A Beautiful Mind

by Sylvia Nasar

  Williams’s approach to management would have made him very much at home in Silicon Valley today: “Williams had a theory,” recalled his deputy, Alexander Mood, also a former Princetonian. “He believed people should be left alone. He was a great believer in basic research. He was a very relaxed administrator. That’s why people thought the math division was pretty weird.”51 Williams’s letter to von Neumann offering the mathematician a two-hundred-dollar-a-month retainer conveys the man’s style. The letter said, “The only part of your thinking we’d like to bid for systematically is that which you spend shaving: we’d like you to pass on to us any ideas that come to you while so engaged.”52 When Williams first arrived, RAND was a tiny annex inside a mammoth Douglas Aircraft factory where thirty thousand workers punched time cards every day. Williams was the one who freed the mathematicians from the clock and then proceeded to demand coffee and blackboards for his mathematicians, explaining that not providing these would guarantee that none of them would produce anything worthwhile. After RAND and Douglas Aircraft parted company, Williams went further. He insisted that the building be open twenty-four hours a day instead of just between eight and five. He got private offices. He set up coffee stations that had their own special full-time maintenance crew. He mollified the engineers and the Air Force generals, who wondered why the hell the mathematicians had to be allowed to be themselves.

  Everyone soon knew Nash by sight. He roamed the halls incessantly.53 He was usually chewing an empty paper coffee cup that was clamped firmly between his teeth. He would glide through the corridors for hours at a time, frowning, lost in thought, shirt untucked, his powerfully built shoulders hunched forward, his sharp Nixonian nose leading the way. Sometimes he wore a small, ironic smile that suggested some secret amusement not likely to be shared with anyone he might encounter. When he did meet someone he knew, he rarely greeted him by name or even acknowledged his presence unless spoken to first, and then not always. When he wasn’t chewing a coffee cup, he whistled, often the same tune, from Bach’s The Art of the Fugue, over and over again.54

  His legend had preceded him. In the eyes of his new colleagues, Arrow recalled, Nash was “a young genius who could do anything, a guy who liked solving problems.”55 Mathematicians who were struggling with tricky problems quickly learned to collar him by planting themselves squarely in his path. Nash’s curiosity was easily piqued, they discovered, provided that the problem struck him as interesting and the speaker mathematically competent. He was usually more than willing to step around to their offices to look at masses of messy equations on their blackboards.

  Williams’s deputy, Alex Mood, was one of the first to try.56 A gentle giant of a man with a dry wit and easy manner, Mood happened to be oppressed by a problem left over from a first, ill-fated thesis attempt at Princeton before the war. He had found a better derivation of a famous solution, he felt, but his proof was overly long, too complicated, and distressingly inelegant. Could Nash come up with something “shorter, simpler”? Nash listened and stared, frowned and walked away. But the very next day, he was back at Mood’s door with a clever and entirely unanticipated solution. Nash had “sidestepped the whole induction by regarding integers as variables and sending them to revealing limits.” As much as anything else, Mood was charmed by Nash’s style. “When he found a problem,” Mood recalled, “he sat down and started attacking it immediately. He didn’t, like some of his colleagues, browse through the library to see what related stuff had already been done.”

  Williams too was immediately taken with Nash and took him under his wing. He frequently told others that Nash had greater insight into mathematical structure than any mathematician he had ever known, an extraordinary remark from a man who spent the late 1930s in Fine Hajl and was an intimate of von Neumann’s. “He knew which factors of a hundred thousand were the most important,” Williams used to say.57 He liked to describe how Nash would come into an office, stare at a blackboard dense with equations, and stand there silently, meditating. “Then,” Williams would say, “he’d solve the whole thing. He could see the structure.”

  However, Nash mostly kept to himself. He talked about his own research rarely and then only with a select few. When he did, it was not usually because he was looking for help. “It wasn’t so much that he sought advice,” another consultant recalled. “You were a reflecting mirror. He was his own creative object.”58 The only person he regularly sought out at RAND was Shapley, and fairly soon people around the mathematics division started to think of the two as a pair, RAND’s Wunderkinder.

  Still, Nash’s eccentricity soon became fodder for RAND’s gossip mill. “He reinforced RAND’s idea that mathematicians were a bit crazy,” Mood said.59 His office, in which he could rarely be found, was a godawful mess. When he left at the end of that summer he did so without bothering to clean out his desk. The staffer who was saddled with the chore found, among other things, “banana peels. Bank statements for Swiss bank accounts with thousands of dollars in them. One or two hundred dollars in cash. Classified documents. The C-1 isometric embedding paper.”60

  Some people found Nash absurdly childish. He was fond of playing adolescent jokes on his colleagues. Knowing that his whistling irritated one particular musicloving mathematician, who frequently asked him to stop, he once left behind a recording of his whistling on the man’s Dictaphone.61 RAND’s blue-collar police force and maintenance crew found Nash an entertaining subject. They would watch him as he left the building walking north on Fourth Avenue. On several occasions some of them complained to a RAND manager that they had seen Nash tiptoing exaggeratedly along the avenue, stalking flocks of pigeons, and then suddenly rushing forward, “trying to kick ’em.”62

  13

  Game Theory at RAND

  We hope [the theory of games] will work, just as we hoped in 1942 that the atomic bomb would work.

