PROPERTY OF THE ESTATE OF JOHN F. NASH JR.
John F. Nash Jr.'s Nobel identification badge reading Dr. John F. Nash | Economics with a medallion portrait of Alfred Nobel (1 1/4 x 3 1/2 in.; 31 x 90 mm). [also accompanied by:]
John F. Nash Jr.'s copy of the 1994 Nobel Prize yearbook: Les Prix Nobel The Nobel Prizes 1994: Nobel Prizes, Presentations, Biographies, and Lectures. Stockholm: Norstedts Tryckeri for the Nobel Foundation, 1995. In 8s (9 1/2 x 6 1/4 in.; 242 x 160 mm). Numerous photographic portraits and diagrams. Publisher's white boards.
A Note on The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel
While the Nobel Prize for outstanding work in peace, chemistry, physics, medicine, and literature has been awarded since 1901 with funds provided for in the will of Alfred Nobel, the prize for economics was not created until 1968, and first awarded in 1969. It was established by the Swedish Central Bank (the Sveriges Riksbank), the oldest central bank in the world, in honor of Alfred Nobel, to celebrate the bank’s three hundredth anniversary. Although the Swedish Academy of Sciences elects the winners using the same criteria as the Nobel Prize, the funds for the economics prize (eight million Swedish kronor, approximately $950,000) do not come from Nobel’s endowment, but are provided by the Riksbank.
The design for the medal was created by the Swedish painter, sculptor, and medalist Gunvor Svensson Lundqvist (1916−2009). While she is credited with having made more than a hundred medals and exhibited her work at a number of Fédération Internationale de la Médaille d’Art (FIDEM) Congresses, she is most well known for her modeling of this medal.
The reverse design is a freely rendered interpretation of the emblem of the Royal Swedish Academy of Sciences which was adopted in 1815. The surfaces of the medal are dimpled to create the impression of hammered metal. According to her obituary on the FIDEM website, the medal was created “in a hurry since the Royal Swedish Academy of Sciences had forgotten that they had to make a new medal for the prize. Gunvor Svensson Lundqvist just had a few days to do the modelling.”
Like all Nobel medals struck since 1980, Nash’s is made from 18 carat green gold and plated with 24 carat gold. While the medals for the other five categories are struck to a weight of 175 g, the weight of the medal for the prize for economic sciences is set at 185 g (6.52 ounces).
The Nobel Prize in Economic Sciences is awarded—with those for Physics, Chemistry, Physiology or Medicine, and Literature—on the festival day of the Nobel Foundation, December 10, the anniversary of the death of Alfred Nobel. The award ceremony takes place at the Stockholm Concert Hall, and each Laureate in these five disciplines receives a medal, a diploma, and a document confirming the amount of his or her financial award from the hand of the King of Sweden. John Nash was given his Nobel by Carl XVI Gustaf, who has occupied the throne since his ascension in 1973. (The Nobel Peace Prize is presented on the same day by the Chairman of the Norwegian Nobel Committee at the Oslo City Hall, in the presence of the King of Norway.)
Being named a Nobel Laureate in 1994 undoubtedly altered the course of my father’s life as well as the lives of his family members. Various sources have commented that John Nash was a somewhat pathetic and nearly forgotten figure impaired from many years by mental illness. Certainly there are some elements of truth to these stories, especially in regards to his mental illness. However, as he grew older, his great work in the areas of game theory and mathematics became increasingly recognized and utilized in widely diverse areas of scientific inquiry. His paper on “Non-Cooperative Games” was both the substance of his PhD dissertation at Princeton University and also, in time, with his discovery of what eventually became known as the Nash Equilibrium, the prime component of his recognition with a Nobel Prize in Economics. In addition to the Nobel Prize recognition, my father’s work in pure mathematics has long been greatly appreciated by his peers and was recognized last year with the Abel Prize for “striking and seminal contributions to the theory of nonlinear partial differential equations and its applications to geometric analysis,” as quoted from the Norwegian Academy of Sciences and Letters.
My father was always intensely focused on his work, and as I came to realize later in life, this focus led to a degree of detachment from everyday human affairs. Yet my father was, in his own inimitable way, always there for my half-brother, Johnny Nash, and me. Towards the end of the 1980s, my father’s mental outlook appeared to be changing in both his family interactions and his view of life. Instead of my receiving cryptographically encoded letters in a numerological code, he began to write to me in plain English. It was even possible to receive a direct response to questions—a clear sign of a major change in his mindset.
When my father was awarded the Nobel Prize for his contributions to game theory and its application to modern economics, the recognition fueled his motivation to continue his work as a scientist and mathematician. His acceptance as an important thinker and scientist at the highest levels meant a great deal to him. After becoming a Nobel Laureate, he immediately returned to work in areas of economics, game theory, mathematics, and even cosmology, another of his long-standing scientific interests. As a Nobel Laureate and an elected member of the highest national scientific academies, he was in constant demand for public-speaking engagements with national and foreign governmental and nongovernmental organizations, businesses, and university groups. At 66 years of age, he had no interest in retirement. He continued to be fully occupied completing grant-funded research, publishing, attending meetings and conferences, working with graduate students in mentorship and collaborative capacities, and donating his time to various organizations and causes supporting the mentally ill. These activities naturally led to filling up his passports with travel stamps from countries around the world. The transformation was simply remarkable to witness.
