeterminate."10 Over the next century, a half-dozen great economists, including the Englishmen John Hicks and Alfred Marshall and the Dane F. Zeuthen, took up Edgeworth's problem, but they, too, ended up throwing up their hands.11 Von Von Neumann and Morgenstern suggested that the answer lay in reformulating the problem as a game of strategy, but they themselves did not succeed in solving it. Neumann and Morgenstern suggested that the answer lay in reformulating the problem as a game of strategy, but they themselves did not succeed in solving it.12 Nash took a completely novel approach to the problem of predicting how two rational bargainers will interact. Instead of defining a solution directly, he started by writing down a set of reasonable conditions that any plausible solution would have to satisfy and then looked at where they took him.

This is called the axiomatic approach - a method that had swept mathematics in the 1920s, was used by von Neumann in his book on quantum theory and his papers on set theory, and was in its heyday at Princeton in the late 1940s.13 Nash's paper is one of the first to apply the axiomatic method to a problem in the social sciences. Nash's paper is one of the first to apply the axiomatic method to a problem in the social sciences.14 Recall that Edgeworth had called the problem of the bargain "indeterminate." In other words, if all one knew about the bargainers were their preferences, one couldn't predict how they would interact or how they would divide the pie. The reason for the indeterminacy would have been obvious to Nash. There wasn't enough information so one had to make additional a.s.sumptions.

Nash's theory a.s.sumes that both sides' expectations about each other's behavior are based on the intrinsic features of the bargaining situation itself. The essence of a situation that results in a deal is "two individuals who have the opportunity to collaborate for mutual benefit in more than one way."15 How they will split the gain, he reasoned, reflects how much the deal is worth to each individual. How they will split the gain, he reasoned, reflects how much the deal is worth to each individual.

He started by asking the question, What reasonable conditions would any solution - any split - have to satisfy? He then posed four conditions and, using an ingenious mathematical argument, showed that, if his axioms held, a unique solution existed that maximized the product of the players' utilities. In a sense, his contribution was not so much to "solve" the problem as to state it in a simple and precise way so as to show that unique solutions were possible.

The striking feature of Nash's paper is not its difficulty, or its depth, or even its elegance and generality, but rather that it provides an answer to an important problem. Reading Nash's paper today, one is struck most by its originality. The ideas seem to come out of the blue. There is some basis for this impression. Nash arrived at his essential idea - the notion that the bargain depended on a combination of the negotiators' back-up alternatives and the potential benefits of striking a deal - as an undergraduate at Carnegie Tech before he came to Princeton, before he started attending Tucker's game theory seminar, and before he had read von Neumann and Morgenstern's book. It occurred to him while he was sitting in the only economics course he would ever attend.16 The course, on international trade, was taught by a clever and young Viennese emigre in his thirties named Bert Hoselitz. Hoselitz, who emphasized theory in his course, had degrees in law and economics, the latter from the University of Chicago.17 International agreements between governments and between monopolies had dominated trade, especially in commodities, between the wars, and Hoselitz International agreements between governments and between monopolies had dominated trade, especially in commodities, between the wars, and Hoselitz was an expert on the subject of international cartels and trade. was an expert on the subject of international cartels and trade.18 Nash took the course in his final semester, in the spring of 1948, simply to fulfill degree requirements. Nash took the course in his final semester, in the spring of 1948, simply to fulfill degree requirements.19 As always, though, the big, unsolved problem was the bait. As always, though, the big, unsolved problem was the bait.

