A Beautiful Mind (25 page)

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.
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For them, the game was simply an interesting challenge to Nash’s idea.

The very afternoon that Dresher and Flood learned of Nash’s equilibrium idea, they ran an experiment using Williams and a UCLA economist, Armen Alchian, as guinea pigs.
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Poundstone says that Flood and Dresher “wondered if real people playing the game — especially people who had rtever heard of Nash or equilibrium points — would be drawn mysteriously to the equilibrium strategy. Flood and Dresher doubted it. The mathematicians ran their experiment one hundred times.”

Nash’s theory predicted that both players would play their dominant strategies, even though playing their dominated strategies would have left both better off. Though Williams and Alchian didn’t always cooperate, the results hardly resembled a Nash equilibrium. Dresher and Flood argued, and von Neumann apparently agreed, that their experiment showed that players tended not to choose Nash equilibrium strategies and instead were likely to “split the difference.”

As it turns out, Williams and Alchian chose to cooperate more often than they chose to cheat. Comments recorded after each player decided on strategy but before he learned the other player’s strategy show that Williams realized that players ought to cooperate to maximize their winnings. When Alchian didn’t cooperate, Williams punished him, then went back to cooperating next round.

Nash, who learned of the experiment from Tucker, sent Dresher and Flood a note — later published as a footnote in their report — disagreeing with their interpretation:
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The flaw in the experiment as a test of equilibrium point theory is that the experiment really amounts to having the players play one large multi-move game. One cannot just as well think of the thing as a sequence of independent games as one can in zero-sum cases. There is too much interaction… . It is really striking however how inefficient [Player One] and [Player Two] were in obtaining the rewards.
One would have thought them more rational.

Nash managed to solve a problem at RAND that he and Shapley had both been working on the previous year. The problem was to devise a model of negotiation
between two parties — whose interests neither coincided nor were diametrically opposed — that the players could use to determine what threats they should use in the process of negotiating. Nash beat Shapley to the punch. “We all worked on this problem,” Martin Shubik later wrote in a memoir of his Princeton experiences, “but Nash managed to formulate a good model of the two-person bargain utilizing threat moves to start with.”
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Instead of deriving the solution axiomatically — that is, listing desirable properties of a “reasonable” solution and then proving that these properties actually point to a unique outcome — as he had in formulating his original model of bargaining, Nash laid out a four-step negotiation.
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Stage One: Each player chooses a threat. This is what Til be forced to do if we can’t make a deal, that is, if our demands are incompatible. Stage Two: The players inform each other of the threats. Stage Three: Each player chooses a demand, that is, an outcome worth a certain amount to him. If the bargain doesn’t guarantee him that amount, he won’t agree to a deal. Stage Four: If it turns out that a deal exists that satisfies both players’ demands, the players get what they ask for. Otherwise, the threats have to be executed. It turns out that the game has an infinite number of Nash equilibria, but Nash gave an ingenious argument for selecting a unique stable equilibrium that coincides with the bargaining solution he previously derived axiomatically. He showed that each player had an “optimal” threat, that is, a threat that ensures that a deal is struck no matter what strategy the other player chooses.

Nash initially wrote up his results in a RAND memorandum dated August 31, 1950, suggesting that he managed to finish the paper just before leaving RAND for Bluefield.
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A longer and more descriptive version of the paper was eventually accepted by
Econometrica,
which had published “The Bargaining Problem” that April. Accepted for publication sometime during the following academic year, “Two Person Cooperative Games” did not in fact appear until January 1953.
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It was Nash’s last significant contribution to the theory of games.

Nobody at RAND solved any big new problems in the theory of noncooperative games. For all intents and purposes, Nash stopped working in the field in 1950. The dominant thrust of game theory at RAND came from the mathematicians, particularly Shapley, and they were guided less by applications than by the mathematics themselves. During the 1950s Shapley focused on cooperative games, which were necessarily of limited interest not only to economists but also to military strategists.

The justification of all mathematical models is that, oversimplified, unrealistic, and even false as they may be in some respect, they force analysts to confront possibilities that would not have occurred to them otherwise. The history of physics and medicine abounds with wrong or incomplete theories that throw just enough light to allow some other big breakthroughs. The atom bomb, for example, was built before physicists understood the structure of particles.

The most significant application of game theory to a military problem grew
straight out of the theory of duels and helped shape what was probably RAND’s single most influential strategic study. The study was the brainchild of Al Wohlstetter, a mathematician who joined RAND’s economics group in early 1951, about six months after Nash joined the mathematics group.

According to Kaplan, the SAC operational plan in the early 1950s-was to fly bombers from the United States to overseas bases and then to mobilize and launch an attack against the Soviet Union from there.
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The Air Force’s whole deterrence strategy was based on the idea of the power of the H-bomb and America’s ability to respond in kind to any attack. Apparently, no one before Wohlstetter had focused on vulnerability to a first strike aimed, not at American cities, but at wiping out the SAC force, then concentrated in a small number of foreign bases within striking distance of the Soviet Union. Kaplan writes:

