by Sylvia Nasar
Nash proved that, on a symmetrical board, the first player can always win. His proof is extremely deft, “marvelously nonconstructive” in the words of Milnor, who plays it very well.22 If the board is covered by black and white pieces, there’s always a chain that connects black to black or white to white, but never both. As Gale put it, “You can walk from Mexico to Canada or swim from California to New York, but you can’t do both.”23 That explains why there can never be a draw as in tic-tac-toe. But as opposed to tic-tac-toe, even if both players try to lose, one will win, like it or not.
The game quickly swept the common room.24 It brought Nash many admirers, including the young John Milnor, who was beguiled by its ingenuity and beauty Gale tried to sell the game. He said, “I even went to New York and showed it to several manufacturers. John and I had some agreement that I’d get a share if it sold. But they all said no, a thinking game would never sell. It was a marvelous game though. I then sent it off to Parker Brothers, but I never got a response.”25 Gale is the one who suggested the name Hex in his letter to Parker Brothers, which Parker used for the Dane’s game. (Kuhn remembers Nash describing the game to him, very likely over a meal at the college, in terms of points with six arrows emanating from each point, proof, in Kuhn’s mind, that his invention was independent of Hein’s.)26 Kuhn made a board for his children, who played it with great delight and saw to it that their children learned it too.27 Milnor still has a board that he made for his children.28 His poignant essay on Nash’s mathematical contributions for the Mathematical Intelligencer, written after Nash’s Nobel Prize, begins with a loving and detailed description of the game.
7
John von Neumann Princeton, 1948–49
JOHN VON 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 intelligent 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.2 He held a dozen consultancies, but his presence in Princeton was much felt.3 “We were all drawn by von Neumann,” Harold Kuhn recalled.4 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 ergodic 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.7 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.
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 Institute 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.11 An alleged model for Dr. Strangelove in the 1963 Stanley Kubrick film,12 he was a passionate Cold Warrior, advocating a first strike against Russia13 and defending nuclear testing.14 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.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.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.18 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 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. 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.25
As a student in Berlin, von Neumann frequently took the three-hour train trip to Göttingen, 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 ergodic 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, Mathematische Grundlagen der Quantenmechanik — the one Nash studied in the original German at Carnegie.27
Von Neumann was a privatdozent first at Berlin and then at Hamburg. He became a half-time professor at Princeton in 1931 and joined the Institute 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 manuscript that became 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.29
In 1948, he was back at the Institute 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.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, 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 passion 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 Institute 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 analytical 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.
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 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
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, Wirtschaftsprognose (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 t
heory 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
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