A Beautiful Mind

by Sylvia Nasar

  The prize itself was a long-overdue acknowledgment by the Nobel committee that a sea change in economics, one that had been under way for more than a decade, had taken place. As a discipline, economics had long been dominated by Adam Smith’s brilliant metaphor of the Invisible Hand. Smith’s concept of perfect competition envisions so many buyers and sellers that no single buyer or seller has to worry about the reactions of others. It is a powerful idea, one that predicted how free-market economies would evolve and gave policymakers a guide for encouraging growth and dividing the economic pie fairly. But in the world of megamergers, big government, massive foreign direct investment, and wholesale privatization, where the game is played by a handful of players, each taking into account the others’ actions, each pursuing his own best strategies, game theory has come to the fore.7

  After decades of resistance — Paul Samuelson used to joke about “the swamp of 77–person game theory”8 — a younger generation of theorists began using game theory in areas from trade to industrial organization to public finance in the late 1970s and early 1980s.9 Game theory opened up “terrain for systematic thinking that was previously closed.” Indeed, as game theory and information economics have become increasingly entwined, markets traditionally seen as fitting the purely competitive mold have increasingly been studied using game-theory assumptions. The latest generation of texts used in top graduate schools today all recast the basic theories of the firm and the consumer, the foundation of economics, in terms of strategic games.10

  “Concepts, terminology and models from game theory have come to dominate many areas of economics,” said Avinash Dixit, an economist at Princeton who uses game theory in work on international trade and is the author of Thinking Strategically. “At last we are seeing the realization of the true potential of the revolution launched by von Neumann and Morgenstern.”11 And because most economic applications of game theory use the Nash equilibrium concept, “Nash is the point of departure.”12

  The revolution has gone far beyond research journals, experimental laboratories at Caltech and the University of Pittsburgh, and classrooms of elite business schools and universities. The current generation of economic policymakers — including Lawrence Summers, undersecretary of the treasury, Joseph Stiglitz, chairman of the Council of Economic Advisers, and Vice-President Al Gore — are steeped in the stuff, which, they say, is useful for thinking about everything from budget proposals to Federal Reserve policy to pollution cleanups.

  The most dramatic use of game theory is by governments from Australia to Mexico to sell scarce public resources to buyers best able to develop them. The radio spectrum, T-bills, oil leases, timber, and pollution rights are now sold in auctions designed by game theorists — with far greater success than that of earlier policies.13’

  Economists like Nobel Laureate Ronald Coase have advocated the use of auctions by government since the 1950s.14 Auctions have long been used in markets where sellers of unusual items — from vintage wines to movie rights — have no idea what bidders are willing to pay. Their basic purpose is to make bidders reveal how much they value the item. But the arguments of Coase and others were stated in abstract, entirely theoretical terms, and little thought was given to how such auctions would actually be conducted. Congress remained skeptical.

  Before 1994, Washington simply gave away licenses for free. Until 1982, it had been up to regulators to decide which companies deserved the -licenses. Needless to say, the process was dominated by political pressures, outrageously expensive paperwork, and long delays. The pace of licensing lagged hopelessly behind market shifts and new technologies. After 1982, Washington awarded licenses using lotteries, with the winners free to resell licenses. Although the reform did speed up the granting of licenses, the process was still hugely inefficient — and unfair. Bidders with no intention of operating an actual telephone business spent millions to get into the game for the purpose of reaping a windfall. Further, although telephone companies were forced to pay the costs of obtaining licenses, Washington (and taxpayers) did not get the benefit of any revenues. There had to be a better way.

  A young generation of game theorists, including Paul Milgrom, John Roberts, and Robert Wilson at the Stanford B-school, came up with that better way.15 Their chief contribution consisted of recognizing, as Milgrom said, that “the mere design of some auction wasn’t enough… . [G]etting the auction design right was also critically important”16 In particular, they concluded that the most obvious auction designs — auctioning licenses one by one in sequence using simultaneous sealed bids — was the way least likely to succeed in getting licenses into the hands of corporations that could use them best — Washington’s stated objective.

