Part 3 (1/2)

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 htest star in Princeton's mathematical firmament and the apostle of the new mathematical era At forty-five, he was universally considered the ent mathematician the twentieth century had produced mathematician the twentieth century had produced1 No one was more responsible for the newly found importance of mathematics in America's intellectual elite Less of a celebrity than Oppenheirapher put it, von Neueneration No one was more responsible for the newly found importance of mathematics in America's intellectual elite Less of a celebrity than Oppenheirapher put it, von Neueneration2 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 felt3 ”We were all drawn by von Neumann,” Harold Kuhn recalled ”We were all drawn by von Neumann,” Harold Kuhn recalled4 Nash was to come under his spell Nash was to come under his spell5 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 hts His ideas ranged froorous proof of the erGodic theore the weather, froa quantu co 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 iant a 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 ame theory7 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 brain8 Unlike the austere and otherworldly G H Hardy, the Caeneration of Aed Hardy abhorred politics, considered applied mathematics repellent, and saw pure mathematics as an esthetic pursuit best practiced for its own sake, like poetry or music9 Von Neumann saw no contradiction between the purestproblems or between the role of the detached thinker and the political activist Von Neumann saw no contradiction between the purestproblems or between the role of the detached thinker and the political activist

He was one of the first of those academic consultants ere always on a train or plane bound for New York, Washi+ngton, or Los Angeles, and whose na when he went to the Institute in 1933 and gave up full-time research in 1955 to becoy Coave up full-time research in 1955 to becoy Commission10 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 atoy 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 atoelove in the 1963 Stanley Kubrick filelove in the 1963 Stanley Kubrick fil a first strike against Russia he was a passionate Cold Warrior, advocating a first strike against Russia13 and defending nuclear testing and defending nuclear testing14 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 jokes15 He was a worka-holic, blunt and even cold at times He was a worka-holic, blunt and even cold at ti joke around Princeton was that von Neumann was really an extraterrestrial who had learned how to imitate a hu joke around Princeton was that von Neumann was really an extraterrestrial who had learned how to ih, von Neuave in his brick mansion on Princeton's fashi+onable Library Place were ”frequent and fa to Paul Halh, von Neuave in his brick mansion on Princeton's fashi+onable Library Place were ”frequent and fa to Paul Halmos, a mathematician who knew von Neues was packed with references to history, politics, and the stock es was packed with references to history, politics, and the stockand so was the speed hich his mind worked He could instantlyelse Stories of von Neu computers in mammoth feats of calculation abound Paul Halmos tells the story in an obituary of the first test of von Neuested a question like ”What is the sit fourth froht 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 twentyat a steady rate of 10 mph At the same time, a fly that travels at a steady 15 mph 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 tays to answer the proble 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 su until one learns that at six, von Neuit numbers in his head22

Born in Budapest to a family of Jewish bankers, von Neuht, he hadworks aimed at professional ht, he hadworks 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 tiineering as a compromise with his father, who feared that his son couldn't ain by enrolling at the University of Budapest and prolectures by Einstein, and returning to Budapest at the end of every semester to take examinations He published his second ave the modern definition of ordinal nue 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 tiineering as a compromise with his father, who feared that his son couldn't ain by enrolling at the University of Budapest and prolectures by Einstein, and returning to Budapest at the end of every semester to take examinations He published his second ave the modern definition of ordinal nue twenty-five he had published ten e twenty-five he had published ten e thirty, nearly three dozen25 As a student in Berlin, von Neuen, where he got to know Hilbert The relationshi+p led to von Neumann's famous 1928 paper on the axiomatization of set theory Later he found the first orous proof of the erGodic theoreroups, invented a new algebra and a new field called ”continuous geoeometry of dimensions that vary continuously (instead of a fourth dimension, one could now speak of three and three-quarters dinew approaches26 Von Neumann was still in his twenties when he wrote his faroundbreaking book on the mathematics of the new quantum physics, Von Neumann was still in his twenties when he wrote his faroundbreaking book on the en der Quantenen der Quanteninal Gerinal Gerie27 Von Neumann was a privatdozent privatdozent first at Berlin and then at Ha 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 caain Halht pure mathematician who understood physics; after that he was an applied mathematician who re 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 caain Halht pure mathematician who understood physics; after that he was an appliedthe war, he collaborated with Morgenstern on a twelve-hundred-pagethe war, he collaborated with Morgenstern on a twelve-hundred-page manuscript 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-bo 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-bo an explosion with nuclear fuel, an idea credited with shortening the time needed to develop the bomb by as much as a year29 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 IAS30 He dropped in at Fine Hall teas from tireat 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 tireat debate over whether the H-bomb, or the Super, as it was known, could and should be built31 He was fascinated byonce that the north and south poles be dyed blue in order to raise the earth's temperature He not only showed the physicists, econoical 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, econoineers that forhs in their fields butla ineers that forhs in their fields butla mathematicians

