2015年5月27日 星期三

John Nash Jr. ; The Triumph (and Failure) of John Nash’s Game Theory

John Nash Jr., renowned for his breakthrough work in mathematics and game theory as well as for his struggle with mental illness, died in an automobile accident May 23. He was 86.

Photo by Robert P. Matthews

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A 'tragic but meaningful' life: Legendary Princeton mathematician John Nash dies

Posted May 27, 2015; 04:30 p.m.
by Morgan Kelly, Office of Communications

John Nash Jr., a legendary fixture of Princeton University's Department of Mathematics renowned for his breakthrough work in mathematics and game theory as well as for his struggle with mental illness, died with his wife, Alicia, in an automobile accident May 23 in Monroe Township, New Jersey. He was 86, she was 82.

During the nearly 70 years that Nash was associated with the University, he was an ingenious doctoral student; a specter in Princeton's Fine Hall whose brilliant academic career had been curtailed by his struggle with schizophrenia; then, finally, a quiet, courteous elder statesman of mathematics who still came to work every day and in the past 20 years had begun receiving the recognition many felt he long deserved. He had held the position of senior research mathematician at Princeton since 1995.

Nash was a private person who also had a strikingly public profile, especially for a mathematician. His life was dramatized in the 2001 film "A Beautiful Mind" in which he and Alicia Nash were portrayed by actors Russell Crowe and Jennifer Connelly. The film centered on his influential work in game theory, which was the subject of his 1950 Princeton doctoral thesis and the work for which he received the 1994 Nobel Prize in economics.

At heart, however, Nash was a devoted mathematician whose ability to see old problems from a new perspective resulted in some of his most astounding and influential work, friends and colleagues said.

Nash and his wife, Alicia (right), who died with him May 23, are pictured at a March 25 reception in honor of Nash's receipt of the 2015 Abel Prize from the Norwegian Academy of Science and Letters, one of the most prestigious honors in mathematics. The couple often attended events in Princeton's Department of Mathematics and were very supportive of undergraduate education. (Photo by Danielle Alio, Office of Communications)

At the time of their deaths, the Nashes were returning home from Oslo, Norway, where John had received the 2015 Abel Prize from the Norwegian Academy of Science and Letters, one of the most prestigious honors in mathematics. The prize recognized his seminal work in partial differential equations, which are used to describe the basic laws of scientific phenomena. For his fellow mathematicians, the Abel Prize was a long-overdue acknowledgment of his contributions to mathematics.

For Nash to receive his field's highest honor only days before his death marked a final turn of the cycle of astounding achievement and jarring tragedy that seemed to characterize his life. "It was a tragic end to a very tragic life. Tragic, but at the same time a meaningful life," said Sergiu Klainerman, Princeton's Eugene Higgins Professor of Mathematics, who was close to John and Alicia Nash, and whose own work focuses on partial differential equation analysis.

"We all miss him," Klainerman said. "It was not just the legend behind him. He was a very, very nice person to have around. He was very kind, very thoughtful, very considerate and humble. All that contributed to his legacy in the department. The fact that he was always present in the department, I think that by itself was very moving. It's an example that stimulated people, especially students. He was an inspiring figure to have around, just being there and showing his dedication to mathematics."

Princeton President Christopher L. Eisgruber said Sunday that the University community was "stunned and saddened by news of the untimely passing of John Nash and his wife and great champion, Alicia."

"Both of them were very special members of the Princeton University community," Eisgruber said. "John's remarkable achievements inspired generations of mathematicians, economists and scientists who were influenced by his brilliant, groundbreaking work in game theory, and the story of his life with Alicia moved millions of readers and moviegoers who marveled at their courage in the face of daunting challenges."

Nash, as shown above in his Graduate School application photo, became associated with Princeton in 1948 when he began his doctoral studies under the late professor Albert Tucker. For the past several decades, Nash was a fixture at Fine Hall, holding the position of senior research mathematician at Princeton since 1995. (Photo courtesy of Princeton University Archives)

Although Nash did not teach or formally take on students, his continuous presence in the department over the past several decades, coupled with the almost epic triumphs and trials of his life, earned him respect and admiration, said David Gabai, the Hughes-Rogers Professor of Mathematics and department chair.

"John Nash, with his long history of achievements and his incredible battle with mental health problems, was hugely inspirational," Gabai said. "It's a huge loss not to have him around anymore."

Gabai said the Nashes regularly attended department events such as receptions, special teas and special dinners, and they also were very supportive of undergraduate education and regularly attended undergraduate events. Gabai, who was with the couple in Norway when John received the Abel Prize, likened their deaths to the department losing two family members.

Even in the 1970s when Nash, still struggling with mental illness, was an elusive presence known as the "Phantom of Fine Hall," his reputation for bravely original thinking motivated aspiring mathematicians, said Gabai, who was a Princeton graduate student at the time. Nash's creativity helped preserve the department's emphasis on risk-taking and exploration, he said.