That problem concerned trade deals between countries with separate currencies, as he told Roger Myerson, a game theorist at Northwestern University, in 1996.20 One of Nash's axioms, if applied in an international trade context, a.s.serts that the outcome of the bargain shouldn't change if one country revalued its currency. Once at Princeton, Nash would have quickly learned about von Neumann and Morgenstern's theory and recognized that the arguments that he'd thought of in Hoselitz's cla.s.s had a much wider applicability. One of Nash's axioms, if applied in an international trade context, a.s.serts that the outcome of the bargain shouldn't change if one country revalued its currency. Once at Princeton, Nash would have quickly learned about von Neumann and Morgenstern's theory and recognized that the arguments that he'd thought of in Hoselitz's cla.s.s had a much wider applicability.21 Very likely Nash sketched his ideas for a bargaining solution in Tucker's seminar and was urged by Oskar Morgenstern - whom Nash invariably referred to as Oskar La Morgue - to write a paper. Very likely Nash sketched his ideas for a bargaining solution in Tucker's seminar and was urged by Oskar Morgenstern - whom Nash invariably referred to as Oskar La Morgue - to write a paper.22 Legend, possibly encouraged by Nash himself, soon had it that he'd written the whole paper in Hoselitz's cla.s.s - much as Milnor solved the Borsuk problem in knot theory as a homework a.s.signment - and that he had arrived at Princeton with the bargaining paper tucked into his briefcase.23 Nash has since corrected the record. Nash has since corrected the record.24 But when the paper was published in 1950, in But when the paper was published in 1950, in Econometrica, Econometrica, the leading journal of mathematical economics, Nash was careful to retain full credit for the ideas: "The author wishes to acknowledge the a.s.sistance of Professors von Neumann and Morgenstern who read the original form of the paper and gave helpful advice as to the presentation." the leading journal of mathematical economics, Nash was careful to retain full credit for the ideas: "The author wishes to acknowledge the a.s.sistance of Professors von Neumann and Morgenstern who read the original form of the paper and gave helpful advice as to the presentation."25 And in his n.o.bel autobiography, Nash makes it clear that it was his interest in the bargaining problem that brought him into contact with the game theory group at Princeton, not the other way around: "as a result of that exposure to economic ideas and problems I arrived at the idea that led to the paper The Bargaining Problem' which was later published in And in his n.o.bel autobiography, Nash makes it clear that it was his interest in the bargaining problem that brought him into contact with the game theory group at Princeton, not the other way around: "as a result of that exposure to economic ideas and problems I arrived at the idea that led to the paper The Bargaining Problem' which was later published in Econometrica. Econometrica. And it was this idea which in turn, when I was a graduate student at Princeton, And it was this idea which in turn, when I was a graduate student at Princeton, led to my interest in the game theory studies there!" led to my interest in the game theory studies there!"26

CHAPTER 10

Nash's Rival Idea Princeton, 194950 Princeton, 194950

I was playing a non-cooperative game in relation to von Neumann rather than simply seeking to join his coalition.

- JOHN F. N F. NASH, J JR., 1993

IN THE SUMMER OF 1949, Albert Tucker caught the mumps from one of his children. 1949, Albert Tucker caught the mumps from one of his children.1 He had planned to be in Palo Alto, California, where he was to spend his sabbatical year, by the end of August. Instead, he was in his office at Fine, gathering up some books and papers, when Nash walked in to ask whether Tucker would be willing to supervise his thesis. He had planned to be in Palo Alto, California, where he was to spend his sabbatical year, by the end of August. Instead, he was in his office at Fine, gathering up some books and papers, when Nash walked in to ask whether Tucker would be willing to supervise his thesis.

Nash's request caught him by surprise.2 Tucker had little direct contact with Nash during the latter's first year and had been under the impression that he would probably write a thesis with Steenrod. But Nash, who offered no real explanation, told Tucker only that he thought he had found some "good results related to game theory." Tucker, who was still feeling out of sorts and eager to get home, agreed to become his adviser only because he was sure that Nash would still be in the early stages of his research by the time he returned to Princeton the following summer. Tucker had little direct contact with Nash during the latter's first year and had been under the impression that he would probably write a thesis with Steenrod. But Nash, who offered no real explanation, told Tucker only that he thought he had found some "good results related to game theory." Tucker, who was still feeling out of sorts and eager to get home, agreed to become his adviser only because he was sure that Nash would still be in the early stages of his research by the time he returned to Princeton the following summer.

