Awarded a Nobel Prize in Economics in 1994 for his pioneering work in game theory, John Nash (born 1928) distinguished himself as one of the foremost mathematical researchers and theorists of the twentieth century.

Game theory was the subject of Nash's doctoral dissertation at Princeton University. Expanding upon the initial game theory of John von Neumann and Oskar Morgenstern, published in their *The Theory of Games and Economic Behavior,* Nash developed what became known as "Nash's equilibrium" to explain how two or more competitors can arrive at a mutually beneficial yet non-cooperative business arrangement. In 1951, he developed the theory that manifolds—objects containing various forms and components—can be described accurately using algebraic equations. He later developed what became known as the Nash-Moser theorem, which explained how it was possible to embed a manifold in a Euclidean space by employing differential calculus instead of algebra and geometry. Nash's subsequent career was diminished by severe mental illness, which was documented in Sylvia Nasar's biography of Nash, *A Beautiful Mind,* and the film of the same name directed by Ron Howard.

## Early Interest in Math

Nash was born on June 13, 1928, in Bluefield, West Virginia, and raised by his parents, John Nash, Sr., an electrical engineer for the Appalachian Power Company, and Margaret Nash, a teacher who retired after her marriage and placed a high value on the education of her two children. As a young man he seemed disinclined to study but displayed a passion for electronics and chemistry experiments that he conducted in his bedroom.

When he was a young teenager, Nash read a book by E. T. Bell, *Men of Mathematics,* to which Nash attributed his eventual passion for number theory. While attending high school and concurrent classes at Bluefield College, Nash collaborated with his father on a paper titled "Sag and Tension Calculations for Cable and Wire Spans Using Catenary Formulas," which was published in a 1945 edition of *Electrical Engineering.* Nash also entered the George Westinghouse competition, winning one of ten nationally awarded full scholarships, which he used to enroll in the Carnegie Institute of Technology in Pittsburgh.

## Mixed Success in Education

Initially aspiring to become an engineer like his father, Nash changed his major to chemistry after performing poorly in mechanical drawing. After he also had trouble with a physical chemistry class, he was convinced by his calculus instructor John Synge to major in mathematics. In 1948, Nash was awarded the John S. Kennedy Fellowship at Princeton University.

At Princeton, Nash was in close proximity to the Institute of Advanced Study, which attracted such notable mathematicians as Albert Einstein, Kurt Godel, Karl Oppenheimer, Hermann Weyl, and John von Neumann. According to Sylvia Nasar, "Princeton in 1948 was to mathematicians what Paris once was to painters and novelists, Vienna to psychoanalysts and architects, and ancient Athens to philosophers and playwrights." In 1949, Nash was awarded an Atomic Energy Commission fellowship to continue his doctoral studies at Princeton. The school's faculty and students admired Nash for his obvious intellect, but his academic career remained undistinguished.

While at Princeton, Nash invented two board games. The first, called "Nash" or "John," was a two-person, zero-sum game, meaning that one player's advantage must result in a proportional disadvantage for the opponent. Unlike other zero-sum games such as chess and tic-tac-toe, however, a tie or draw was impossible in Nash's game. The game had been invented independently from Nash and eventually was marketed in the 1950s as Hex. Nash also collaborated with several students to create the game "So Long, Sucker," a multiple-player game that rewarded the player most skilled at deception.

## Battled Mental Illness

After graduating from Princeton, Nash taught mathematics at the Massachusetts Institute of Technology in Cambridge. He had a son with Eleanor Stier before marrying Alicia Larde in 1957, with whom he also fathered a son. Along with teaching at MIT, Nash worked at the RAND Corporation think tank in Santa Monica, California. Nash was fired in 1954 after being arrested for indecent exposure in a public restroom during a Santa Monica police sting against homosexuals. Nasar wrote: "The biggest shock to Nash may not have been the arrest itself, but the subsequent expulsion from RAND." In 1957, he divided his time between the Institute for Advanced Study and the Courant Institute of Mathematical Sciences at New York University.