  — ANONYMOUS PENTAGON SCIENTIST to Fortune, 1949

  NASH’S NOVEL IDEA about games with many players had preceded him at RAND by several months. The first version of his elegant proof of the existence of equilibrium for games with many players — two skimpy pages in the November 1949 issue of the National Academy of Sciences proceedings — swept through the white stucco building at Fourth and Broadway like a California brushfire.1

  The biggest appeal of the Nash equilibrium concept was its promise of liberation from the two-person zero-sum game. The mathematicians, military strategists, and economists at RAND had focused almost exclusively on games of total conflict — my win is your loss or vice versa — between two players. Shapley and Dresher’s 1949 review of game theory research at RAND refers to the organization’s “preoccupation with the zero-sum two person game.”2 That preoccupation was natural, given that these were games for which the von Neumann theory was both sound and reasonably complete. Zero-sum games also seemed to fit the problem — nuclear conflict between two superpowers — which absorbed most of RAND’s attention.

  Only it really didn’t. At least some of the researchers at RAND were already chafing at the central assumption of a fixed payoff in such games, Arrow recalled.3 As weapons got ever more destructive, even all-out war had ceased to be a situation of pure conflict in which opponents had no common interest whatever. Inflicting the greatest amount of damage on an enemy — bombing him back to the Stone Age — no longer made any sense, as American strategists realized during the final phase of the campaign against Germany when they decided not to destroy the coal mines and industrial complexes of the Ruhr.4 As Thomas C. Schelling, one of RAND’s nuclear strategists, would put it a decade later,5

  In international affairs, there is mutual dependence as well as opposition. Pure conflict, in which the interests of two antagonists are completely opposed, is a special case; it would arise in a war of complete extermination, otherwise not even in war. Thepossibility of mutual accommodation is as important and dramatic as the element of conflict. Concepts like deterrence, limited war, and disarmament, as well as negotiation, are concern
ed with the common interest and mutual dependence that can exist between participants in a conflict.

  Schelling goes on to say why this is so: “These are games in which, though the element of conflict provides the dramatic interest, mutual dependence is part of the logical structure and demands some kind of collaboration or mutual accommodation — tacit, if not explicit — even if only in the avoidance of mutual disaster.”6

  In 1950, at least the economists at RAND were aware that if game theory were to evolve into a descriptive theory that could be usefully applied to real-life military and economic conflicts, one had to focus on games that allowed for cooperation as well as conflict. “Everybody was already bothered by the zero-sum game,” Arrow recalled. “You’re trying to decide whether to go to war or not. You couldn’t say that the losses to the losers were gains to the winner. It was a troublesome thing.”7

  Military strategists were the first to seize on the ideas of game theory. Most economists ignored The Theory of Games and Economic Behavior and the few that didn’t, like John Kenneth Galbraith writing in Fortune and Carl Kaysen, later director of the Institute for Advanced Study, turn out to have had significant contact with military strategists during the war.8 An article in Fortune in 1949 by John McDonald made it clear that the military hoped to use von Neumann’s theory of games to work out intelligence missions, bombing patterns, and nuclear defense strategy.9 On the lookout for new ideas and with plenty of money to spend, the Air Force embraced game theory with the same enthusiasm with which the Prussian military had embraced probability theory a couple hundred years earlier.10

  Game theory had already made its debut in military planning rooms. It had been used during the war to develop antisubmarine tactics when German submarines were destroying American military transports. As McDonald reported in Fortune:11

  The military application of “Games” was begun early in the last war, some time in fact before the publication of the complete theory, by ASWOEG (Anti-Submarine Warfare Operations Evaluation Group). Mathematicians in the group had got hold of von Neumann’s first paper on poker, published in 1928.

  But von Neumann actually spent his frenetic visits to Santa Monica almost exclusively with the computer engineers and the nuclear scientists.12 His enormous prestige and Williams’s deft salesmanship led to a major concentration on game theory at RAND from 1947 into the 1950s. The hope was that game theory would provide the mathematical underpinning for a theory of human conflict and spread to disciplines other than mathematics. Williams convinced the Air Force to let RAND create two new divisions, economics and social science. By the time Nash arrived, a “trust” of game theory research had grown up at RAND including such game theorists as Lloyd S. Shapley, J. C. McKinsey, N. Dalkey, F. B. Thompson, and H. F. Bohnenblust, such pure mathematicians as John Milnor, statisticians David Blackwell, Sam Karlin, and Abraham Girschick, and economists Paul Samuelson, Kenneth Arrow, and Herbert Simon.13

  Most of the RAND military applications of game theory concerned tactics. Air battles between fighters and bombers were modeled as duels.14 The strategic problem in a duel is one of timing. For each opponent, having the first shot maximizes the chance of a miss. But having the better shot also maximizes the chance for being hit. The question is when to fire. There’s a tradeoff. By waiting a little longer each opponent improves his own chance of scoring a hit, but also increases the risk of being shot down. Such duels can be both noisy and silent. With “silent guns,” the duelist doesn’t know the other has fired unless he is hit. Therefore, neither participant knows whether the other still has a bullet or has fired and missed and is now defenseless.