Last year was very traumatic with the shockingly sudden, tragic deaths of my father and stepmother, Alicia. It was unbelievable that two people so very much alive and enjoying some of the best years of their lives could suddenly and undeniably be gone. Although it still feels difficult at times to accept, I am comforted to know that my father and Alicia had twenty truly wonderful years since the time of traveling to Stockholm, Sweden, to accept the Nobel Prize. In an encouraging letter, written shortly after being named for his Nobel, my father told me: “well, everyone has a little luck in their life. …” I was lucky to have been his son and to have known him. My memories of him will continue to amuse and inspire me.
Over the years, my father’s reputation and all of his accomplishments and scientific work became most intimately connected with his magnificent Nobel Prize. Being named a Nobel Laureate marks the recipient for life as a person who has achieved something of greatness on behalf of the world. My father and his entire family cherished his Nobel Prize and the accomplishment it represents. Now, upon my father’s death, the estate is required to conduct a careful accounting by placing a valuation on all personal property including his Nobel Prize. The estate is offering the Nobel Prize for sale because of the impossibility of dividing this physical object. It is also of the utmost importance to both my brother and me, to see our father’s Nobel Prize protected, enjoyed, and possibly shared with a wider audience that appreciates my father’s accomplishments. We believe my father would appreciate and approve of this decision as well.
John D. Stier
The life of John Forbes Nash Jr. was the antithesis of those long-lived racers described in A. E. Housman’s “To an Athlete Dying Young”:
… lads that wore their honours out,
Runners whom renown outran
And the name died before the man.
Nash did indeed win his academic laurels as a youth, but the significance of his work was never completely forgotten. Across a wide spectrum of hard and social sciences, his name, particularly as preserved in the eponym “Nash equilibrium,” gained currency in the decades after the appearance of his path-breaking dissertation, “Non-Cooperative Games” in 1950. But the man—wracked by a paranoid schizophrenia that for the better part of thirty years turned his reality into what he later characterized as dream-like delusions—seemed transformed into an apparition. To many authors citing his publications, and even former colleagues, John F. Nash Jr. was dead.
But remarkably, and in the words of a fellow Nobel Laureate (albeit in Literature), Nash did not simply endure, he prevailed. On his own terms, eventually eschewing medication and other conventional treatments, Nash emerged from his madness to long-deferred personal acclaim, capped by his award of the 1994 Sveriges Risbank Prize in Economic Sciences in Memory of Alfred Nobel for his “pioneering analysis of equilibria in the theory of non-cooperative games”—seminal work that he had largely completed as a graduate student in Princeton’s Department of Mathematics.
Nash shared the Nobel Prize with John C. Harsanyi and Reinhard Selton, economists who developed significant applications of the Nash equilibrium and other of Nash’s concepts of game theory. Receiving the Nobel Prize was a transformative event for Nash, who wrote frankly of his struggles with mental illness in his Nobel biography and whose life inspired the Academy Award-winning film A Beautiful Mind (2001). During an interview at a 2004 meeting of Nobel Laureates in Economic Sciences, Nash acknowledged that the Nobel Prize “had a tremendous impact on my life, more than on the life of most Prize winners because I was in an unusual situation. I was unemployed at the time. … And so I was in a position to be very much influenced by the recognition of my earlier work. … I had become widely known, but in a sense it wasn't officially recognised. I was quoted very frequently in the literature of economics and mathematics, but it's quite different to get official recognition.” One particularly meaningful mark of recognition that Nash received in wake of his Nobel was his appointment, in 1995, as Senior Research Mathematician at Princeton
As a young man, John Nash seemed destined for recognition. A handsome but somewhat aloof prodigy, Nash entered Carnegie Tech (now Carnegie Mellon) as a seventeen-year-old in the autumn of 1945 on a Westinghouse scholarship. He majored initially in chemical engineering, transferring after his first semester to chemistry. But Nash found both disciplines too regimented and circumscribed. He then answered the Siren call of mathematics, which allowed him the freedom to think and learn and understand in the intuitive and individual manner he preferred. Three years later, Nash left Carnegie with bachelor's and master's degrees in mathematics and a succinct letter of recommendation from his faculty advisor, Prof. Richard J. Duffin: "Mr. Nash is nineteen years old and is graduating from Carnegie Tech in June. He is a mathematical genius."
While not as well known as Duffin’s, the letter of recommendation sent to Princeton by John L. Synge, the Head of Carnegie’s Department of Mathematics shows an even better understanding of Nash’s abilities, character, and methodologies: “Mr. Nash is unique in my experience of students. I would rank him among the best I have had, and possibly he is the very best. At first impression, he might appear inferior, since he does not write out his work in polished form, nor does he lecture impressively. However, this external clumsiness is more than compensated by quickness of understanding, originality, and capacity for seeing the inner meaning of an argument, all unrivalled in my experience. While still a junior, he was capable of assimilating the most advanced work available in graduate lectures. On account of his understanding he has been able to take many more courses than an ordinary undergraduate. Consequently his training is very wide, both in pure and applied mathematics. … From the standpoint of character and industry, Mr. Nash is all that can be desired. Also, he has a pleasant personality, and although stubborn in mathematical argument and full of confidence, his pre-eminence among the undergraduates here has not made him conceited.
Princeton and Harvard, among other universities, both offered Nash a fellowship to continue his graduate studies; he chose Princeton for a variety of reasons, including its relative proximity to his hometown. While at Carnegie Tech, Nash had taken an elective course in International Economics, which led to his first published paper, “The Bargaining Problem,” which appeared in Econometrica (1950). “The Bargaining Problem” proposed a definitive solution to a bargaining situation involving two individuals who have the opportunity to collaborate for mutual benefit in more than one way. This idea sparked Nash’s interest in game theory, which was kindled further by John von Neumann and Oskar Morgenstern’s Theory of Games and Economic Behavior (1944).
