Sveriges Riksbank Prize in Economic Sciences /Nobel Prize in Economics: 1994
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 1994 was awarded jointly to John C. Harsanyi, John F. Nash Jr. and Reinhard Selten "for their pioneering analysis of equilibria in the theory of non-cooperative games."
John F. Nash is the only person who has been awarded both the Economic Sciences Nobel Prize and the Abel Prize in Mathematics. He received the Economic Sciences Prize for his work in game theory, more specifically the Nash equilibrium theory. He introduced the distinction between cooperative games in which binding agreements can be made and non-cooperative games where binding agreements are not feasible. He also did ground-breaking work in other mathematics and developed the Nash embedding theorem.
"Through their contributions to equilibrium analysis in non-cooperative game theory, the three laureates constitute a natural combination," noted the Swedish Royal Academy. “Nash provided the foundations for the analysis, while Selten developed it with respect to dynamics and Harsanyi with respect to incomplete information." Nash put forth his key idea—the Nash equilibrium—in the PhD thesis he submitted to the Princeton Mathematics Department in 1950, when he was 22 years old.
[Curator’s note: The following material quotes and paraphrases extensively from articles posted by the Nobel Prize Committee, the Princeton University Office of Communication, Princeton Alumni Weekly, and The Daily Princetonian; see Sources below for details.]
John Forbes Nash Jr. was born on June 13, 1928, in Bluefield, West Virginia. His father and namesake, John Forbes Nash, was an electrical engineer for the Appalachian Electric Power Company. His mother, Margaret Virginia (née Martin) Nash had been a schoolteacher before she was married. Nash attended kindergarten and public school, and he learned from books provided by his parents and grandparents. Nash's parents pursued opportunities to supplement their son's education and arranged for him to take advanced mathematics courses at a local community college during his final year of high school.
He attended Carnegie Institute of Technology from 1945 to 1948 (that later became Carnegie Mellon University) through a full benefit of the George Westinghouse Scholarship, initially majoring in chemical engineering. He switched to a chemistry major and eventually, at the advice of his teacher John Lighton Synge switched to mathematics. After graduating in 1948, with both a B.S. and M.S. in mathematics, Nash accepted a fellowship to Princeton University, where he pursued further graduate studies in mathematics and sciences.
Nash's adviser and former Carnegie professor Richard Duffin wrote a letter of recommendation for Nash's entrance to Princeton stating, "He is a mathematical genius." Nash was also accepted at Harvard University. However, the chairman of the mathematics department at Princeton, Solomon Lefschetz, offered him the John S. Kennedy fellowship, convincing Nash that Princeton valued him more. Further, he considered Princeton more favorably because of its proximity to his family in Bluefield. At Princeton, he began work on his equilibrium theory, later known as the Nash equilibrium.
He did his doctoral work at Princeton in two years. His dissertation thesis was entitled "Non-cooperative Games." In the thesis, Nash defined a new concept of equilibrium and used methods from topology to prove the existence of an equilibrium point for n-person, finite, non-cooperative games. The "n" referred to the number of players; "finite" meant that the number of possible strategies were limited; and "non-cooperative" meant that no communication and therefore no collusion or side payments were allowed between players.
The Nash Equilibrium Point
In the abstract formulation of game theory, a contest consisted of each player choosing a pure strategy, which was a complete plan for every possible situation that he or she might encounter during the course of play. When the pure strategies of all players are submitted to an umpire, the entire course of play and the payoffs to the players is determined. But all games cannot be solved with pure strategies; therefore, players must use a mix of pure strategies by choosing the probabilities with which each pure strategy is played. Thus, in the game of "Matching Pennies," the pure strategies are "heads" or "tails," and the mixed strategies are the random frequencies with which a player chooses to play these pure strategies. Nash proved that there existed at least one set of mixed strategies, with one for each player, a Nash-equilibrium point, such that no player could improve his or her position by changing his or her strategy and therefore that profile of mixed strategies enjoyed an essential property of stability. Although Nash proved the existence of such a profile of mixed strategies, it needed not to be unique.