  Game theorists treat an auction like a game with rules and try to evaluate how a given set of rules, taken together, is apt to affect the bidders’ behavior. They take stock of the options the rules allow, the payoffs to the bidders associated with the options, and bidders’ expectations about their competitors’ likely choices.

  Why did these economists conclude that traditional auction formats would not work? Mainly because the value of each individual license to a user depends — as is the case with a Rembrandt or a Picasso — on what other licenses the user is able to obtain. Some licenses are perfect substitutes for one another. That would be the case for similar spectrum bands to provide a given service. But others are complements. That would be the case for licenses to provide paging services in different parts of the country.

  “To permit the efficient license assignment, an auction must allow bidders to consider various packages of licenses, combining complements and switching among substitutes during the course of the auction. Designing an auction to allow this is quite difficult,” writes Paul Milgrom, one of the economists who designed the FCC auction of which Gore was speaking.17

  A second source of complexity, Milgrom says, is that the purpose of the licenses is to create businesses for new services with unknown technology and unknown consumer demand. Since bidders’ opinions are bound to be wildly divergent, it is possible that license assignment would depend more on bidders’ optimism than on their ability to create a desired service.18 Ideally, an auction design can minimize that problem.

  As Congress and the FCC inched closer to the notion of auctioning off spectrum rights, Australia and New Zealand both conducted spectrum auctions.19 That they proved to be costly flops and political disasters illustrated that the devil really was in the details. In New Zealand, the government ran a so-called second price auction, and newspapers were full of stories about winners who paid far below their bids. In one case, the high bid was NZ$7 million, the second bid NZ$5,000, and the winner paid the lower price. In another, an Otago University student bid NZ$1 for a television license in a small city. Nobody else bid, so he got it for one dollar. The government expected the cellular licenses to fetch NZ$240 million. The actual revenue was NZ$36 million, one-seventh of the advance estimate. In Australia, a botched auction, in which parvenu bidders pulled the wool over the government’s eyes, delayed the introduction of pay television by almost a year.

  The FCC’s chief economist was an advocate of auctions, but no game theorists were involved in the first stage of the FCC auction design. The theorists’ phones started ringing only by accident after the FCC issued a tentative proposal for an auction format with dozens of footnotes to the theoretical literature on auctions.20 That was how Milgrom and his colleague Robert Wilson, leading auction theorists, got into the game.

  Milgrom and Wilson proposed that the FCC adopt a simultaneous, multipleround auction.21 In a simultaneous auction, a bunch of licenses are sold at the same time. Multiple rounds mean that, after the first round of bidding, prices are announced, and bidders have a chance to withdraw or raise one another’s bids. This is repeated round after round until the auction is over. The chief advantage of this format is that it allows bidders to take account of interdependencies among licenses. Just as sequential, closed-bid auctions let sellers disco
ver what bidders are willing to pay for individual items, the simultaneous, ascending-bid auction lets them discover the market value of different groupings of items.

  This early proposal — which the FCC eventually adopted — did not cover seemingly small but critical details.22 Should there be deposits? Minimum bid increments? Time limits? Should the bidding system be wholly computerized or executed by hand? and so forth. Milgrom, Roberts, and another game theorist, Preston McAfee, an adviser to AirTouch, provided proposals on these issues. The FCC hired another game theorist, John McMillan, of the University of California at San Diego, to help evaluate the effect of every proposed rule. According to Milgrom, “Game theory played a central role in the analysis of the rules. Ideas of Nash equilibrium, rationalizability, backward induction, and incomplete information, though rarely named explicitly, were the real basis of daily decisions about the details of the auction process.”23

  By late spring 1995, Washington had raised more than $10 billion from spectrum auctions. The press and the politicians were ecstatic. Corporate bidders were largely able to protect themselves from predatory bidding and were able to assemble an economically sensible set of licenses. It was, as John McMillan said, “a triumph for game theory.”24

  50

  Reawakening Princeton, 1995–97

  Mathematics is a young man s game. Yet it is not bearable to contemplate a brief distinction and burgeoning of activity... followed by a lifetime of boredom.