Nash’s 1950 Princeton dissertation, “Non-Cooperative Games,” introduced what came to be called the Nash equilibrium: the concept of a stable solution to a game involving two or more players which provides all players with their best outcome without adjusting their strategies to account for the other players’ own strategies. Nash’s revolutionary breakthrough was to discover the mechanism through which all competitors in a zero-sum game could benefit without forming coalitions or otherwise cooperating. As Nash’s dissertation (published the following year in Annals of Mathematics) explained, “The notion of an equilibrium point is the basic ingredient of our theory. This notion yields a generalization of the concept of the solution of a two-person zero-sum game. It turns out that the set of equilibrium points of a two-person zero-sum game is simply the set of all pairs of opposing ‘good strategies.’ … [I] shall define equilibrium points and prove that a finite non-cooperative game always has at least one equilibrium point.” The Nash equilibrium is now applied to fields as diverse as economics, banking, defense, politics, evolutionary biology, and even auctions.
After receiving his doctorate, John Nash served as an instructor at Princeton for a year before accepting an appointment as a C. L. E. Moore instructor at MIT in 1951. The Moore instructorships are given to recent PhDs with the potential to conduct significant research in pure mathematics. Nash remained on the mathematics faculty at MIT until his resignation in 1959 due to his deteriorating mental health. While at MIT, he continued to publish in the field of game theory, including “Two-Person Cooperative Games” in Econometrica (1953) and, with others, “Some Experimental n-Person Games” in Decision Processes (1954). In addition to mathematical and game-theory research, Nash taught at MIT and one of his students, Alicia Larde, became his wife in 1957.
It was for this discovery—as well as his introduction of the distinction between cooperative and non-cooperative games—that Royal Swedish Academy of Sciences awarded Nash the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel, some forty-five years after his dissertation was written. The principal reason for this delay is well known from the film A Beautiful Mind, based on Sylvia Nasar’s 1998 biography of the same title. (The title came from a remark by Lloyd Shapley, a fellow Princeton graduate student and Nobel Laureate, who said of Nash, “He was obnoxious. What redeemed him was a keen, beautiful, logical mind.”) Nash himself wrote of his “change from scientific rationality of thinking into the delusional thinking characteristic of persons who are psychiatrically diagnosed as ‘schizophrenic’ or ‘paranoid schizophrenic.’”
For more than three decades, including periods of hospitalization, Nash lived a largely solitary life, trapped by auditory delusions. But he somehow emerged from his mental illness and reintegrated with a world that had largely forgotten him, despite his work being increasingly remembered. In the last two decades of his life Nash became an inspirational advocate for mental health care, even while continuing his mathematical work. Nash and his wife, Alicia, were killed in a taxi accident in May 2015 when they were returning home from the airport after he had received the Abel Prize from the Norwegian Academy of Sciences and Letters.
Nash’s insights into game theory have become so pervasive that they affect our daily lives in many ways, whether or not we are fully cognizant of the operation of the “Nash equilibrium.” And through the film A Beautiful Mind and Nash’s later work in the area of mental health, he has become a symbol of the triumph of the human spirit as well as an icon of twentieth-century mathematical thought. In both his personal and professional lives—difficult as they often were—John Nash truly embodied Alfred Nobel’s stipulation that the prizes awarded in his name be given to “those who ... shall have conferred the greatest benefit on mankind.”
The John F. Nash Jr. Nobel Prize is one of the most complete documentations of a Nobel Prize to ever be offered for sale, including, in addition to the gold medal and its original red morocco case, the accompanying calligraphic diploma with an original watercolor drawing by Bengt Landin, the original box and attaché case for the diploma, the original letters from the Nobel Foundation and the Royal Swedish Academy of Sciences announcing the award and his prize monies, and even Nash’s Nobel lapel identification badge.
John Nash, Game Theory, and the Nobel Prize
by Ken Binmore CBE FBA PhD BSc
Nobel Prizes in chemistry, physics, literature, peace and medicine were first awarded in 1901. Nobel Prizes are awarded annually to at most three individuals in each discipline. The winners receive a substantial sum of money and a gold medal, but the true reward lies in being recognized as the leading researcher in the world within your field.
Sweden’s central bank celebrated its 300th anniversary by establishing a parallel Nobel Memorial Prize in Economic Sciences in 1968. Laureates are chosen—along with the winners of the Nobel Prizes in chemistry and physics—by the Royal Swedish Academy. Famous winners include Paul Samuelson, Ken Arrow, Friedrich von Hayek and Milton Friedman. The 26th Nobel Prize in economics was awarded to John Forbes Nash Jr. in 1994 for his work on game theory, which had by that time become a basic workhorse in economic theory.
Nash shared the prize with the game theorists John Harsanyi and Reinhard Selten, whose work is also superlative, but media attention, then and now, inevitably focused on Nash. The scientific reason lies in the fact that his research on the idea of an equilibrium in games turns out to be fundamental not only in economic applications but also in evolutionary biology and elsewhere. The popular reason lies in a life history that is so extraordinary that it inspired a book and a movie on his life, both called A Beautiful Mind. Few people can have experienced such highs and lows: from a figure of fun on the Princeton campus to a folk hero with a Nobel Prize within one short year. Even his death was dramatic. In 2015, he and his wife, Alicia, were killed when their taxi crashed on the New Jersey Turnpike after returning from a ceremony in Norway, at which he had been awarded the Abel Prize for his mathematical work.
