Princeton University Ph.D.

A less obvious type of application (of non-cooperative games) is to the study of cooperative games. By a cooperative game we mean a situation involving a set of players, pure strategies, and payoffs as usual; but with the assumption that the players can and will collaborate as they do in the von Neumann and Morgenstern theory.

John Forbes Nash Jr. was an American mathematician who made fundamental contributions to game theory, differential geometry, and the study of partial differential equations. Nash's work has provided insight into the factors that govern chance and decision-making inside complex systems found in everyday life.

His theories are widely used in economics. Serving as a senior research mathematician at Princeton University during the later part of his life, he shared the 1994 Nobel Memorial Prize in Economic Sciences with game theorists Reinhard Selten and John Harsanyi. In 2015, he also shared the Abel Prize with Louis Nirenberg for his work on nonlinear partial differential equations. John Nash is the only person to be awarded both the Nobel Memorial Prize in Economic Sciences and the Abel Prize.

In 1959, Nash began showing clear signs of mental illness, and spent several years at psychiatric hospitals being treated for paranoid schizophrenia. After 1970, his condition slowly improved, allowing him to return to academic work by the mid-1980s. His struggles with his illness and his recovery became the basis for Sylvia Nasar's biography, A Beautiful Mind, as well as a film of the same name starring Russell Crowe as Nash.

On May 23, 2015, Nash and his wife Alicia died in a car crash while riding in a taxi on the New Jersey Turnpike.

## Early life and education

Nash was born on June 13, 1928, in Bluefield, West Virginia. His father, 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. He was baptized in the Episcopal Church. He had a younger sister, Martha born November 16, 1930.

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 which 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, to mathematics. After graduating in 1948 at age 19 with both a B.S. and M.S. in mathematics, Nash accepted a scholarship to Princeton University, where he pursued further graduate studies in mathematics.

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.

## Major contributions

### Game theory

Nash earned a PhD in 1950 with a 28-page dissertation on non-cooperative games.

The thesis, written under the supervision of doctoral advisor Albert W. Tucker, contained the definition and properties of the Nash equilibrium, a crucial concept in non-cooperative games. It won Nash the Nobel Memorial Prize in Economic Sciences in 1994.

Publications authored by Nash relating to the concept are in the following papers :

- Nash, John Forbes 1950. "Equilibrium Points in N-person Games". Proceedings of the National Academy of Sciences of the United States of America. 36 1: 48–49. Bibcode:1950PNAS...36...48N. doi:10.1073/pnas.36.1.48. MR 0031701. PMC 1063129. PMID 16588946.
- Nash, John Forbes 1950. "The Bargaining Problem" PDF. Econometrica. Econometric Society. 18 2: 155–62. doi:10.2307/1907266. JSTOR 1907266. MR 0035977.
- Nash, John Forbes 1951. "Non-cooperative Games". Annals of Mathematics. Princeton, New Jersey: Princeton University. 54 2: 286–95. doi:10.2307/1969529. JSTOR 1969529. MR 0043432.
- Nash, John Forbes 1953. "Two-person Cooperative Games" PDF. Econometrica. Econometric Society. 21 1: 128–40. doi:10.2307/1906951. JSTOR 1906951. MR 0053471.

### Other mathematics

Nash did groundbreaking work in the area of real algebraic geometry:

- Nash, John Forbes 1952. "Real algebraic manifolds". Annals of Mathematics. 56 3: 405–21. doi:10.2307/1969649. JSTOR 1969649. MR 0050928. See "Proc. Internat. Congr. Math". AMS. 1952: 516–17. Cite journal requires |journal= help

His work in mathematics includes the Nash embedding theorem, which shows that every abstract Riemannian manifold can be isometrically realized as a submanifold of Euclidean space. He also made significant contributions to the theory of nonlinear parabolic partial differential equations and to singularity theory.