"In those days, he was very present, but rarely said anything and just wandered benignly through Fine Hall. Nevertheless, we all knew that the mathematics he did was really spectacular," Gabai said. "It went beyond proving great results. He had a profound originality as if he somehow had insights into developing problems that no one had even thought about.

"I think he prided himself that he had his way of thinking about things," Gabai continued. "He was such an extraordinary exemplar of the things that this department strives for. Beyond great originality, he demonstrated tremendous tenacity, courage and fearlessness."

Since winning the Nobel Prize, Nash had entered a long period of renewed activity and confidence — which coincided with Nash's greater control of his mental state — that allowed him to again put his creativity to work, Klainerman said. He met Nash upon joining the Princeton faculty in 1987, but his doctoral thesis had made use of a revolutionary method introduced by Nash in connection to the Nash embedding theorems, which the Norwegian Academy described as "among the most original results in geometric analysis of the twentieth century."

"When he got the Nobel Prize, there was this incredible transformation," Klainerman said. "Prior to that we didn't realize he was becoming normal again. It was a very slow process. But after the prize he was like a different person. He was much more confident in himself."

During their frequent talks in recent years, Nash would offer unique perspectives on numerous topics spanning mathematics and current events, Klainerman said. "Even though his mind wasn't functioning as it did in his youth, you could tell that he had an interesting point of view on everything. He was always looking for a different angle than everybody else. He always had something interesting to say."

Nash's life was dramatized in the 2001 film "A Beautiful Mind," which centered on his influential work in game theory. The subject of his 1950 Princeton doctoral thesis, the work earned Nash the 1994 Nobel Prize in economics. Nash is pictured above at a 1994 press conference following his receipt of the Nobel Prize. Seated next to Nash is the late Princeton mathematics professor Harold Kuhn, a lifelong friend of Nash who was central to having Nash's work recognized by the Nobel Prize committee. (Photo by Denise Applewhite, Office of Communications)

Nash's quick and distinctive mind still shone in his later years, said Michail Rassias, a visiting postdoctoral research associate in mathematics at Princeton who was working with Nash on the upcoming book, "Open Problems in Mathematics." He and Nash had just finished the preface of their book before Nash left for Oslo. They agreed upon a quote from Albert Einstein that resonated with Nash (although Nash pointed out that Einstein was a physicist, not a mathematician, Rassias said): "Learn from yesterday, live for today, hope for tomorrow. The important thing is not to stop questioning."

"Even at 86, his mind was still open," Rassias said. "He still wanted to have new ideas. Of course, he couldn't work like when he was 20, but he still had this spark, the soul of a young mathematician. The fact that he moved slowly and talked with a quiet voice had nothing to do with the enthusiasm with which he did mathematics. It was very inspirational."

Sixty years younger than Nash, Rassias said his work with Nash began with a conversation in the Fine Hall commons room in September.

"I could tell there was mathematical chemistry between us and that led to this intense collaboration. He was very simple, very open to discussing ideas with new people if you said something that attracted his interest," Rassias said. "Nash gave this impression that he was distant, but when you actually had the opportunity to talk to him he was not like that. He tended to walk alone, but if you got the courage to talk him it would be very natural for him to talk to you."

At 86, Nash (center) continued to inspire and work with younger researchers such as Michail Rassias (left), a Princeton visiting postdoctoral research associate in mathematics who was working on an upcoming book with Nash. Although 60 years Nash's junior, Rassias admired Nash's quick thinking and curiosity. This photo, taken during a special departmental reception in honor of Nash's winning the Abel Prize, includes Yakov Sinai (right), a Princeton professor of mathematics, who was awarded the 2014 Abel Prize. (Photo by Danielle Alio, Office of Communications)

Rassias has been inspired by the enthusiasm and willingness with which a person of Nash's stature dedicated months of his time to working with a young mathematician. It was an example Rassias hopes to emulate during his own career.

"Remembering what John Nash did for me, I will definitely try to give all my heart and soul to younger people in all steps of their careers," Rassias said. "I also will try to keep my mind and enthusiasm for math alive to the end. That is something I will try to achieve like him."

Born in Bluefield, West Virginia, in 1928, Nash received his doctorate in mathematics from Princeton in 1950 and his graduate and bachelor's degrees from Carnegie Institute of Technology (now Carnegie Mellon University) in 1948.

His honors included the American Mathematical Society's 1999 Leroy P. Steele Prize for Seminal Contribution to Research and the 1978 John von Neumann Theory Prize. Nash held membership in the National Academy of Sciences and in 2012 was an inaugural fellow of the American Mathematical Society.