  — NORBERT WIENER

  ON THE AFTERNOON of the Nobel announcement, after the press conference, a small champagne party was in progress in Fine Hall. Nash made a short speech.1 He was not inclined to give speeches, he said, but he had three things to say. First, he hoped that getting the Nobel would improve his credit rating because he really wanted a credit card. Second, he said that one is supposed to say that one is really glad he is sharing the prize, but he wished he had won the whole thing because he really needed the money badly. Third, Nash said that he had won for game theory and that he felt that game theory was like string theory, a subject of great intrinsic intellectual interest that the world wishes to imagine can be of some utility. He said it with enough skepticism in his voice to make it funny.

  All the Swedes’ fears — not to mention Harold Kuhn’s own private worries — about how Nash would cope with the pomp in Stockholm proved groundless. Everything went swimmingly. The receptions. The press briefings. The Nobel award ceremony itself. The lecture in Uppsala afterward. Indeed, in the weeks between the announcement of the prize and the ceremony, Nash did and felt things that had lain beyond his grasp for decades. When he first arrived in Stockholm, Jorgen Weibull recalled, he behaved pretty much as Weibull had remembered from Princeton a few years before: “He didn’t look you in the eye. He mumbled. Socially he was very tentative, very uncertain. But his mood went up from day to day. He got less and less unhappy.”2

  Harold Kuhn, who was to lead a Nobel seminar honoring Nash’s work, and his wife Estelle accompanied Nash and Alicia to Stockholm.3 It was exhilarating. The nicest moment of the week, so full of grand scenes and ceremonies, came when Nash had his much-dreaded private audience with the King. By tradition, the King spends a couple of minutes alone with each Laureate. When Nash’s turn came, he grimaced and frowned so much that Harold was afraid he might refuse to go into the King’s chambers at the last minute, but finally he followed the aide inside.

  Five minutes passed, then seven. Finally, after a full ten minutes, Nash emerged, looking relaxed, even amused. “What did you talk about?” everybody asked at once. Quite a bit, it turned out. In 1958, John told Harold and Estelle, he and Alicia had taken a grand tour of Europe and had driven up into the south of Sweden in their new Mercedes 180. The King had been a student in Uppsala then, addicted to fast sports cars. Around that time, the Swedes were shifting from driving on the left to driving on the right. Nash and the King had spent ten minutes chatting about the pitfalls of driving fast on the lefthand side of the road.

  At dusk, Nash and Weibull were riding in a limousine through the countryside north of Stockholm. The farmhouses were lighting up one at a time, the sky was beginning to glimmer. Nash reached over to Weibull and said, “Look, Jorgen. It’s so beautiful.”4

  They were on their way back from Uppsala where Nash had given a talk — his first in three decades.5 Nash hadn’t been asked to give the customary hour-long Nobel lecture in Stockholm. The lecture at the University of Uppsala was arranged by Christer Kiselman.6 Nash’s chosen topic was a problem that had interested him before his illness and that he had taken up again since his remission: developing a mathematically correct theory of a non-expanding universe that is consistent with known physical observations. The conventional view, of course, is that the universe is expanding, and attempting to overturn the consensus is exactly the kind of contrarian intellectual bet that Nash has always enjoyed.

  Nash’s talk on “the possibility that the universe isn’t expanding” began with tensor calculus and general relativity — stuff so difficult that Einstein used to say he understood it only in moments of exceptional mental clarity. Though he later confessed to nervousness, he spoke without notes, clearly and convincingly, according to Weibull, who has a doctorate in physics7 Physicists and mathematicians in the audience said afterward that Nash’s ideas were interesting, made sense, and were expressed with the appropriate degree of skepticism.