John Nash was born in Bluefield, West Virginia, in 1928. His high school career was idiosyncratic; he was somewhat aloof and disliked regimentation. But nobody doubted that he was a very clever boy. He read a great deal. He played chess. He whistled pieces by Bach. To help out his father, who was an electrical engineer with the Appalachian Power Company, he used the theory of the catenary to calculate how much slack would be needed in hanging the heavy power cables then being used to bring electricity to rural areas.
His mathematical ambition was kindled early by reading Bell’s famous Men of Mathematics. Seeking to emulate such mathematical prodigies as Karl Friedrich Gauss and Blaise Pascal, he found an independent proof of a classic theorem of Pierre de Fermat (whose famous “last theorem” was recently finally proved by Andrew Wiles). It is not surprising that his fellow students, both in school and later at university, found him both odd and arrogant.
Nash went to study chemical engineering at Carnegie Mellon University (then the Carnegie Institute of Technology) in Pittsburgh with a full Westinghouse scholarship. But he found the coursework and teaching methodologies uninteresting, and his math professors encouraged him to switch to mathematics, which he pursued with distinction at both the degree and the masters level at Carnegie.
It is typical of his individualistic genius that he was inspired to create the Nash Bargaining Solution as a result of attending an undergraduate course on International Trade—the only course on economics that he ever attended! Heaven knows what was taught in the course at that time, but Nash could see that traders need to bargain with each other, but no theory of bargaining was offered. Indeed, bargaining was thought to be outside the scope of economics in those days—skill at bargaining supposedly being nothing more than a personality trait and therefore best left to psychologists. It is not clear exactly when Nash finally put together his axiomatic theory of rational bargaining that eventually led to the subject being so absorbed into economics that it now seems almost absurd that there was a time that economists held that rationality had nothing to say on the subject. This was not the subject for which he was awarded his Nobel Prize—the Nash bargaining solution is totally different from a Nash equilibrium—but many Nobel Prizes for economics have been awarded for research less worthy than Nash’s work on bargaining.
He began his graduate work at Princeton in 1948, having come from Carnegie with superlative recommendations. By this time he was tall and handsome in appearance, but retained his odd habits: whistling and pacing, walking off in the middle of conversations. People comment on his fierce ambition and intellectual arrogance, but that was nothing very unusual in those days. This was a time after the Second World War, when nothing seemed impossible for young Americans.
In the mathematics common room located in Princeton’s famous Fine Hall, one could take tea with Albert Einstein or John von Neumann. Here he found people he could admire—people he could seek to emulate and perhaps even outperform. However his first contribution, inspired by the hexagonal tiling in the Fine Hall men’s room, was merely to invent a parlor game nowadays called Hex that was played obsessively in the mathematics department for some time. Hopefully, the Princeton mathematicians played Hex better than Go, which they had learned during the war. Only after the war did they get a chance to pit their wits against a passing Japanese master. He beat them all by a few stones, which they thought was doing well until someone explained that it was considered rude in Japan to win by too much.
John von Neumann was a particularly important person for Nash, although he offered little encouragement. Perhaps he felt the same as Einstein when Nash visited him in his office to tell him how to do physics better. Von Neumann—a polymath who made major contributions in both mathematics and theoretical physics—was important to Nash because he was the creator of game theory . His insight was that the problems raised by strategic parlor games like Hex or Poker are no different in essence from a whole range of situations in which people interact in real life. Drivers maneuvering in heavy traffic are playing a driving game. Bargain hunters bidding on eBay are playing an auctioning game. A firm and a union negotiating next year's wage are playing a bargaining game. When opposing candidates choose their platform in an election, they are playing a political game. The owner of a grocery store deciding today's price for corn flakes is playing an economic game. Adolf Hitler and Josef Stalin played a game that killed off a substantial fraction of the world's population. Nikita Krushchev and Jack Kennedy played a game during the Cuban missile crisis that might have wiped us out altogether. Game theory does not have the answers to all such problems, but it takes us a long way by figuring out how people would play such games if they chose their strategies rationally.
In their monumental Theory of Games and Economic Behavior, von Neumann and Oskar Morgenstern  explain how games can be modeled as mathematical entities about which theorems can be proved. Before this book, economic science had nothing useful to say about either handling risk or making decisions with restricted information. From the point of view of Nash’s Nobel Prize, it is the first half of the book that matters. In this part of the book, von Neumann’s minimax theorem provides a complete rational solution for any two-person zero-sum game. Alice should assume that Bob will guess her strategy accurately and choose his strategy to minimize her reward. She should then choose whatever strategy maximizes her payoff subject to this constraint. The need to keep the opponent guessing in such situations will often result in the players using a “mixed strategy” in which they randomize over their pure strategies. For example, in the playground game in which Alice has to guess which way up Bob has placed a hidden coin, everybody knows that both players should choose heads or tails with equal probability.
Zero-sum games like Poker—in which what one player gains the other necessarily loses—have limited interest. They seldom arise in an economic context because all the parties to a trade expect to gain something by interacting or they wouldn’t be trading at all. So Nash set himself the problem of extending the theory to games in general. The result was a doctoral thesis of 27 pages that he completed at the age of 21—at which age most of us have no more ambition than to avoid failing our next exam. In this thesis, he succeeded in locating the vital philosophical needle in a haystack of irrelevancies that had concealed it from others. We nowadays refer to the principle that he discovered as a Nash equilibrium. This is simply a profile of strategies, one for each player, each of which is simultaneously a best reply to the strategies of the other players. His theorem is that every finite game has at least one Nash equilibrium when mixed strategies are allowed.