Nash is survived by his sister, Martha Nash Legg, and sons John David Stier and John Charles Martin Nash. He had his younger son, John Nash, with Alicia shortly after their marriage in 1957, which ended in divorce in 1963. They remarried in 2001.

Despite their divorce, Alicia, who was born in El Salvador in 1933, endured the peaks and troughs of Nash's life alongside him, Klainerman said. Their deaths at the same time after such a long life together of highs and lows seemed literary in its tragedy and romance, he said.

"They were a wonderful couple," Klainerman said. "You could see that she cared very much about him, and she was protective of him. You could see that she cared a lot about his image and the way he felt. I felt it was very moving.

"Coming home from Oslo, he must have been extremely happy, and she must have been extremely happy for him," he continued. "They went for the apotheosis of his career, and died in this terrible way on the way back. But they were together."

A memorial service for Nash will be planned at the University in the fall.


The Triumph (and Failure) of John Nash’s Game Theory


The mathematician John Nash, in Beijing, China, in 2011.CREDITPHOTOGRAPH BY CHINAFOTOPRESS/GETTY

Thanks to the sterling efforts of Sylvia Nasar, Ron Howard, and Russell Crowe, many people are aware that John Nash, the Princeton mathematician who was killed over the weekend in a car crash on the New Jersey Turnpike, lived a remarkable life. It included early academic stardom, decades of struggling with schizophrenia, and, in 1994, a shared Nobel Memorial Prize in Economic Sciences. But outside the field of economics, Nash’s contribution to game theory, for which he was awarded the Nobel, remains rather less well understood.

Although it is often used in economics, game theory can be applied to any venue where people, or other decision makers, interact strategically and follow rules-based behavior. The setting could be nuclear negotiations, such as the ones currently taking place between Iran and the great powers. It could be a product market, in which a number of firms compete for business. Or it could be a political campaign, in which various candidates try to outdo each other. The word “strategically” is important, because the various players, in choosing from a variety of possible moves, take account of one another’s actions, or likely actions. And the phrase “rules-based” means that the players are acting purposefully and seeking to maximize their own advantages, rather than behaving passively, or randomly.

On one level, Nash’s contribution to game theory was highly mathematical, and, ultimately, somewhat trivial. That is how his intellectual rival at Princeton, John von Neumann, reputedly described it back in 1949, anyway, and he had a point. In co-authoring the 1944 magnum opus “Theory of Games and Economic Behavior,” von Neumann had virtually invented a new subject, complete with its own language. Nash, in diverting from his studies in pure mathematics to this nascent field, showed that in a certain class of games a certain set of outcomes exists: those outcomes are now called “Nash equilibria.”

Many of Nash’s fellow mathematicians were more impressed by his work in algebraic geometry. Over time, though, the game-theoretical methods he pioneered became widely used in the social sciences, and especially in economics. Indeed, in a 2004 article for the National Academy of Sciences that reviewed the genesis and development of Nash-based game theory, the economists Charles Holt and Alvin Roth noted, “Students in economics classes today probably hear John Nash’s name as much as or more than that of any economist.”

To understand why that is, you need to know a bit about the history of economics. Before game theory was invented, economists had a workable account of the dynamics of competitive markets with many buyers and sellers, such as the markets for grain and other commodities. This was the theory of supply and demand, which Alfred Marshall and others developed. Economists also had a workable theory of how the economy as a whole operates: Keynesian economics. But in studying the dynamics of industries where a handful of businesses compete against one another, or how corporations respond to regulation, or how bidders in an auction decide how much to bid, they hadn’t made much progress.

Enter von Neumann and his co-author, Oskar Morgenstern, who provided an intellectual framework for analyzing such situations: game theory. But despite the monumental nature of their achievement, von Neumann and Morgenstern succeeded in showing that definitive solutions, or “equilibria,” existed for only a fairly narrow category of interactions: so-called zero-sum games, in which one person’s gain is another person’s loss. (Poker is a zero-sum game; so is coin tossing.) Often in real-world situations, though, such as how to divide a market among a few competitors, there is a positive economic surplus to be divided: the question is who gets what, and that depends on which actions (or “strategies”) are adopted.

This is where Nash came in. He started out by defining a particular solution to games—one marked by the fact that each player is making out the best he or she (or it) possibly can, given the strategies being employed by all of the other players. Then, applying a deep-mathematical theory that had been developed earlier by the Dutch mathematician L. E. J. Brouwer, Nash demonstrated that such an equilibrium exists in any game with a finite number of players and a finite number of moves to choose from.

It was this derivation that von Neumann dismissed as trivial: analytically, Brouwer’s “fixed point theorem” did much of the work for Nash. But the lasting importance of Nash’s contribution wasn’t the existence proof, it was the idea of a “Nash equilibrium,” or, as it is sometimes called, a best-response equilibrium. Over time, this concept would become almost as familiar in economics textbooks as supply-and-demand curves.

