A Beautiful Mind (19 page)

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.

— J
OHN
F. N
ASH,
J
R
., 1993

I
N 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 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 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.

Many mathematicians, most famously the French genius Henri Poincaré, 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 passage from a 1908 essay about the genesis of mathematical discovery, Poincaré writes:
⁸

For fifteen days I struggled to prove that no functions analogous 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 participate 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 passed his generals.
⁹
He wanted, he had told the secretary cockily, 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 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.
¹²
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 cocked 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.”
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 café 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 Nobel autobiography that his ideas
“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 clusters 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.
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.

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.”
¹⁸
Gale recalls Nash’s saying, “I’d call this an equilibrium point.” The idea of equilibrium is that it is a natural resting point that tends to persist. Unlike von Neumann, Gale saw Nash’s point. “Hmm,” he said, “that’s quite a thesis.” Gale realized that Nash’s idea applied to a far broader class of real-world situations than von Neumann’s notion of zero-sum games. “He had a concept that generalized to disarmament,” Gale said later. But Gale was less entranced by the possible applications of Nash’s idea than its elegance and generality. “The mathematics was so beautiful. It was so right mathematically.”