Journalists sometimes congratulate Nash on having invented an entirely different theory to von Neumann, not understanding that anyone playing a Nash equilibrium in a two-person zero-sum game will necessarily honor the minimax theorem and vice versa. What is special about two-person zero-sum games is that there is no issue about which of their Nash equilibria—of which there are often many—should be regarded as the rational solution of the game. It is perhaps for this reason that von Neumann was dismissive of Nash’s efforts. Or perhaps he was just responding as most experts respond when it turns out that they have gone astray for some simple reason rather than because their complicated calculations are wrong.
The equilibrium selection problem remains troublesome to this day, but it turns out that one doesn’t need to be able to identify a rational solution to all games in order for the notion of a Nash equilibrium to become basic in economics. Indeed, when it works in practice, it is seldom because the players are rational at all. It works because people commonly play the same games over and over again as they progress through life. Each time they play, they take into account their previous experience and adjust their choice of strategy in the direction of a higher reward. If this process should ever stop, it can only be at a Nash equilibrium, because only then is everybody doing as well as they can given the behavior of the other players. It is for the same reason, that Nash equilibrium has become so important in evolutionary biology. It is somehow typical of Nash’s own progress through life that the editor of Annals of Mathematics to whom Nash sent his thesis for publication should have chosen to publish the whole thesis entire—except for the vital evolutionary interpretation, which he dismissed as “uninteresting.”
By the turn of the century, society had begun to reap the rewards of a theory of economic behavior that doesn’t assume that all markets are perfect. It is nowadays hard to imagine how governments would regulate imperfectly competitive industries without the theory of games. New laws change the rules of the economic game being played. Eventually the players adapt their behavior to the new game. How they will change their behavior is predicted using the idea of a Nash equilibrium. Mechanism design is about choosing the rules of the new game so that the resulting Nash equilibrium is socially optimal. (A third Nobel Prize for game theory was awarded for this subject in 2007. The second was awarded in 2005.)
The most spectacular examples of the use of mechanism design occurred in a series of big-money telecom auctions in which governments sold off the right to the exclusive use of chunks of hugely valuable radio spectrum. One cannot take an auction design off the shelf for this purpose. It would be reckless, for example, to auction radio spectrum as Sotheby’s will auction Nash’s Nobel medal. Each different situation requires a different auction, whose rules are tailored to the underlying economic environment. In the year 2000, I was in the hot seat for the (3G) telecom auction that made a total of $35 billion, for which reason Newsweek magazine described me as the ruthless, poker-playing economist who destroyed the telecom industry. The media would have done better to lay this charge at Nash’s door, although he was neither ruthless nor an economist. (Nor was the telecom industry destroyed by being made to pay more than peanuts for the public assets on which they had previously being making money hand over fist.)
After leaving Princeton, Nash worked as a consultant to the RAND Corporation and as an instructor at MIT. His aim now was to solve some important problem in mathematics with a view to having a shot at the Fields Medal—which was then regarded as the mathematical equivalent of the Nobel Prize. (Nobel did not establish a prize in mathematics, supposedly because his fiancée ran off with a mathematician.) In particular, he solved a problem in differential (smooth) geometry that others thought to be hopeless. He showed that the abstract geometric spaces called Riemann manifolds can be embedded in Euclidean space without any stretching or tearing. This is a problem that isn’t even easy to write down in a precise way, but it is the kind of stuff that gets mathematicians excited. It was perhaps a pity that he was not awarded the recently established Abel Prize just before he died for this work rather than his equally worthy but less glamorous contribution to the theory of partial differential equations.
Niels Abel, incidentally, was a brilliant Norwegian mathematician who showed amongst other great work that it is impossible to write down a formula for the solution of quintic equations in the same way that one can for quadratic equations (and for cubics and quartics). He died largely unknown at the age of 26. Gauss apparently threw Abel’s paper unread into his waste basket, commenting that he had no time for such rubbish. But Nash’s story shows that it isn’t always true that it is a mistake to be too far ahead of your contemporaries if you want your work to be appreciated while you are still alive.
While Nash’s mathematical research was going well, his personal life was degenerating into chaos. Eventually, he was lucky enough to marry Alicia Larde in 1957. He was good-looking and intelligent and so was she, but two years into their marriage with a child on the way, he began finally to fall apart. The voices in his head which he reported to have told him in the past how to solve difficult mathematical problems now began to tell him crazy things—that he was the Emperor of Antarctica and suchlike. He remained unable to cope with life in a normal way for the next forty years or so. It is hard to know how he would have survived had Alicia not turned out to be loyal beyond all reasonable expectation.
In April of 1959 he was eventually diagnosed with paranoid schizophrenia and forcibly hospitalized, not once but several times. Resenting his treatment, he initiated a divorce in 1963, but Alicia took him back into her home in 1970, by which time he had quieted down somewhat. She had some financial help from friends and relatives, and Nash had a small inheritance from his mother, but basically she carried the burden on her own shoulders by working as a computer programmer. He roamed the Princeton campus as a kind of mathematical phantom expounding on computational problems that made sense only to himself. It is not even clear that he was aware that he had become a figure of fun to the younger generation. Age apparently often mitigates schizophrenic illnesses, but a recovery like Nash experienced forty years on is unusual. He believed that he eventually learned to use his rationality to distinguish when his voices made sense and when they did not.