Economists learned of game theory through the 1944 publication of Theory of Games and Economic Behavior by John yon Neumann and Oskar Morgenstern. Their analysis was largely limited to games involving only two players. When analyzing games with more than two players, they assumed that players would form coalitions (for example, two ganging up against the third), thus reducing the number of players to two. This theory was called "cooperative" game theory.
Nash, however, was able to deal with the general n-player game and to prove that points of equilibrium existed even if there was no cooperation or collusion between the players, which led game theory in a direction that yon Neumann and Morgenstern had not imagined. Nash’s proof of the existence of at least one equilibrium point in this very wide class of non-cooperative games had a major impact on modern economic theory. If we think of economic behavior as a game in which there were well-defined rules, and all the players tried to maximize their payoffs, then in general it would be possible for any given player to improve his or her position by changing his or her strategy. Consequently, players would keep changing their strategies until they reached a Nash equilibrium point at which no player could improve his or her position. This analysis made it possible in some cases to predict the likely strategies that economic actors would adopt in the long run —namely, those at a Nash equilibrium point at which no player could change to improve his or her outcome.
A short version of Nash’s PhD thesis appeared as an announcement in Proceedings of the National Academy of Sciences in 1950; that version with its one-page proof was entitled "Equilibrium points in n-person games." The thesis, revised by Nash was published in 1951 as "Non-cooperative Games" in Annals of Mathematics. Initially, in the thesis, Nash used Brouwer's fixed point theorem (1926) to prove the existence of an equilibrium point. Both the Proceedings announcement and the Annals version use the more general Kakutani's Fixed Point Theorem (1946) in place of Brouwer. Nash credited this simplification of the proof to a suggestion by David Gale, the professor emeritus of mathematics at the University of California, Berkeley, then a graduate student in mathematics at Princeton. Nash’s thesis was supervised by Mathematics Professor Albert Tucker, a topologist turned game theorist, the same year as Nash’s thesis. Tucker created the paradox known as the "Prisoner's Dilemma."
Nash was appointed research assistant and instructor at Princeton in 1950 and worked as a consultant for the Rand Corp. during the summers of 1950 and 1952. In 1951, he was appointed a C.L.E. Moore Instructor in mathematics at the Massachusetts Institute of Technology, where he was promoted to assistant professor in 1953 and associate professor in 1957. He left the Institute in 1959 and returned to Princeton.
Nash returns to Princeton
Since the mid-1960s, Nash lived in the Princeton area. As a visiting research collaborator in the Mathematics Department, he made use of computing and library facilities in a program of independent research. Recipient of the yon Neumann Theory Prize from the Operations Research Society of America, he was a fellow of the Econometric Society.
For the several decades, Nash was a fixture at Fine Hall, holding the position of senior research mathematician at Princeton since 1995. Although Nash did not teach or formally take on students, his continuous presence in the department over 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 at the time.
“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, Sergiu Klainerman, Princeton’s Eugene Higgins Professor of Mathematics, 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. (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 to him it would be very natural for him to talk to you.”
At 86, Nash continued to inspire and work with younger researchers. In the photo at right, Nash (center) is attending a special departmental reception in honor of his winning the Abel Prizes. At left is Michail Rassias. On the right is Yakov Sinai, Princeton professor of mathematics, who was awarded the 2014 Abel Prize.
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.”
Nash and his wife, Alicia are pictured at right 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.
Taxi crash in New Jersey
John Nash, whose front-page New York Times obituary reported that he “was widely regarded as one of the great mathematicians of the 20th century,” was killed with his wife, Alicia, May 23, 2015, in a taxi crash on the New Jersey Turnpike. He was 86. They were returning to their home in Princeton Junction from Norway, where he was a co-recipient of the Abel Prize, a top award for mathematicians 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 Klainerman, 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 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.”
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.