The reason is its broad applicability, which extends well beyond economics. Take, for example, the problem of deciding which side of the road to drive on—a question that clearly involves trying to figure out what everybody else will do. If you are living in the United States, where custom and law dictate using the right lane, sticking to that lane is a Nash equilibrium: it gives you the best chance of getting to your destination in one piece. And since the same logic applies to everybody else, the “stay on the right” solution is pretty stable.

Like the intersection of a supply curve and a demand curve, the concept of a Nash equilibrium appeared to pick out a distinct point where things would inevitably end up. Indeed, once you grasped the idea, it was hard to see how an outcome that wasn’t a Nash equilibrium could be sustained for very long in the absence of coercion or misinformation. If there were a better response available, such as driving down the median or zigzagging from left to right, at least some of the players would eventually adopt it. And that would mean that the original solution wasn’t a stable solution at all.

In the nineteen-sixties, seventies, and eighties, economic theorists worked on extending Nash’s approach. At the same time, however, it became clear that his concept of equilibrium has some serious drawbacks that limit its usefulness. To start with, there is often more than one best-response equilibrium, and, in some cases, there is a very large (or even infinite) number of them. For a methodology that is designed to pick out particular solutions, this non-uniqueness property is a serious problem, especially since there is usually no obvious way of deciding which Nash equilibrium will end up being selected.

To return to the driving example, a moment’s reflection should persuade you that driving on the left can also be a best-response equilibrium. In the United Kingdom and many other countries, it’s the one that has been adopted and enshrined into law. But why do Americans drive on the right and Brits on the left? And if we were starting out from scratch, which convention would be adopted: left or right? Nash, and the many economists who have followed in his footsteps over the past sixty-five years, can’t necessarily provide an answer.

This is just one of the drawbacks of Nash’s approach. Another problem is that game theory is mentally taxing. In many games, including some that initially seem pretty simple, finding the Nash equilibria can be very difficult, at least for ordinary mortals. And when the rules of the game aren’t clear, or when some information is hidden, or when the passage of time is introduced into the analysis, even seasoned game theorists sometimes have a hard time figuring things out. To be sure, various refinements of the Nash equilibrium can be called upon to deal with some of these complications, and there are also refinements of the refinements—but they are even more complicated.

The long and the short of it is that if the purpose of economic theories is to predict which of many possible outcomes will occur, Nash’s methodology often isn’t much help—a point acknowledged by David Kreps, an economic theorist at Stanford, back in 1990. But asking any theory in the social sciences to correctly predict the future is a very demanding requirement. And asking that it accomplish this task across a wide range of areas, such as the ones to which Nash’s approach has been applied, is surely too much.

That’s partly because Nash-influenced game theory isn’t actually a testable scientific theory at all. It is an intellectual tool—a way of organizing our thoughts systematically, applying them in a consistent manner, and ruling out errors. Like Marshallian supply-and-demand analysis or Bayesian statistics, it can be applied to many different problems, and its utility depends on the particular context. But while appealing to the Nash criteria doesn’t necessarily give the correct answer, it often rules out a lot of implausible ones, and it usually helps pin down the logic of the situation.

For these reasons, studying game theory, and learning how to recognize a Nash equilibrium, are highly worthwhile exercises. Once you learn the basics, it is amazing how broadly they can be applied. (Much of the advanced stuff can be safely skipped.) For example, in writing a book about the economics of the financial crisis and trying to figure out why so many people on Wall Street and elsewhere did things that ultimately blew up in their faces, I relied heavily on the Prisoner’s Dilemma, a simple game involving a particular type of Nash equilibrium that shows how certain incentive schemes can promote self-destructive behavior.

My experience wasn’t out of the ordinary. These days, political scientists, evolutionary biologists, and even government regulators are obliged to grasp best-response equilibria and other aspects of game theory. Whenever a government agency is considering a new rule—a set of capital requirements for banks, say, or an environmental regulation—one of the first questions it needs to ask is whether obeying the rules leads to a Nash equilibrium. If it doesn’t, the new policy measure is likely to prove a failure, because those affected will seek a way around it.

John Nash, in writing his seminal 1951 article, “Non-Cooperative Games,” which was published in The Annals of Mathematics, surely didn’t predict any of this. He was then a brilliant young mathematician who saw some interesting theoretical problems in a new field and solved them. But one thing led to another, and it was he, rather than von Neumann, who ended up as an intellectual celebrity, the subject of a Hollywood movie. Life, as Nash discovered in tragic fashion, often involves the unexpected. Thanks to his work, though, we know it is possible to impose at least some order on the chaos.

John Cassidy has been a staff writer at The New Yorker since 1995. He also writes acolumn about politics, economics, and more, for newyorker.com.