A Beautiful Mind

by Sylvia Nasar

  The essence of von Neumann and Morgenstern’s message was that economics was a hopelessly unscientific discipline whose leading members were busily peddling solutions to pressing problems of the day — such as stabilizing employment — without the benefit of any scientific basis for their proposals.30 The fact that much of economic theory had been dressed up in the language of calculus struck them as “exaggerated” and a failure.31 This was not, they said, because of the “human element” or because of poor measurement of economic variables.32 Rather, they claimed, “Economic problems are not formulated clearly and are often stated in such vague terms as to make mathematical treatment a priori appear hopeless because it is quite uncertain what the problems really are.”33

  Instead of pretending that they had the expertise to solve urgent social problems, economists should devote themselves to “the gradual development of a theory.”34 The authors argued that a new theory of games was “the proper instrument with which to develop a theory of economic behavior.”35 The authors claimed that “the typical problems of economic behavior become strictly identical with the mathematical notions of suitable games of strategy.”36 Under the heading “necessary limitations of the objectives,” von Neumann and Morgenstern admitted that their efforts to apply the new theory to economic problems had led them to “results that are already fairly well known,” but defended themselves by contending that exact proofs for many well-known economic propositions had been lacking.37

  Before they have been given the respective proofs, theory simply does not exist as a scientific theory. The movements of the planets were known long before their courses had been calculated and explained by Newton’s theory… .

  We believe that it is necessary to know as much as possible about the behavior of the individual and about the simplest forms of exchange. This standpoint was actually adopted with remarkable success by the founders of the marginal utility school, but nevertheless it is not generally accepted. Economists frequently point to much larger, more burning questions and brush everything aside which prevents them from making statements about them. The experience of more advanced sciences, for example, physics, indicates this impatience merely delays progress, including the treatment of the burning questions.

  When the book appeared in 1944, von Neumann’s reputation was at its peak. It got the kind of public attention — including a breathless front-page story in The New York Times — that no other densely mathematical work had ever received, with the exception of Einstein’s papers on the special and general theories of relativity.38 Within two or three years, a dozen reviews appeared by top mathematicians and economists.39

  The timing, as Morgenstern had sensed, was perfect. The war had unleashed a search for systematic attacks on all sorts of problems in a wide variety of fields, especially economics, previously thought to be institutional and historical in character. Quite apart from the new theory of games, a major transformation was under way — led by Samuelson’s Foundations of Economic Theory — making economic theory more rigorous through the use of calculus and advanced statistical methods.40 Von Neumann was critical of these efforts, but they surely prepared the ground for the reception of game theory.41

  Economists were actually somewhat standoffish, at least compared to mathematicians, but Morgenstern’s antagonism to the economics profession no doubt contributed to that reaction. Samuelson later complained to Leonard, the historian, that although Morgenstern made “great claims, he himself lacked the mathematical wherewithal to substantiate them. Moreover [Morgenstern] had the irksome habit of always invoking the authority of some physical scientist or another.”42 In Princeton, Jacob Viner, the chairman of the economics department, heaped scorn on the unpopular Morgenstern by saying that if game theory couldn’t even solve a game like chess, what good was it, since economics was far more complicated than chess?43

  It must have become obvious to Nash fairly early on that “the bible,” as The Theory of Games and Economic Behavior was known to students, though mathematically innovative, contained no fundamental new theorems beyond von Neumann’s stunning min-max theorem.44 He reasoned that von Neumann had succeeded neither in solving a major outstanding problem in economics using the new theory nor in making any major advance in the theory itself.45 Not a single one of its applications to economics did more than restate problems that economists had already grappled with.46 More important, the best-developed part of the theory — which took up one-third of the book — concerned zero-sum two-person games, which, because they are games of total conflict, appeared to have little applicability in social science.47 Von Neumann’s theory of games of more than two players, another large chunk of the book, was incomplete.48 He couldn’t prove that a solution existed for all such games.49 The last eighty pages of The Theory of Games and Economic Behavior dealt with non-zero-sum games, but von Neumann’s theory reduced such games formally to zero-sum games by introducing a fictitious player who consumes the excess or makes up the deficit.50 As one commentator was later to write, “This artifice helped but did not suffice for a completely adequate treatment of the non-zero-sum case. This is unfortunate because such games are the most likely to be found useful in practice.”51

  To an ambitious young mathematician like Nash, the gaps and flaws in von Neumann’s theory were as alluring as the puzzling absence of ether through which light waves were supposed to travel was to the young Einstein. Nash immediately began thinking about the problem that von Neumann and Morgenstern described as the most important test of the new theory.

  9

  The Bargaining Problem 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 NEUMANN AND MORGENSTERN, The Theory of Games and Economic Behavior, second edition, 1947

  NASH WROTE HIS FIRST PAPER, one of the great classics 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.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 analysis.

  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 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.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 tradition of physics, writes Robert Heilbroner in 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 quantities and because anything that dealt with quantities could be translated into mathematics.7

  Edgeworth thought of people as so many profit-and-loss calculators
and recognized that the world of perfect competition had “certain properties peculiarly favorable to mathematical calculation; namely a certain indefinite multiplicity and dividedness, analogous 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 competitive 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 competition, but the consequences of cooperation proved elusive.9

  Is it peace or war? asks the lover of “Maud” of economic competition. 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 competition 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 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.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 assumptions.

  Nash’s theory assumes 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.

  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 émigré 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 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.19 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, asserts 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 class 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.22

  Legend, possibly encouraged by Nash himself, soon had it that he’d written the whole paper in Hoselitz’s class — much as Milnor solved the Borsuk problem in knot theory as a homework assignment — and that he had arrived at Princeton with the bargaining paper tucked into his briefcase.23 Nash has since corrected the record.24 But when the paper was published in 1950, in Econometrica, the leading journal of mathematical economics, Nash was careful to retain full credit for the ideas: “The author wishes to acknowledge the assistance 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 Nobel 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. 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!”26

  10

  Nash’s Rival Idea Princeton, 1949–50

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

  — JOHN F. NASH, JR., 1993

  IN THE SUMMER OF 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.

  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 Nas
h 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 basement of the Nassau Inn — as tradition demanded of men who had just passed their generals.3 The mathematicians were growing more boisterous and drunken by the minute. A limerick competition 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 shaggy 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 passage, 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.6 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.