By the end of the war, von Neuh 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, Herraital computer, and a system for weather prediction The theoretically oriented Institute had no interest in building a co that the Normandy invasion had almost failed because of poor weather predictions He promoted the MANIAC, as theh, von Neuin a speech in Montreal in 1945 that ”reat need of co instruments to break the present stalemate created by the failure of the purely analytical 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, Herraital computer, and a system for weather prediction The theoretically oriented Institute had no interest in building a co that the Normandy invasion had almost failed because of poor weather predictions He promoted the MANIAC, as theh, von Neuin a speech in Montreal in 1945 that ”reat need of co instruments to break the present stalemate created by the failure of the purely analytical approach to nonlinear problela fearlessly into fields far beyondthe siht for younger men ere 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 analysis If a biophysicist can usefully eist simplified models of the universe then we can reasonably expect that siames may prove to be useful models for more complicated conflicts

- JOHN W WILLIAMS, The Coyst

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 systeas 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 systeas for the exercise of human rationality

The first edition of The Theory of Games and Economic Behavior The Theory of Gaenstern caenstern ca a popular new se a popular new sea the war in antisubame theory research at Princeton The Navy, which hadthe war in antisubame theory research at Princeton3 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 ”declasse,” 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 ”declasse,”4 but to la associated with von Neumann but to la associated with von Neu about von Neuenstern's book6 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 seued by the apparent wealth of interesting, unsolved probleulars at the se he was identified as a ued by the apparent wealth of interesting, unsolved probleulars at the se he was identified as a meaaan to interest en, the Princeton of its time, in the 1920s9 Von Neumann was the first to provide a coame and to prove a fundamental result, the min-max theorem Von Neumann was the first to provide a coame and to prove a fundamental result, the min-max theorem10 Von Neumann's 1928 paper, Zur Theorie der Gesellschaftspiele, Zur Theorie der Gesellschaftspiele, suggests that the theory of gaiven the external conditions and the participants in the situation (provided that the'latter are acting of their own free will) - y if one looks at the effect it has on the participants,” adding, in a footnote, ”[this] is the principal problem of classical econo to act under given external circuht have applications to econoiven the external conditions and the participants in the situation (provided that the'latter are acting of their own free will) - y if one looks at the effect it has on the participants,” adding, in a footnote, ”[this] is the principal problem of classical econo to act under given external circumstances”11 But the focal point of the theory - in von Neumann's lectures and in discussions inthe 1930s - basically reames like chess and poker But the focal point of the theory - in von Neumann's lectures and in discussions inthe 1930s - basically reames like chess and poker12 It was not until von Neure, in Princeton in 1938 that the link to econoenstern, a fellow ere, in Princeton in 1938 that the link to econo expatriate frorandson of the Kaiser's father, Friedrich III of Gerray eyes and a sensuous ure on horseback, and caused a sensation a a beautiful redhead named Dorothy, a volunteer for the World Federalists ray eyes and a sensuous ure on horseback, and caused a sensation a a beautiful redhead named Dorothy, a volunteer for the World Federalists many years his junior15 Born in Silesia, Gerrew up and was educated in Vienna in a period of great intellectual and artistic ferrew up and was educated in Vienna in a period of great intellectual and artistic ferment16 After a three-year fellowshi+p abroad financed by the Rockefeller Foundation, he became a professor and, until the Anschluss, was head of an institute for business cycle research When HitlerPrinceton, and he decided it made sense to stay He joined the university's econoues He gravitated to the Institute, where Einstein, von Neu for, but never receiving, an appoint,” he wrote disdainfully to a friend, referring to