Mikhail Leonidovich Gromov writes about Nash's work:

Nash was solving classical mathematical problems, difficult problems, something that nobody else was able to do, not even to imagine how to do it. ... But what Nash discovered in the course of his constructions of isometric embeddings is far from 'classical' — it is something that brings about a dramatic alteration of our understanding of the basic logic of analysis and differential geometry. Judging from the classical perspective, what Nash has achieved in his papers is as impossible as the story of his life ... [H]is work on isometric immersions ... opened a new world of mathematics that stretches in front of our eyes in yet unknown directions and still waits to be explored.

John Milnor gives a list of 21 publications.

In the Nash biography A Beautiful Mind, author Sylvia Nasar explains that Nash was working on proving Hilbert's nineteenth problem, a theorem involving elliptic partial differential equations when, in 1956, he suffered a severe disappointment. He learned that an Italian mathematician, Ennio de Giorgi, had published a proof just months before Nash achieved his. Each took different routes to get to their solutions. The two mathematicians met each other at the Courant Institute of Mathematical Sciences of New York University during the summer of 1956. It has been speculated that if only one had solved the problem, he would have been given the Fields Medal for the proof.

In 2011, the National Security Agency declassified letters written by Nash in the 1950s, in which he had proposed a new encryption–decryption machine. The letters show that Nash had anticipated many concepts of modern cryptography, which are based on computational hardness.

## Mental illness

Although Nash's mental illness first began to manifest in the form of paranoia, his wife later described his behavior as erratic. Nash seemed to believe that all men who wore red ties were part of a communist conspiracy against him. He mailed letters to embassies in Washington, D.C., declaring that they were establishing a government. Nash's psychological issues crossed into his professional life when he gave an American Mathematical Society lecture at Columbia University in early 1959. Originally intended to present proof of the Riemann hypothesis, the lecture was incomprehensible. Colleagues in the audience immediately realized that something was wrong.

He was admitted to McLean Hospital in April 1959, staying through May of the same year. There, he was diagnosed with paranoid schizophrenia. According to the Diagnostic and Statistical Manual of Mental Disorders, or DSM, a person suffering from the disorder is typically dominated by relatively stable, often paranoid, fixed beliefs that are either false, over-imaginative or unrealistic, and usually accompanied by experiences of seemingly real perception of something not actually present. Further signs are marked particularly by auditory and perceptional disturbances, a lack of motivation for life, and mild clinical depression.

In 1961, Nash was admitted to the New Jersey State Hospital at Trenton. Over the next nine years, he spent periods in psychiatric hospitals, where he received both antipsychotic medications and insulin shock therapy.

Although he sometimes took prescribed medication, Nash later wrote that he did so only under pressure. After 1970, he was never committed to a hospital again, and he refused any further medication. According to Nash, the film A Beautiful Mind inaccurately implied he was taking the new atypical antipsychotics of the time period. He attributed the depiction to the screenwriter who was worried about the film encouraging people with the disorder to stop taking their medication.

Nash felt psychotropic drugs were overrated and that the adverse effects were not given enough consideration once someone was deemed mentally ill. According to Sylvia Nasar, author of the book A Beautiful Mind, on which the movie was based, Nash recovered gradually with the passage of time. Encouraged by his then former wife, de Lardé, Nash lived at home and spent his time in the Princeton mathematics department where his eccentricities were accepted even when his condition was poor. De Lardé credits his recovery to maintaining "a quiet life" with social support.

Nash dated the start of what he termed "mental disturbances" to the early months of 1959, when his wife was pregnant. He described a process of change "from scientific rationality of thinking into the delusional thinking characteristic of persons who are psychiatrically diagnosed as 'schizophrenic' or 'paranoid schizophrenic'". For Nash, this included seeing himself as a messenger or having a special function of some kind, of having supporters and opponents and hidden schemers, along with a feeling of being persecuted and searching for signs representing divine revelation. Nash suggested his delusional thinking was related to his unhappiness, his desire to feel important and be recognized, and his characteristic way of thinking, saying, "I wouldn't have had good scientific ideas if I had thought more normally." He also said, "If I felt completely pressureless I don't think I would have gone in this pattern". He did not draw a categorical distinction between schizophrenia and bipolar disorder. Nash reported he did not hear voices until around 1964, and later engaged in a process of consciously rejecting them. He further stated he was always taken to hospitals against his will. He only temporarily renounced his "dream-like delusional hypotheses" after being in a hospital long enough to decide he would superficially conform — to behave normally or to experience "enforced rationality". Only gradually on his own did he "intellectually reject" some of the "delusionally influenced" and "politically oriented" thinking as a waste of effort. By 1995, however, even though he was "thinking rationally again in the style that is characteristic of scientists," he said he felt more limited.