Six weeks later, Nash and another student were buying beers for a crowd of graduate students and professors in the bar in the bas.e.m.e.nt of the Na.s.sau Inn - as tradition demanded of men who had just pa.s.sed their generals.3 The mathematicians were growing more boisterous and drunken by the minute. A limerick compet.i.tion was in full swing. The object was to invent the cleverest, dirtiest rhyme about a member of the Princeton mathematics department, preferably about one of the ones present, and shout it out at the top of one's lungs. The mathematicians were growing more boisterous and drunken by the minute. A limerick compet.i.tion was in full swing. The object was to invent the cleverest, dirtiest rhyme about a member of the Princeton mathematics department, preferably about one of the ones present, and shout it out at the top of one's lungs.4 At one point, a s.h.a.ggy Scot aptly named Macbeath jumped to his feet, beer bottle in hand, and began to belt out stanza after stanza of a popular and salacious drinking song, with the others chiming in for the chorus: "I put my hand upon her breast/She said, Young man, I like that best'/(Chorus) Gosh, gore, blimey, how ashamed I was." At one point, a s.h.a.ggy Scot aptly named Macbeath jumped to his feet, beer bottle in hand, and began to belt out stanza after stanza of a popular and salacious drinking song, with the others chiming in for the chorus: "I put my hand upon her breast/She said, Young man, I like that best'/(Chorus) Gosh, gore, blimey, how ashamed I was."5 That night, with its quaint, masculine rite of pa.s.sage, marked the effective end of Nash's years as a student. He had been trapped in Princeton for an entire hot and sticky summer, forced to put aside the interesting problems he had been thinking about, to cram for the general examination. hot and sticky summer, forced to put aside the interesting problems he had been thinking about, to cram for the general examination.6 Luckily, Lefschetz had appointed a friendly trio of examiners: Church, Steenrod, and a visiting professor from Stanford, Donald Spencer. Luckily, Lefschetz had appointed a friendly trio of examiners: Church, Steenrod, and a visiting professor from Stanford, Donald Spencer.7 The whole nerve-racking event had gone rather well. The whole nerve-racking event had gone rather well.

Many mathematicians, most famously the French genius Henri Poincare, have testified to the value of leaving a partially solved problem alone for a while and letting the unconscious work behind the scenes. In an oft-quoted pa.s.sage from a 1908 essay about the genesis of mathematical discovery, Poincare writes:8 For fifteen days I struggled to prove that no functions a.n.a.logous to those I have since called Fuchsian functions could exist. I was then very ignorant. Every day I sat down at my work table where I spent an hour or two; I tried a great number of combinations and arrived at no result... .

I then left Caen where I was living at the time, to partic.i.p.ate in a geological trip sponsored by the School of Mines. The exigencies of travel made me forget my mathematical labors; reaching Coutances we took a bus for some excursion or another. The instant I put my foot on the step the idea came to me, apparently with nothing whatever in my previous thoughts having prepared me for it.

Nash's "wasted" summer, with its enforced break from his research, proved unexpectedly fruitful, allowing several vague hunches from the spring to crystallize and mature. That October, he started to experience a virtual storm of ideas. Among them was his brilliant insight into human behavior: the Nash equilibrium.

Nash went to see von Neumann a few days after he pa.s.sed his generals.9 He wanted, he had told the secretary c.o.c.kily, to discuss an idea that might be of interest to Professor von Neumann. It was a rather audacious thing for a graduate student to do. He wanted, he had told the secretary c.o.c.kily, to discuss an idea that might be of interest to Professor von Neumann. It was a rather audacious thing for a graduate student to do.10 Von Neumann was a public figure, had very little contact with Princeton graduate students outside of occasional lectures, and generally discouraged them from seeking him out with their research problems. But it was typical of Nash, who had gone to see Einstein the year before with the germ of an idea. Von Neumann was a public figure, had very little contact with Princeton graduate students outside of occasional lectures, and generally discouraged them from seeking him out with their research problems. But it was typical of Nash, who had gone to see Einstein the year before with the germ of an idea.

Von Neumann was sitting at an enormous desk, looking more like a prosperous bank president than an academic in his expensive three-piece suit, silk tie, and jaunty pocket handkerchief.11 He had the preoccupied air of a busy executive. At the time, he was holding a dozen consultancies, "arguing the ear off Robert Oppenheimer" over the development of the H-bomb, and overseeing the construction and programming of two prototype computers. He had the preoccupied air of a busy executive. At the time, he was holding a dozen consultancies, "arguing the ear off Robert Oppenheimer" over the development of the H-bomb, and overseeing the construction and programming of two prototype computers.12 He gestured Nash to sit down. He knew who Nash was, of course, but seemed a bit puzzled by his visit. He gestured Nash to sit down. He knew who Nash was, of course, but seemed a bit puzzled by his visit.