the University ”It is too provincial” After a three-year fellowshi+p abroad financed by the Rockefeller Foundation, he became a professor and, until the Anschluss, was head of an institute for business cycle research When HitlerPrinceton, and he decided it made sense to stay He joined the university's econoues He gravitated to the Institute, where Einstein, von Neu for, but never receiving, an appoint,” he wrote disdainfully to a friend, referring to the University ”It is too provincial”17 Morgenstern was, by tenose (Econonose (Econo the ups and downs of the economy was a futile endeavor was an atte the ups and downs of the economy was a futile endeavor18 One reviewer called it as ”remarkable for its pessimism as it is for anytheoretical innovation” One reviewer called it as ”remarkable for its pessimism as it is for anytheoretical innovation”19 Unlike those in astronoe outcomes Unlike those in astronoe outcoe, and businesses and consue, and businesses and consuer theme was the failure of econo economic actors He saw interdependence as the salient feature of all econo other econo it21 Robert Leonard, the historian, writes: ”To soly harsh views of economic theory were the product of mathematicians' critical stance on the subject” Robert Leonard, the historian, writes: ”To soly 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 theto one of von Neuenstern ”interested him in aspects of econooods between two orto one of von Neuenstern ”interested him in aspects of econooods between two or opoly 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 foropoly 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 for in the truly scientific spirit”25 He convinced von Neuaenstern, who had studied philosophy, not mathematics, could not contribute to the elaboration of the theory, but played muse and producer He convinced von Neuaenstern, who had studied philosophy, not mathematics, could not contribute to the elaboration of the theory, but played muse and producer26 Von Neue 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 coe 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 community27 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 subsuround the theory in individual psychology It was also an effort to refore of scientific logic, in particular set theory and combinatorial methods The authors wrapped the new theory in thetheir 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 subsuround the theory in individual psychology It was also an effort to refore of scientific logic, in particular set theory and combinatorial methods The authors wrapped the new theory in thetheir treatise to Newton's Principia Principia and the effort to put econo to Newton'shis invention of the calculus, of physics and the effort to put econo to Newton'shis invention of the calculus, of physics28 One reviewer, Leo Hurwicz, wrote, ”Ten more such books and the future of economics is assured” One reviewer, Leo Hurwicz, wrote, ”Ten more such books and the future of econoenstern's e was that econoproble employment - without the benefit of any scientific basis for their proposals30 The fact that e of calculus struck theerated” and a failure The fact that e of calculus struck theerated” and a failure31 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 variables32 Rather, they claimed, ”Economic probleue terms as to make mathematical treatment Rather, they claimed, ”Economic probleue 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 proble that they had the expertise to solve urgent social probleradual developames was ”the proper instrument hich to develop a theory of econoames was ”the proper instrument hich to develop a theory of economic behavior”35 The authors claimed that ”the typical problems of economic behavior become strictly identical with the y” The authors claimed that ”the typical problems of economic behavior become strictly identical with the y”36 Under the heading ”necessary lienstern 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 the ”necessary lienstern 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 the that exact proofs forcontending that exact proofs for 37 Before they have been given the respective proofs, theory simply does not exist as a scientific theory Thebefore 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 sie This standpoint was actually adopted with reinal utility school, but nevertheless it is not generally accepted Econo questions and brush everything aside which prevents the statements about them The experience of more advanced sciences, for exaress, including the treat questions