Nash wrote in 1994:

I spent times of the order of five to eight months in hospitals in New Jersey, always on an involuntary basis and always attempting a legal argument for release. And it did happen that when I had been long enough hospitalized that I would finally renounce my delusional hypotheses and revert to thinking of myself as a human of more conventional circumstances and return to mathematical research. In these interludes of, as it were, enforced rationality, I did succeed in doing some respectable mathematical research. Thus there came about the research for "Le problème de Cauchy pour les équations différentielles d'un fluide général"; the idea that Prof. [Heisuke] Hironaka called "the Nash blowing-up transformation"; and those of "Arc Structure of Singularities" and "Analyticity of Solutions of Implicit Function Problems with Analytic Data".

But after my return to the dream-like delusional hypotheses in the later 60s I became a person of delusionally influenced thinking but of relatively moderate behavior and thus tended to avoid hospitalization and the direct attention of psychiatrists.

Thus further time passed. Then gradually I began to intellectually reject some of the delusionally influenced lines of thinking which had been characteristic of my orientation. This began, most recognizably, with the rejection of politically oriented thinking as essentially a hopeless waste of intellectual effort. So at the present time I seem to be thinking rationally again in the style that is characteristic of scientists.

## Recognition and later career

In 1978, Nash was awarded the John von Neumann Theory Prize for his discovery of non-cooperative equilibria, now called Nash Equilibria. He won the Leroy P. Steele Prize in 1999.

In 1994, he received the Nobel Memorial Prize in Economic Sciences along with John Harsanyi and Reinhard Selten as a result of his game theory work as a Princeton graduate student. In the late 1980s, Nash had begun to use email to gradually link with working mathematicians who realized that he was the John Nash and that his new work had value. They formed part of the nucleus of a group that contacted the Bank of Sweden's Nobel award committee and were able to vouch for Nash's mental health ability to receive the award in recognition of his early work.

Nash's later work involved ventures in advanced game theory, including partial agency, which show that, as in his early career, he preferred to select his own path and problems. Between 1945 and 1996, he published 23 scientific studies.

Nash has suggested hypotheses on mental illness. He has compared not thinking in an acceptable manner, or being "insane" and not fitting into a usual social function, to being "on strike" from an economic point of view. He advanced views in evolutionary psychology about the value of human diversity and the potential benefits of apparently nonstandard behaviors or roles.

Nash developed work on the role of money in society. Within the framing theorem that people can be so controlled and motivated by money that they may not be able to reason rationally about it, he criticized interest groups that promote quasi-doctrines based on Keynesian economics that permit manipulative short-term inflation and debt tactics that ultimately undermine currencies. He suggested a global "industrial consumption price index" system that would support the development of more "ideal money" that people could trust rather than more unstable "bad money". He noted that some of his thinking parallels economist and political philosopher Friedrich Hayek's thinking regarding money and a nontypical viewpoint of the function of the authorities.

Nash received an honorary degree, Doctor of Science and Technology, from Carnegie Mellon University in 1999, an honorary degree in economics from the University of Naples Federico II on March 19, 2003, an honorary doctorate in economics from the University of Antwerp in April 2007, an honorary doctorate of science from the City University of Hong Kong on November 8, 2011, and was keynote speaker at a conference on game theory. He was also a prolific guest speaker at a number of world-class events, such as the Warwick Economics Summit in 2005 held at the University of Warwick. In 2012, he was elected as a fellow of the American Mathematical Society.

On May 19, 2015, a few days before his death, Nash, along with Louis Nirenberg, was awarded the 2015 Abel Prize by King Harald V of Norway at a ceremony in Oslo.