Up to that point, most military applications of game theory had focused on tactics — the best way to plan a fighter-bomber duel, how to design bomber formations or execute anti-submarine warfare campaigns. But Wohlstetter would carry it further. It was this insistence on figuring out one’s own best moves in light of the enemy’s best moves that provoked Wohlstetter to look at a map and to conclude that the closer we are to them, the closer they are to us — the easier it is for us to hit them, the easier it is for them to hit us. Wohlstetter and his team estimated that a mere 120 bombs … could destroy 75 to 85 percent of the B-47 bombers while they casually sat on overseas bases. The SAC, seemingly the most powerful strike force in the world, was appearing to be so vulnerable in so many ways that merely putting the plan into action … created a target so concentrated that it invited a pre-emptive attack from the Soviet Union.
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Wohlstetter’s study had an electrifying effect on the Air Force establishment. With its focus on American vulnerability and the temptation of a Soviet surprise attack, the study also rationalized a paranoia in the military establishment that seeped into the body politic and wound up as national hysteria over the supposed “missile gap” in the second half of the 1950s. The RAND report, Fred Kaplan writes, “legitimized a basic fear of the enemy and the unknown through mathematical calculation and rational analysis, providing the techniques and the general perspective through which the new and rather scary situation — the Soviet Union’s acquisition of long range nuclear weapons — could be discussed and acted upon.”
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The golden age at RAND, from the point of view of the mathematicians, strategic thinkers, and economists, was already coming to a close.
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After a time, RAND’s sponsors grew less enthusiastic about pure research, less tolerant of idiosyncrasies, and more demanding. Mathematicians got bored and frustrated with game theory. Consultants stopped coming and permanent staffers drifted to universities. Nash never returned after the summer of 1954. Flood left for Columbia University in
1953. Von Neumann, who in any case had played a very small role in the group after inspiring it, dropped his RAND consultancy in 1954 when he accepted an appointment as a member of the Atomic Energy Commission.

Game theory, in any case, was going out of vogue at RAND. R. Duncan Luce and Howard Raiffa concluded in their 1957 book,
Games and Decisions: “We have
the historical fact that many social scientists have become disillusioned with game theory. Initially there was a naive band-wagon feeling that game theory solved innumerable problems of sociology and economics, or that, at least it made their solution a practical matter of a few years’ work. This has not turned out to be the case.”
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The military strategists were of the same mind. “Whenever we speak of deterrence, atomic blackmail, the balance of terror … we are evidently deep in game theory,” Thomas Schelling wrote in 1960, “yet formal game theory has contributed little to the clarification of these ideas.”
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14
The Draft
Princeton, 1950–51

N
EITHER THE PROSPECT
of playing military strategist, nor living in Santa Monica, nor earning a handsome salary tempted Nash to accept Williams’s offer of a permanent post at the think tank. Nash shared little of RAND’s camaraderie or sense of mission. He wanted to work on his own and to have the freedom to roam all over mathematics. To do that, he would have to obtain a faculty position at a leading university.

For the moment, he planned to spend the upcoming academic year in Princeton. Tucker had arranged for his support by assigning him to teach a section of undergraduate calculus
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and making him a research assistant on his Office of Naval Research grant.
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In fact, Nash intended to devote most of his energy to his own research and to looking for an academic opening for the following fall. But before he could turn to these matters, he was forced to confront an immediate threat to his career plans, namely, the Korean War.

North Korea had invaded the South on June 25, 1950, about the time that Nash was flying to Santa Monica.
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A week later Truman promised to send American troops to repel the invasion. The first reinforcements landed July 19. By July 31, Truman had issued an order to the Selective Service to call up one hundred thousand young men right away, twenty thousand immediately. A week or two later, John Sr. and Virginia wrote that Nash might be in imminent danger of being drafted. Like most Republicans, they disliked Truman and had their doubts about the war. They urged Nash to come to Bluefield as soon as practical to talk with members of the local draft board personally to sound them out about a II-A. Surely, they said, Nash was more valuable at RAND or at Princeton than in uniform.

When Nash left RAND at the very end of August, he flew from Los Angeles to Boston and spent a day at the world mathematical congress, which was meeting in Cambridge.
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He presented his algebraic manifolds result to a small audience there, a nice distinction for a young mathematician. But he was anxious to get back to Bluefield and didn’t stay for most of the meetings.

He was determined to do all he could to avoid the draft. With a war on, even an unpopular and undeclared war, who knew how long he would have to serve? Any interruption of his research could jeopardize his dream of joining a top-ranked
mathematics department. Returning World War II veterans had flooded the job market and enrollments were falling because of the draft. In two years there would be another crop of brilliant youngsters clamoring for the handful of instructorships. His game theory thesis had been greeted with a mix of indifference and derision by the pure mathematicians, so his only hope of a good offer, he felt, was to finish his paper on algebraic manifolds.

Besides, he had no wish to become part of someone else’s larger design and dreaded the thought of military life — his hawkish instincts and southern background notwithstanding. He had been one of the few boys at Beaver High who hadn’t prayed for World War II to last long enough so that he would have a chance to serve. Life in the army, with its mindless regimentation, stultifying routines, and lack of privacy, revolted him, and he had heard enough stories from other mathematicians to dread being herded together with the kind of rude, uneducated young men whose company he had been only too happy to escape when he left Bluefield for Carnegie Tech.

Nash proceeded methodically. Once back in Bluefield, he called on two members of the board, including its chairman, a retired attorney named T. H. Scott, whom he later described as “a rock-ribbed Republican (Truman = moron = Roosevelt),” and a Dr. H. L. Dickason, the president of Bluefield State, a black junior college on the far side of the town.
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He made it his business to find out as much as he could about the men who would be deciding his fate. As it turned out, the board had only a fuzzy sense of what Nash was doing. Until he showed up at the Peery building, they had no idea that he had already received his doctorate and had assumed he was returning to Princeton that fall as a student. His student deferment had not yet been canceled.