A Beautiful Mind

by Sylvia Nasar

  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.”

  Once again, Gale acted as Nash’s agent. “I said this is a great result,” Gale recalled. “This should get priority.” He told Nash that he was sure that Nash had a brilliant thesis in hand. But he also urged Nash to take credit for the result right away before someone else came up with a similar idea. Gale suggested asking a member of the National Academy of Sciences to submit the proof to the academy’s monthly proceedings. “He was spacey. He would never have thought of doing that,” Gale said recently, “so he gave me his proof and I drafted the NAS note.” Lefschetz submitted the note immediately and it appeared in the November proceedings.19 Gale added later, “I certainly knew right away that it was a thesis. I didn’t know it was a Nobel.”20

  Almost fifty years later, two months before his death, Tucker could not recall getting Nash’s first draft of the thesis, which Nash mailed to him at Stanford, or his own reaction on reading it, other than being surprised that Nash had produced a result so quickly. He was certain, however, that he had not been bowled over. He said: “Whether or not this was of any interest to economists wasn’t known.”21

  Nash used to say that Tucker was “a machine,” implying that Tucker was methodical but unimaginative.22 But, in fact, Nash was quite astute to have chosen him as an adviser. Tucker, a Canadian, Methodist rigidity notwithstanding, possessed a rare willingness to defend unconventional ideas and individuals. A truly fine teacher, he firmly believed that students should choose research topics they felt passionate about, not ones they merely believed would appeal to their professors. A few years later, it was Tucker who convinced another young, offbeat genius who would go on to become one of the fathers of artificial intelligence, Marvin L. Minsky, to drop the mainstream but boring mathematics problem he had chosen as a thesis topic and instead to write on his real passion, the structure of the brain.23 Tucker always claimed that he did little more than sign off on N
ash’s slender, twenty-seven-page dissertation — “There was no essential role played by me,” Tucker said — but he encouraged Nash to get it out quickly and defended its merits within the department.24 Kuhn, who was close to Tucker at the time, later recalled: “The thesis itself was completed and submitted after the persistent urging and counsel of Professor Tucker. John always wanted to add more material, and Tucker had the wisdom to say, ’Get the results out early’ ”25

  Tucker responded to Nash’s first draft by demanding that Nash include a concrete example of his equilibrium idea. He also suggested a number of changes in Nash’s presentation. “I urged him to deal with a particular case rather than only a general case,”26 Tucker said. The recommendation, to his mind, was largely esthetic. “When you deal with the general case you have to deal with sophisticated notation that is very hard to read,” he said.27 Nash responded with a prolonged silence that was in fact a measure of his fury. “He reacted unfavorably, largely by expressing nothing. I didn’t hear from him again for a long time,” Tucker recalled.28

  Nash was actually considering dropping the thesis with Tucker and pursuing another topic, an ambitious problem in algebraic geometry, with Steenrod instead.29 He chose to interpret Tucker’s demands for revisions — along with von Neumann’s coldly dismissive reaction — as signs that the department would not accept his work on game theory for a dissertation. However, Tucker, who could be surprisingly forceful, eventually convinced Nash to stick with his original conception — and to make the requested changes. “Nash had an answer for everything,” he said. “You couldn’t catch him out in a mathematical fault.”30 A May 10 letter to Lefschetz reads: “It is not necessary that I see the revised draft, for he has kept me informed (almost daily) of the progress of the revision.”31 Tucker adds, “I was delighted to notice a pleasant change of attitude in Nash during the course of our long correspondence on his work. He became much more cooperative and appreciative towards the end. I wrote to him like a Dutch uncle, but I suspect you or someone else at the Princeton end had some influence in effecting the change.”32

  The entire edifice of game theory rests on two theorems: von Neumann’s min-max theorem of 1928 and Nash’s equilibrium theorem of 1950.33 One can think of Nash’s theorem as a generalization of von Neumann’s, as Nash did, but also as a radical departure. Von Neumann’s theorem was the cornerstone of his theory of games of pure opposition, so-called two-person zero-sum games. But two-person zero-sum games have virtually no relevance to the real world.34 Even in war there is almost always something to be gained from cooperation. Nash introduced the distinction between cooperative and noncooperative games.35 Cooperative games are games in which players can make enforceable agreements with other players. In other words, as a group they can fully commit themselves to specific strategies. In contrast, in a noncooperative game, such collective commitment is impossible. There are no enforceable agreements. By broadening the theory to include games that involved a mix of cooperation and competition, Nash succeeded in opening the door to applications of game theory to economics, political science, sociology, and, ultimately, evolutionary biology.36

  Although Nash used the same strategic form as von Neumann had proposed, his approach is radically different. More than half of the von Neumann and Morgenstern book deals with cooperative theory. In addition, von Neumann and Morgenstern’s solution concept — something called a stable set — does not exist for every game. By contrast, Nash proved on page six of his thesis that every noncooperative game with any number of players has at least one Nash equilibrium point.