Meanwhile, the world of economics had also moved on. There had been much excitement about von Neumann and Morgenstern’s book in the late 1940s, but disillusionment set in when researchers realized that there were only solid results for zero-sum games. Few people were therefore ready for the new start provided by Nash’s  paper proving the existence of Nash equilibria, although interest was kept alive by the splendid book of Howard Raiffa and Duncan Luce . It was not until the late 1970s that a new generation of economic theorists realized what a treasure had been lying neglected in their own backyard. Ten years later, a complete revolution had substituted Nash’s version of game theory for the nebulous foundations on which economic theory had subsisted in the past. But recognition for Nash was still slow in coming. Even getting him elected to the Econometric Society was strongly opposed by some establishment figures on the grounds that we cannot have mad people in our cozy little club. Similar problems were apparently raised at the Swedish Academy when the question arose of who should be awarded a Nobel Prize for game theory in 1994, and he would undoubtedly not have been chosen as one of the three winners if the committee had not been reliably assured of his recovery, although it is not at all clear why somebody should be denied the recognition they deserve because they have later fallen ill.
The award of the Nobel Prize was a real-life Cinderella story for John and Alicia Nash. Sylvia Nasar’s  biography A Beautiful Mind made him into a folk hero. It was followed by an Oscar-winning movie with the same name, starring Russell Crowe as John and Jennifer Connelly as Alicia. I was present at a dinner in Princeton at which John and Alicia arrived a bit late as a consequence of their attending a showing of the rushes of the movie. We asked them how accurate was the movie, but their delight in being played by such good-looking film stars overcame all their scruples. However, I wouldn’t rely on the scene in the bar in which Russell Crowe explains the relevance of Nash equilibrium to the dating game if you really want to understand what Nash equilibrium is all about.
John and Alicia’s lives were transformed in the years that followed the Nobel Prize. They were now financially secure, and John took pleasure in his celebrity. Nobel Laureates receive many invitations to lecture and Nash was generous with his time and ready to speak to a wide range of audiences. He even spoke in Sweden on the ideas on physics about which Einstein had been dismissive long ago. Some people still found him odd but all arrogance had been leached away. The recognition that came at last from the mathematical world with the award of the Abel Prize must have been particularly sweet. Nash's switchback career therefore ended on a long upbeat note, although I found it hard to empathize when he told me that his intellectual achievements were an adequate compensation for all the accompanying pain and anguish of the past.
Bristol, September 2016
Luce, R. and H. Raiffa, Games and Decisions, Wiley, New York, 1957.
Nasar, S. A Beautiful Mind, Simon and Schuster, New York, 1998.
Nash, J. “The Bargaining Problem”, Econometrica 18 (1950), 155-162.
Nash, J. “Non-Cooperative Games”, Annals of Mathematics 54 (1951), 286-295.
Nash, J. “Two-Person Cooperative Games”, Econometrica 21 (1953), 128-140.
Von Neumann, J. and O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, Princeton, 1944.
About the author:
Ken Binmore is a mathematician turned economist and philosopher. He has held Chairs at London School of Economics, the University of Michigan, and University College London. A range of applied work includes the design of major telecom auctions in many countries across the world. As a consequence of the $35 billion raised by the telecom auction he organized in the UK, he was described by Newsweek magazine as the “ruthless, poker-playing economist who destroyed the telecom industry.” He has contributed to game theory, experimental economics, evolutionary biology and moral philosophy. He currently works in decision theory. His books include Natural Justice (OUP), Does Game Theory Work? (MIT Press), A Very Short Introduction to Game Theory (OUP), Rational Decisions (PUP), and the introduction to John Nash’s Essays on Game Theory (Edward Elgar). He is a Visiting Professor of Economics at the universities of Bristol and Warwick, and a Visiting Professor of Philosophy at LSE. A short CV is available at http://www.homepages.ucl.ac.uk/~uctpa97/
Appendix: Some Personal Recollections of John F. Nash Jr. by Friends and Colleagues
The equilibrium John Nash discovered changed game theory by allowing for stable solutions in games to be found. Prior to his work, many problems couldn’t be solved because the mechanisms for finding a solution were lacking. I spent about a year trying to figure out just what he had done and how he did it. It finally came to me. It was a truly genius and innovative solution, and not too complicated once you understood the whole thing. But it was dang impressive work.
Since his discovery others have found ways to further improve and refine his theory. These are all just added bells and whistles to what was already there. The biggest theme in game theory now is combining it with behavioral economics, such as what Ariel Rubenstein is doing. (Rubenstein taught for a while at Princeton, where he met Nash.) Again, what they are doing would not be possible without the basis of having a Nash equilibrium for the games.
John Nash invented something that is so basic that the rest of the field relies on it. It’s like he invented something as important as multiplication. Without it, the rest of the work could not be done. So his legacy is solid and will be lasting. I think he understood that and felt that in his later years.
Yvan J. Kelly, PhD
Associate Vice President of Academic Affairs
Most of the movie-going public are aware of the first and second acts of John Nash Jr.'s life, first as a brilliant but brash young mathematician at Princeton and MIT, then as a mentally ill recluse haunting the hallways of the Princeton math department. After he regained control of his mind and was awarded the Nobel Prize, Nash was famously the subject of the book and later popular movie A Beautiful Mind.