  It is a quiet life, despite the fairytale of Stockholm and the lofty status of Laureate. The Nashes still live in the Insulbrick house with the hydrangeas out front, next to the alley and across from the Princeton train station. There is a new boiler, a new roof, a few new items of furniture, but that’s about it. (Nash was also able to pay down his half of the mortgage.) The few friends they see regularly, among them Jim Manganaro, Felix and Eva Browder, and of course Armand and Gaby Borel, are pretty much the people they have been seeing for some years. Their daily routines have changed less than one might think, dominated as they are by the twin needs of earning a living and caring for Johnny. Alicia takes the train to Newark every day. Nash, who no longer drives, rides the “Dinky” into town, eats lunch at the Institute, and spends the afternoons in the library or, on rare occasions, in his new office. Very often, when Johnny is not in the hospital or on the road, he takes Johnny with him.

  It is a life resumed, but time did not stand still while Nash was dreaming. Like Rip Van Winkle, Odysseus, and countless fictional space travelers, he wakes to find that the world he left behind has moved on in his absence. The brilliant young men that were are retiring or dying. The children are middle-aged. The slender beauty, his wife, is now a mature woman in her sixties. And there is his own seventieth birthday fast approaching.

  There are days when he feels that he has escaped the ravages of time, when he believes he can pick up where he left off, when he feels “like a person who wants to do the research he might have done in his 30s and 40s at the delayed time of his 60s and 70s!” In his Nobel autobiography, he writes:

  Statistically, it would seem improbable that any mathematician or scientist, at the age of 66, would be able through continued research efforts to add to his or her previous achievements. However, I am still making the effort, and it is conceivable that with the gap period of 25 years of partially deluded thinking providing a sort of vacation, my situation may be atypical. Thus I have hopes of being able to achieve something of value through my current studies or with any new ideas that come in the future.8

  But many days he is not able to work. As Nash once told Harold Kuhn, “The Phantom was not in until very late, after 6:00 P.M. because even a Phantom can have ordinary human problems and need to go to a doctor.”9 And there are other days when he discovers an error in his calculations or learns that a promising idea has already been mined by someone else, or when he hears of new experimental data that seem to make certain speculations of his seem less interesting.

  On such days, he is full of regrets. The Nobel cannot restore what has b
een lost. For Nash, the primary pleasure in life had always come from creative work rather than from emotional closeness to other people. Thus, recognition for his past achievements, while a balm, has also cast a harsh light on the vexing issue of what he is capable of doing now. As Nash put it in 1995, getting a Nobel after a long period of mental illness was not impressive; what would be impressive is “persons who AFTER a time of mental illness achieve a high level of mental functioning (and not just a high level of social respectability.)”10

  Nash gave the starkest assessment of his own situation in front of an audience of psychiatrists to whom he had been introduced as “a symbol of hope.” In answer to a question at the end of his 1996 Madrid lecture, he said, “To recover rationality after being irrational, to recover a normal life, is a great thing!” But then he paused, stepped back, and said in a far stronger, more assertive voice: “But maybe it is not such a great thing. Suppose you have an artist. He’s rational. But suppose he cannot paint. He can function normally. Is it really a cure? Is it really a salvation? … I feel I am not a good example of a person who recovered unless I can do some good work,” adding in a wistful, barely audible whisper, “although I am rather old.”11

  These thoughts were much in Nash’s mind when he turned down an offer of thirty thousand dollars from the Princeton University Press in 1995 to publish his collected works. “Psychologically I have a problem since I have been, unfortunately, a long time without any publications,” he said to Harold Kuhn. He was saying, in short, that he doesn’t want to close the door on future work by acknowledging that his lifetime oeuvre is complete.