In early 1959, Nash began exhibiting symptoms of paranoid schizophrenia. After losing his ability to teach and do research, he underwent insulin coma therapy during several stays in psychiatric hospitals, including one where he shared a room with poet Robert Lowell. When not institutionalized, he made several trips to Europe, where he attempted to establish a world government and resign his United States citizenship because he was convinced he was a political prisoner. He also declared himself the emperor of Antarctica and tried to establish a defense fund for what he believed was an impending extra-terrestrial attack.

In 1962, Alicia Nash filed for divorce, and Nash lived with his widowed mother until her death in 1969. He then moved back into the house he shared with Alicia Nash. For the next 15 years, Nash spent much of his time wandering freely on the Princeton campus. In the late 1980s, however, he showed signs of remission from mental illness. He accepted the Nobel Prize for economics in 1994 and spent much of the 1990s attending to his second son's schizophrenia. He and Alicia Nash eventually remarried.

## Developed Game Theory

While at RAND, Nash participated in developing new technologies, theories, and strategies for the United States military through a private nonprofit organization that employed many of the nation's most prominent intellectuals. One of the strategies that RAND was beginning to explore for modern warfare was game theory, which expressed itself in such Cold War strategies as mutual deterrence and the arms race. Whereas John von Neumann and Oskar Morgenstern had conceived of game theory as a zero-sum relationship between non-cooperating competitors, Nash argued that some competitors could benefit from an adversarial relationship by seeking an equilibrium point that would either minimize negative repercussions or maximize positive outcomes. Jeremy Bernstein, writing in *Commentary,* noted: "Part of Nash's contribution was to allow one to relax the assumptions of von Neumann's theorem; the game does not have to be zero-sum or involve only two players. … What he showed was that in a very wide range of such 'games,' there must be at least one such strategy leading to equilibrium, and if there are several, one must decide among them." Assuming that all competitors behave in a rational manner, Nash hypothesized that each party would apply its dominant strategy to yield mutually beneficial results.

In 1950, Nash submitted his equilibrium theory as his Princeton doctoral thesis. It has since become widely used in military and economic strategies, as well as in biology. According to animal behaviorist Peter Hammerstein, quoted by Robert Pool in *Science,* "The Nash equilibrium turns out to be terribly important in biology. … Such concepts are proving vital in analyzing a range of biological data, from sex ratios to animals' decisions about whether to fight each other for territory or food." The theory earned him the Nobel Prize, shared with fellow game theorists John Harsanyi and Reinhard Selten.

Following his work in game theory, Nash focused on, among other things, manifolds. According to Nasar: "In one dimension, a manifold may be a straight line, in two dimensions a plane, or the surface of a cube, a balloon, or a doughnut." Although the object remains the same, it appears different when viewed from different perspectives. Because of their mutability, manifolds seemingly defied accurate depictions until Nash employed polynomial algebraic equations to describe them in 1950 and 1951.

In September 1951, beginning his tenure at MIT, Nash combined his work with manifolds with an interest in fluid dynamics. Nash applied the results of this research to his next mathematical theory, which asserted that it is possible to embed a Riemannian manifold in a Euclidean space. An eighteenth-century German mathematician, G.F.B. Riemann theorized that previous Euclidean notions of geography were inaccurate, due to the curvature of the earth's surface, therefore making all parallel lines subject to intersection and the sums of any triangle's angles unequal to 180 degrees. Rather than employ geometry or algebra to solve the problem, Nash developed a new method of applying 19th-century differential calculus. Jurgen Moser later applied the breakthrough to celestial mechanics, resulting in its eventual name: the Nash-Moser theorem.

## Books

Nasar, Sylvia, *A Beautiful Mind: The Life of Mathematical Genius and Nobel Laureate John Nash,* Simon & Schuster, 1998.

## Periodicals

*Commentary,* August 1998.

*Forbes,* July 3, 1995.

*Science,* October 21, 1994.

*Time,* October 24, 1994.

*Washington Post,* December 18, 2001.