What most people do not know about is his third act, where at age 70 he applied for and was awarded a grant from the National Science Foundation to study a novel problem he invented at the intersection of economics and game theory. With these funds, he hired me as an undergraduate research assistant to carry out some of the computational and numerical work. Soon after the project began, the movie came out in the theaters; Nash had to change his email address from the one published on the department website, and began receiving mail literally by the bucket-load. In many ways, our relationship was a standard collaboration between a senior scientist and an aspiring young mathematician. Though of course there were quirky aspects as well, such as the time I emailed Nash about some new progress and asked if he wanted to meet on Monday to discuss—he replied that he unfortunately had to attend the Oscars that weekend, but would be back Tuesday afternoon.
It was amazing to witness an elderly man who was still just as thirsty for knowledge as a giddy teenager; in this sense, Nash's third act was much like the first. For example, he came to the department one day with an extra skip in his step, eager to show me something. He had been reading about elliptic curve cryptography and wrote his own program to factor large integers; the program worked, and indeed factored a rather large number. The algorithm was not new, but it was new to him, and brought him a great sense of satisfaction to see it run and work. One cannot but admire a man who has already accomplished so much, yet still finds pleasure in small everyday victories.
Nash kept in frequent touch after I went off to grad school, updating me on progress with the project (he hired two other assistants in succession), meeting me for lunch on my periodic trips to Princeton, and even attending my concerts in town (I played in a Klezmer band, the Klez Dispensers, at the time). In 2008, at age 80, his paper on the project appeared in print. It would be his last publication. Nash was a kind and generous man, and I miss him very much.
Associate Professor, Department of Mathematics
When the Nobel Prize was awarded to my brother, John Nash, it transformed his life. The truth is it gave him a second chance at life. Many people have had much to say about John and all the aspects of his life and his accomplishments. I speak only as his sister.
I wish the award had come earlier so that our parents could have known how he fulfilled their hopes and dreams. We were a small family with devoted parents. Bluefield, West Virginia, was a good place to live and nothing like some of the negative stereotypes of West Virginia. John enjoyed the best of a small city with church, social, and cultural events. As a family, I recall trips and the wonderful things we did together like visiting New York and seeing a Broadway play. We attended public schools and were well prepared to deal with college.
After John went to college, he was not back in Bluefield often but I know he continued to care deeply for his mountain home. He always followed the news from Bluefield and especially the weather reports. Bluefield is called “nature’s air-conditioned city” and lemonade is handed out downtown when the temperature reaches 90⁰. He also spoke on occasion of possibly being buried there someday. One of the first events he chose to attend after being named a Nobel Laureate was the National Youth Science Camp in West Virginia as a special guest.
To me, he is still basically my brother, John, and not some glorified being. John’s brilliant mind made possible the achievements for which he has been honored but I do not doubt that his solid family background supported him along the way.
Martha Nash Legg
I entered MIT in the fall of 1957. MIT had the policy that each professor would teach a class in their specialty as well as teaching an undergraduate class, even a freshman class. I was lucky to have Prof. Nash for freshman calculus. This particular year he taught freshman calc in the fall and honors calc in the spring, as well as his game theory class.
One day before the start of freshman calculus class, one of the chemistry brains in the class brought in liquid nitrogen triiodide which he spread on the floor the full length of the blackboard. Several minutes later Dr. Nash walks in and takes one step: Pow!! Nash swings around in the smallest part of a second imaginable pointing an accusatory finger at the class. “Triiodide?!” he exclaims. It was as if nitrogen triiodide had been completely occupying his thoughts for the prior 10 hours. It was at this point that I knew that I was in another league; this kind of thing would have caused confusion and turmoil and infinite discussion in high school, but here, in the twinkling of an eye, it was all over.
Nash then picked up a book (as it happened a chemistry lab notebook) from a student whose desk was closest at hand. This, by dropping it face down on the floor, he used the book to detonate the remaining triiodide up and down in front of the blackboard: Pow!, Pow!, Bang!, Pop! … Bits of book and vapors exploding. The classroom full of techno-lunatics, starved for the slightest amusement, is near hysteria as Dr. Nash, delicately holding the unlucky lab book at one corner between his thumb and forefinger, its outer covering rubblized with smoking shards hanging down, hands the book back to the hapless, dismayed, if not disgruntled student in the front row. Nash, on seeing the student’s reaction, shrugs his shoulders as if to convey the notion “don’t blame me, it’s not my fault and you might be well advised to choose your classmates with more care in the future.” This did little to bank the general hilarity.
The most striking thing, which came immediately apparent, about having a class with John Nash was his precise use of the English language. I had never heard the language used with such clarity. The precision resulted in great clarity—his lessons were very transparent. The course could have been one in English. The pace of the class was really ideal, not being so fast that little is understood so the only hope was notes, but rather it was possible to actually understand the material right in class, with ample opportunity to ask questions—this was ideal.
At John’s 80th birthday celebration by the economics department of Princeton, I had the great pleasure of meeting his vivacious sister, Martha. I told her that being a student in John’s freshman class at MIT, the thing that strongly struck me was that I had never heard the English language used with such precision before. She explained that his father was an engineer and his mother was an English teacher who also taught Latin and knew German and some other languages—having taken summer courses until she married. Martha related that when they were growing up if she had a math problem that she want solved she would go to John, but if she wanted it explained she would go to her father.
On one occasion at MIT, Prof. Nash announced that we were to have a special treat. His friend Prof. Donald Newman was going to give the proof that pi was irrational. This took all period. I can’t recall if this tour de force was presented using notes or not. Most likely not, since at that time at MIT there was a prevailing opinion that if you used notes to teach a class you weren’t prepared. As I remember, Prof. Nash was sitting in the back row.