  A report by Dresher and Shapley summarizing RAND’s game theory research between the fall of 1947 and the spring of 1949 gives the flavor.15 The mathematicians describe a problem of staggered attacks in a bombing mission:

  Problem

  A single intercepter base, having I fighters, is located on a base line. Each fighter has a given endurance. If a fighter, vectored out against a bomber attack, has not yet engaged his original target, then at the option of the ground controller he may be vectored back to engage a second attack.

  The attacker has a stock of N bombers and A bombs. The attacker chooses two points to attack and sends N1 bombers including A1 bomb carriers on the first attack and t minutes later he sends N2 = N – N1 bombers including A2 = A – A1 carriers on the second attack.

  The payoff to the attacker is the number of bomb carriers that are not destroyed by the fighters.

  Solution

  Both players have pure optimal strategies. An optimal strategy of the attacker is to attack both targets simultaneously and distribute the A bomb carriers in proportion to the number of bombers in each attack. An optimum strategy of the defender is to dispatch interceptors in proportion to the number of attacking bombers and not to revector fighters. The value of the game to the attacker will be

  V = max (0,A(1-1/Nk))

  where k is the kill probability of the fighter

  The game Nash had in mind could be solved without communication or collaboration. Von Neumann had long believed that the RAND researchers ought to focus on cooperative games, conflicts in which players have the opportunity to communicate and collaborate and are able “to discuss the situation and agree on a rational joint plan of action, an agreement that is assumed to be enforceable.”16 In cooperative games, players form coalitions and reach agreements. The key assumption is that there’s an umpire around to enforce the agreement. The mathematics of cooperative games, like the mathematics of zero-sum games, is rich and elegant. But most economists, like Arrow, were cool to the idea.17 It was like saying, they thought, that the only hope for preventing a dangerous and wasteful nuclear arms race lay in appointing a world government with the power to enforce simultaneous disarmament. World government, as it happens, was a popular idea among mathematicians and scientists at the time. Albert Einstein, Bertrand Russell, and indeed much of the world’s intellectual elite subscribed to some version of “one worldism.”18 Even von Neumann tipped his hat to the notion, conservative hawk that he was. But most social scientists were dubious that any nation, much less the Soviets, would cede sovereignty to such an extent. Cooperative game theory also seemed to have little relevance to most economic, political, and military problems. As Arrow jokingly put it, “You did have cooperative game theory. But I couldn’t force the other side to cooperate.”19

  By demonstrating that noncooperative games, games that did not involve joint actions, had stable solutions, said Arrow, “Nash suddenly provided a framework to ask the right questions.” At RAND, he added, it immediately led “a lot of people to calculate equilibrium points.”

  News of Nash’s equilibrium result also inspired the most famous game of strategy in all of social science: the Prisoner’s Dilemma. The Prisoner’s Dilemma was partly invented at RAND, some months before Nash arrived, by two RAND mathematicians who responded to Nash’s idea with more skepticism than appreciation of the revolution that Nash’s concept of a game would inspire.20 The actual tale of prisoners used to illustrate the game’s significance was invented by Nash’s Princeton mentor, Al Tucker, who used it to explain what game theory was all about to an audience of psychologists at Stanford.21

  As Tucker told the story, the police arrest two suspects and question them in separate rooms.22 Each one is given the choice of confessing, implicating the other, or keeping silent. The central feature of the game is that no matter what the other suspect does, each (considered alone) would be better off if he confessed. If the other confesses, the suspect in question ought to do the same and thereby avoid an especially harsh penalty for holding out. If the other remains silent, he can get especially lenient treatment for turning state’s witness. Confession is the dominant strategy. The irony is that both prisoners (considered together) would be better off if neither confessed — that is, if they cooperated — but since each is aware of the other’s incentive to confess, it is “rational” for both to confes
s.

  Since 1950, the Prisoner’s Dilemma has spawned an enormous psychology literature on determinants of cooperation and defection.23 On a conceptual level; the game highlights the fact that Nash equilibria — defined as each player’s following his best strategy assuming that the other players will follow their best strategy — aren’t necessarily the best solution from the vantage point of the group of players.24 Thus, the Prisoner’s Dilemma contradicts Adam Smith’s metaphor of the Invisible Hand in economics. When each person in the game pursues his private interest, he does not necessarily promote the best interest of the collective.

  The arms race between the Soviet Union and the United States could be thought of as a Prisoner’s Dilemma. Both nations might be better off if they cooperated and avoided the race. Yet the dominant strategy is for each to arm itself to the teeth. However, it doesn’t appear that Dresher and Flood, Tucker, or, for that matter, von Neumann, thought of the Prisoner’s Dilemma in the context of superpower rivalry.25 For them, the game was simply an interesting challenge to Nash’s idea.