## Personal life

In 1951, Massachusetts Institute of Technology MIT hired Nash as a C. L. E. Moore instructor in the mathematics faculty. About a year later, Nash began a relationship in Massachusetts with Eleanor Stier, a nurse he met while admitted as a patient. They had a son, John David Stier, but Nash left Stier when she told him of her pregnancy. The film based on Nash's life, A Beautiful Mind, was criticized during the run-up to the 2002 Oscars for omitting this aspect of his life. He was said to have abandoned her based on her social status, which he thought to have been beneath his.

In Santa Monica, California, in 1954, while in his twenties, Nash was arrested for indecent exposure in a sting operation targeting gay men. Although the charges were dropped, he was stripped of his top-secret security clearance and fired from RAND Corporation, where he had worked as a consultant.

Not long after breaking up with Stier, Nash met Alicia Lardé Lopez-Harrison January 1, 1933 – May 23, 2015, a naturalized U.S. citizen from El Salvador. Lardé graduated from MIT, having majored in physics. They married in February 1957; although Nash was an atheist, the ceremony was performed in an Episcopal church. They had a son together, John Charles Martin Nash, who earned a Ph.D. in Mathematics from Rutgers University and also suffered from schizophrenia as an adult.

In 1958, Nash earned a tenured position at MIT, and his first signs of mental illness were evident in early 1959. At this time, his wife was pregnant with their first child. He resigned his position as a member of the MIT mathematics faculty in the spring of 1959 and his wife had him admitted to McLean Hospital for treatment of schizophrenia that same year. Their son, John Charles Martin Nash, was born soon afterward. The child was not named for a year because his wife felt Nash should have a say in the name given to the boy. Due to the stress of dealing with his illness, Nash and Lardé divorced in 1963. After his final hospital discharge in 1970, Nash lived in Lardé's house as a boarder. This stability seemed to help him, and he learned how to consciously discard his paranoid delusions. He stopped taking psychiatric medication and was allowed by Princeton to audit classes. He continued to work on mathematics and eventually was allowed to teach again. In the 1990s, Lardé and Nash resumed their relationship, remarrying in 2003.

## Death

On May 23, 2015, Nash and his wife, Alicia Nash, died in a car accident on the New Jersey Turnpike near Monroe Township, New Jersey. They were on their way home from Newark Airport after a visit to Norway, where Nash had received the Abel Prize, when their taxicab driver, Tark Girgis, lost control of the vehicle and struck a guardrail. Both passengers were ejected from the car upon impact. State police revealed that it appeared neither passenger was wearing a seatbelt at the time of the crash. At the time of his death, the 86-year-old Nash was a longtime resident of New Jersey. He was survived by two sons, John Charles Martin Nash, who lived with his parents at the time of their death, and elder child John Stier.

Following his death, obituaries appeared in scientific and popular media throughout the world. In addition to their obituary for Nash, The New York Times published an article containing quotes from Nash that had been assembled from media and other published sources. The quotes consisted of Nash's reflections on his life and achievements.

## Legacy

At Princeton, Nash became known as "The Phantom of Fine Hall" Princeton's mathematics center, a shadowy figure who would scribble arcane equations on blackboards in the middle of the night.

He is referred to in a novel set at Princeton, The Mind-Body Problem, 1983, by Rebecca Goldstein.

Sylvia Nasar's biography of Nash, A Beautiful Mind, was published in 1998. A film by the same name was released in 2001, directed by Ron Howard with Russell Crowe playing Nash; it won four Academy Awards, including Best Picture.

## Awards

- 1978 – INFORMS John von Neumann Theory Prize
- 1994 – Nobel Memorial Prize in Economic Sciences
- 1999 – Leroy P Steele Prize
- 2002 class of Fellows of the Institute for Operations Research and the Management Sciences
- 2010 – Double Helix Medal
- 2015 – Abel Prize

Nash embedding theorem

Nash functions

Nash–Moser theorem

Hilbert's nineteenth problem

Nobel Memorial Prize in Economic Sciences (1994)

Member of the National Academy of Sciences (1996)

Abel Prize (2015)

Siffer (2011-)