He listened carefully, with his head c.o.c.ked slightly to one side and his fingers tapping. Nash started to describe the proof he had in mind for an equilibrium in games of more than two players. But before he had gotten out more than a few disjointed sentences, von Neumann interrupted, jumped ahead to the vet unstated conclusion of Nash's argument, and said abruptly, "That's trivial, you know. That's just a fixed point theorem." tapping. Nash started to describe the proof he had in mind for an equilibrium in games of more than two players. But before he had gotten out more than a few disjointed sentences, von Neumann interrupted, jumped ahead to the vet unstated conclusion of Nash's argument, and said abruptly, "That's trivial, you know. That's just a fixed point theorem."13 It is not altogether surprising that the two geniuses should clash. They came at game theory from two opposing views of the way people interact. Von Neumann, who had come of age in European cafe discussions and collaborated on the bomb and computers, thought of people as social beings who were always communicating. It was quite natural for him to emphasize the central importance of coalitions and joint action in society. Nash tended to think of people as out of touch with one another and acting on their own. For him, a perspective founded on the ways that people react to individual incentives seemed far more natural.

Von Neumann's rejection of Nash's bid for attention and approval must have hurt, however, and one guesses that it was even more painful than Einstein's earlier but kindlier dismissal. He never approached von Neumann again. Nash later rationalized von Neumann's reaction as the naturally defensive posture of an established thinker to a younger rival's idea, a view that may say more about what was in Nash's mind when he approached von Neumann than about the older man. Nash was certainly conscious that he was implicitly challenging von Neumann. Nash noted in his n.o.bel autobiography that his ideas "deviated somewhat from the 'line' (as if of 'political part lines') of von Neumann and Morgenstern s book. "deviated somewhat from the 'line' (as if of 'political part lines') of von Neumann and Morgenstern s book."14 Valleius, the Roman philosopher, was the first to offer a theory for why geniuses often appeared, not as lonely giants, but in cl.u.s.ters in particular fields in particular cities. He was thinking of Plato and Aristotle, Pythagoras and Archimedes, and Aeschylus, Euripides, Sophocles, and Aristophanes, but there are many later examples as well, including Newton and Locke, or Freud, Jung, and Adler. He speculated that creative geniuses inspired envy as well as emulation and attracted younger men who were motivated to complete and recast the original contribution.15 In a letter to Robert Leonard, Nash wrote a further twist: "I was playing a non-cooperative game in relation to von Neumann rather than simply seeking to join his coalition. And of course, it was psychologically natural for him not to be entirely pleased by a rival theoretical approach."16 In his opinion, von Neumann never behaved unfairly. Nash compares himself to a young physicist who challenged Einstein, noting that Einstein was initially critical of Kaluza's five-dimensional unified theory of gravitational and electric fields but later supported its publication. In his opinion, von Neumann never behaved unfairly. Nash compares himself to a young physicist who challenged Einstein, noting that Einstein was initially critical of Kaluza's five-dimensional unified theory of gravitational and electric fields but later supported its publication.17 Nash, so often oblivious to the feelings and motivations of other people, was quick, in this case, to pick up on certain emotional undercurrents, especially envy and jealousy. In a way, he saw rejection as the price genius must pay. Nash, so often oblivious to the feelings and motivations of other people, was quick, in this case, to pick up on certain emotional undercurrents, especially envy and jealousy. In a way, he saw rejection as the price genius must pay.

A few days after the disastrous meeting with von Neumann, Nash accosted David Gale. "I think I've found a way to generalize von Neumann's min-max theorem," he blurted out. "The fundamental idea is that in a two-person zero-sum solution, the best strategy for both is ... The whole theory is built on it. And it works with any number of people and doesn't have to be a zero-sum game." David Gale. "I think I've found a way to generalize von Neumann's min-max theorem," he blurted out. "The fundamental idea is that in a two-person zero-sum solution, the best strategy for both is ... The whole theory is built on it. And it wor