When the book appeared in 1944, von Neuot 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 relativity38 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 enstern 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 econoht to be institutional and historical in character Quite apart froames, a major transformation was under way - led by Samuelson's Foundations of Econo econoh the use of calculus and advanced statistical h the use of calculus and advanced statistical methods40 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 theory41 Economists were actually somewhat standoffish, at least coonism to the economics profession no doubt contributed to that reaction Samuelson later coenstern reat claims, he himself lacked the enstern] had the irkso the authority of some physical scientist or another”42 In Princeton, Jacob Viner, the chairman of the econoenstern by saying that if gaood was it, since economics was far more complicated than chess? In Princeton, Jacob Viner, the chairman of the econoenstern by saying that if gaood 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 Gah mathematically innovative, contained no funda h mathematically innovative, contained no funda min-max theorem44 He reasoned that von Neumann had He reasoned that von Neu proble any major advance in the theory itself succeeded neither in solving athe new theory nor in le one of its applications to economics did rappled with Not a single one of its applications to economics did rappled with46 More important, the best-developed part of the theory - which took up one-third of the book - concerned zero-suames 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-suames of total conflict, appeared to have little applicability in social science47 Von Neue chunk of the book, was incoae chunk of the book, was incomplete48 He couldn't prove that a solution existed for all such gaaes of The Theory of Games and Economic Behavior The Theory of Gaaa a fictitious player who consumes the excess or aa a fictitious player who consumes the excess or makes up the deficit50 As one commentator was later to write, ”This artifice helped but did not suffice for a completely adequate treatment of the non-zero-suames 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-suames are the most likely to be found useful in practice”51 To an aaps and flaws in von Neu absence of ether through which light waves were supposed to travel was to the young Einstein Nash i about the probleenstern described as the the most important test of the new theory most important test of the new theory

CHAPTER 9

The Bargaining Proble 1949

We hope however to obtain a real understanding of the problele; that is, froy”

- 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 classics ofhis second terreat classics ofhis second ter Problem” is a remarkably down-to-earth work for amathematician Yet no one but a brilliant mathematician could have conceived the idea In the paper, Nash, whose econoraduate course taken at Carnegie, adopted ”an altogether different angle” on one of the oldest proble solution ”The Bargaining Problem” is a remarkably down-to-earth work for amathematician Yet no one but a brilliant mathematician could have conceived the idea In the paper, Nash, whose econoraduate course taken at Carnegie, adopted ”an altogether different angle” on one of the oldest proble solution2 By so doing, he showed that behavior that econoy, and therefore beyond the reach of econo, was, in fact, a, he showed that behavior that econoy, and therefore beyond the reach of econo, was, in fact, ae, the basis of econo has been the stuff of legend since the Levantine kings and the pharaohs traded gold and chariots for weapons and slaves3 Despite the rise of the great impersonal capitalist marketplace, with its millions of buyers and sellers who neverwealthy individuals, powerful governiant corporations - dominates the headlines But two centuries after the publication of Adareat impersonal capitalist marketplace, with its millions of buyers and sellers who neverwealthy individuals, powerful governiant 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 econoain would interact, or how they would split up the pie there were still no principles of econoain would interact, or how they would split up the pie4 The econoain was a reclusive Oxford don, Francis Ysidro Edgeworth, in 18815 Edgeworth and several of his Victorian contemporaries were the first to abandon the historical and philosophical tradition of Smith, Ricardo, and Marx and to atteeworth 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 Philosophers6 Edgeworth was not fascinated with economics because it justified or explained or condeloomy, into the future This odd soul was fascinated by economics because econo that dealt with quantities could be translated into and because anything that dealt with quantities could be translated into ht of people as so nized that the world of perfect competition had ”certain properties peculiarly favorable to mathematical calculation; namely a certain indefinite ous to that infinity and infinitesie 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 competitive fashi+on 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 fore enterprises and cartels His mathematical models captured the results of competition, but the consequences of cooperation proved elusive9 Is it peace or war? asks the lover of ”Maud” of economic competition It is both, pax or pact between contractors during contract, hen some of the contractors without consent of others contract

The first principle of Econoent is actuated only by self-interest The workings of this principle ent 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