On a test subsequent to this class, one of the questions was to prove one of the following was irrational: (a) pi, (b) “e”, or (c) gamma (Euler’s constant). Choosing pi would mean that you wouldn’t have time to do any other problem on the test. Proving “e” irrational wasn’t that bad, as Nash had given a quick proof in one of the classes. Also it was mentioned in class that gamma had never been proven irrational. If you took gamma you could still be working on the quiz 40 years later.
A friend reminded me—“don’t you remember, you had to bring colored pens into Nash’s class to take notes.” The MIT math department was too cheap to have colored chalk, so when Prof. Nash came to class he extracted a box of colored chalk from his jacket pocket, swung his jacket over one of those ugly chairs that MIT had at the teacher’s desk and proceeded to draw careful, beautiful multicolor diagrams when he was teaching complex variables of the Brewer fixed point theorem.
Through the MIT Alumni Club, I had the good fortune to know Alicia and through her to renew acquaintance with John in Princeton. We had many good family gatherings and meals together. Once when we were at dinner, the question came up regarding the incompleteness theorem of Gödel and the assertion that there are some statements that can’t be proved true or false. We asked John for such an example. He immediately replied “This statement is false.” I had seen this referred to in Rebecca Goldstein’s book (Incompleteness: The Proof and Paradox of Kurt Gödel, 2005), but really didn’t appreciate its significance. This was an example of how very close to the surface all these thoughts were. He seemed to be able to give you an answer (very far from a glib answer) without having the need for apparent thinking. Everything seemed to be right there with no need to probe his mind.
It is quite a privilege to be able to see first-hand the workings of an extraordinary mind. If a statement was made, Nash received this information and immediately checked it for logic; how does it fit in with what he knows, and what are the implications of the statement. I would not immediately see the implications, or if I ever did it would be hours or days later, but with him it was essentially immediate. Also, most minds of well above average capability are usually either strongly analytical or they are very creative; usually not both. Nash’s mind had a great analytical capability but also with a strong creative vein.
But Nash was also pragmatic and very human. My son Bob, an alumnus of Seton Hall, arranged to have John give an interview with two professors. After the interview he was surrounded by students like a rock star. When we came back that night after the event I said to him that he was a hero; he said that tomorrow he would be a patient—he had a doctor’s appointment the next day.
And while driving John home from his office one day, we were talking of death. I said it is a mystery of where we came from. John said, from a biological viewpoint, of course, it is not a mystery—but from a philosophical view point it is a different story.
Here's the John Nash I knew. Yes, Nobel Prize winner. Yes, Abel Prize winner. Yes, THE Beautiful Mind. But John was also such a gentle soul, a longtime family friend who was always there for family events to celebrate and participate. I’ll never forget how John once drove me to the train station. We arrived 15 minutes before the train. He insisted on waiting with me until the train came so I didn’t have to wait alone. It was such a thoughtful and paternal act. When I asked John to speak on his latest work, "Ideal Money," to my graduate school he came without hesitation. What an honor to share a piece of his brilliance with my fellow classmates. I won't forget John’s laugh. His eyes and face scrunched up like a little kid when something humored him. His laugh always made me smile too.
Jane Manganaro Fuller
John Forbes Nash is honored and revered not only for his mathematical genius and the seminal contributions he made to game theory, for which he shared the 1994 Nobel Prize in Economics, but also as a person with the mental illness schizophrenia, who made truly extraordinary intellectual advances in an academic discipline before the onset of this usually devastating and lifelong brain disorder. Some 30 years after he first became overtly ill, Professor Nash learned to control his psychotic symptoms, without medication, motivated by his desire to care for himself and his family. I was privileged to spend considerable private and public time with him from 1995 until his accidental death, in 2015, and came to know him as exceptionally compassionate, witty, modest, and engaged in life.
By this remembrance, I hope to both pay tribute to him and to destigmatize schizophrenia, which did not preclude Professor Nash from utilizing his exceptional genius and creativity to produce breakthroughs in social science which are acknowledged to be comparable in significance to the very top discoveries in the biological and physical sciences. I have suggested elsewhere that the thought processes associated with schizophrenia may even have contributed to some of his most important mathematical insights (Meltzer, “John Forbes Nash: A Remembrance,” in Games and Economic Behavior, Nash Memorial Special Issue: In press, 2016).
Herbert Y. Meltzer, MD
Professor of Psychiatry and Behavioral Sciences,
Feinberg School of Medicine, Northwestern University
John had a circle of close friends that numbered about twenty. Almost without exception, whenever he would go on one of his numerous speaking trips, he would send to us from his hotel a group-addressed e-mail, in which he would describe in detail some interesting event (or events) that had occurred during his visit. Without exception he always had a different, unique way of looking at things that was very refreshing and insightful.
One day in the course of conversation with John I said (as perhaps almost any mathematician chosen at random would say), “Andrew Wiles proved the Fermat Conjecture.” Whereupon John very sharply reprimanded me by pointing out that what I had just said was very clearly, totally incorrect. The correct statement would be, “Ken Ribet proved the Fermat Conjecture; Andrew Wiles proved the Shimura-Taniyama Conjecture for semistable elliptic curves.” Of course John was correct, but later I reflected on the fact that he was the only person ever to be so genuinely concerned (upset actually) about what I had said many times to many people, to be moved to take the time and effort to first reprimand me and then to "correct" me on the matter.
John and Alicia were among the nicest people I have known, and I shall miss them greatly.
J. Mozzochi, PhD