  To understand the beauty of Nash’s result, write Avinash Dixit and Barry Nalebuff in Thinking Strategically, one begins with the notion that interdependence is the distinguishing feature of games of strategy.37 The outcome of a game for one player depends on what all the other players choose to do and vice versa. Games like tic-tac-toe and chess involve one kind of interdependence. The players move in sequence, each aware of the other’s moves. The principle for a player in a sequential-move game is to look ahead and reason back. Each player tries to figure out how the other players will respond to his current move, how he will respond in turn, and so forth. The player anticipates where his initial decision will ultimately lead and uses the information to make his current best choice. In principle, any game that ends after a finite sequence of moves can be solved completely. The player’s best strategy can be determined by looking ahead to every possible outcome. For chess, in contrast to tic-tac-toe, the calculations are too complex for the human brain — or even for computer programs written by humans. Players look a few moves ahead and try to evaluate the resultant positions on the basis of experience.

  Games like poker, on the other hand, involve simultaneous moves. “In contrast to the linear chain of reasoning for sequential games, a game with simultaneous moves involves a logical circle,” write Dixit and Nalebuff. “Although players act at the same time, in ignorance of other players’ current actions, each is forced to think about the fact that there are other players who in turn are similarly aware.”38

  Poker is an example of, ’I think he thinks that I think that he thinks that I think …’ Each must figuratively put himself in the shoes of all and try to calculate the outcome. His own best action is an integral part of the calculation.”

  Such circular reasoning would seem to have no conclusion. Nash squared the circle using a concept of equilibrium whereby each player picks his best response to what the others do. Players look for a set of choices such that each person’s strategy is best for him when all others are playing their best strategies.

  Sometimes one person’s best choice is the same no matter what the others do. That is called a dominant strategy for that player. At other times, one player has a uniformly bad choice — a dominated strategy — in the sense that some other choice is best for him irrespective of what the others do. The search for equilibrium should begin by looking for dominant strategies and eliminating dominated ones. But these are special and relatively rare cases. In most games each player’s best choice does depend on what the others do, and one must turn to Nash’s construct.

  Nash defined equilibrium as a situation in which no player could improve his or her position by choosing an alternative available strategy, without implying that each person’s privately held best choice will lead to a collectively optimal result. He proved that for a certain very broad class of games of any number of players, at least one equilibrium exists — so Tong as one allows mixed strategies. But some games have many equilibria and others, relatively rare ones that fall outside the class he defined, may have none.

  Today, Nash’s concept of equilibrium from strategic games is one of the basic paradigms in social sciences and biology.39 It is largely the success of his vision that has been responsible for the acceptance of game theory as, in the words of The New Palgrave, “a powerful and elegant method of tackling a subject that had become increasingly baroque, much as Newtonian methods of celestial mechanics had displaced the primitive and increasingly ad hoc methods of the ancients.”40 Like many great scientific ideas, from Newton’s theory of gravitation to Darwin’s theory of natural selection, Nash’s idea seemed initially too simple to be truly interesting, too narrow to be widely applicable, and, later on, so obvious that its discovery by someone was deemed all but inevitable.41 As Reinhard Selten, the German economist who shared the 1994 Nobel with Nash and John C. Harsanyi, said: “Nobody would have foretold the great impact of the Nash equilibrium on economics and social science in general. It was even less expected that Nash’s equilibrium point concept would ever have any significance for biological theory.”42 Its significance was not immediately recognized, not even by the brash twenty-one-year-old author himself, and certainly not by the genius who inspired Nash, von Neumann.43

  11

  Lloyd Princeton, 1950

  All mathematicians live in two different worlds. They live in a crystalline world of perfect platonic forms. An ice palace. But they al
so live in the common world where things are transient, ambiguous, subject to vicissitudes. Mathematicians go backward and forward from one world to another. They’re adults in the crystalline world, infants in the real one.

  — S. CAPPELL, Courant Institute of Mathematics, 1996

  AT TWENTY-ONE, Nash the mathematical genius had emerged and connected with the larger community of mathematicians around him, but Nash the man remained largely hidden behind a wall of detached eccentricity. He was quite popular with his professors, but utterly out of touch with his peers. His interactions with most of the men his own age seemed motivated by an aggressive competitiveness and the most cold considerations of self-interest. His fellow students believed that Nash had felt nothing remotely resembling love, friendship, or real sympathy, but as far as they were able to judge, Nash was perfectly at home in this arid state of emotional isolation.

  This was not the case, however. Nash, like all human beings, wanted to be close to someone, and at the beginning of his second year at Princeton he had finally found what he was looking for. The friendship with Lloyd Shapley, an older student, was the first of a series of emotional attachments Nash formed to other men, mostly brilliant mathematical rivals, usually younger. These relationships, which usually began with mutual admiration and intense intellectual exchange, soon became one-sided and typically ended in rejection. The relationship with Shapley foundered within a year, although Nash never completely lost touch with him over the decades to follow — all through his long illness and after he began to recover — when he and Shapley became direct competitors for the Nobel Prize.