# INFORMS NEWS: In Memoriam - John F. Nash Jr. (1928-2015)

“This man is a genius.”

Professor Richard Duffin of the Carnegie Institute of Technology wrote a single sentence reference, recommending John Nash for the doctoral program in mathematics at Princeton in 1948. Nash graduated in 1950 and took a job in the mathematics department at MIT in 1951. From his thesis he published two papers: (1) “Equilibrium Points in n-Person Games (1950), Proceedings of the National Academy Sciences (The total text in what turned out to be a Nobel Prize paper was 317 words); (2) “Non-Cooperative Games” (1951), Annals of Mathematics. The latter developed Nash’s results further and contrasted them with Von Neumann and Morgenstern’s famous book, “Theory of Games and Economic Behavior” (1944), which Nash described as about “cooperative games” and his own work about “non-cooperative games,” noting that the former are zero-sum whereas the latter need not be.

Nash’s initial paper proved that a finite non-cooperative game always has at least one equilibrium point, today called a “Nash equilibrium.” This has the property that no player can obtain a better payoff by changing her/his strategy, if the other players do not change theirs.

The important new field of non-cooperative games was born. It is a better model of many real-world situations than a zero-sum assumption can produce. It is in common use today to analyze business, economic and political phenomena.

In 1978, INFORMS awarded the John von Neumann Theory Prize to John Nash and Carlton Lemke for their contribution to the theory of games. The theory of games was von Neumann’s most distinctive contribution to the field of OR/MS. It seems fitting that the John von Neumann Prize should be shared by Nash and Lemke, who were major contributors to the theory of non-cooperative games, the principal extension of von Neumann’s original idea. Furthermore, the prevailing trend among mathematicians has been to search for “elementary” (i.e., algebraic) proofs in new results. This tended to treat all game theory as a branch of the theory of linear inequalities. Nash, however, had introduced a different approach with his idea of non-cooperative n-person games and his general existence theorem.

Nash’s equilibrium proofs, however, were non-constructive, and for many years it seemed that the nonlinearity of the problem would prevent the actual numerical solution of any but the simplest non-cooperative games. The breakthrough came in 1964 with an ingenious method for solving finite, two-person games. Carlton Lemke and J. T. Howson devised it. It provided a path-following algorithm that was both a constructive existence proof and a practical means of calculation. Lemke took the lead in exploiting its many applications. The game theory aspect was strengthened because the path-following methodology is a way of finding and calculating Nash equilibria.

In 1994, the Nobel Prize in Economic Sciences was shared by John C. Harsanyi, John F. Nash Jr. and Reinhard Selten “for their pioneering analysis of equilibria in the theory of non-cooperative games.” Nash’s contributions were: “to introduce the distinction between cooperative games, in which binding agreements are not feasible and to develop an equilibrium concept for non-cooperative games that is now called Nash equilibrium.” Harsanyi and Selten each founded new subfields with new literatures and applications, but they both trace their ancestry to Nash’s first existence theorem.

Other important Nash research includes two path-breaking papers in 1954 and 1956. They prove that “every Riemannian manifold can be isometrically embedded into some Euclidean space.” These provided the basis of much subsequent mathematics. Later, after bouts with mental illness from which he gradually recovered, Nash did important work in partial differential equations.

In 1957, Nash married an MIT physics student from El Salvador, Alicia Lopez-Harrison de Larde’. However, in 1959 he was stricken by mental illness, diagnosed as paranoid schizophrenia. Alicia had him admitted to McLean Hospital near Boston. He continued to have delusions that took him in and out of mental hospitals near Princeton until 1970. He was largely supported mentally and financially by the mathematics community and by Alicia Nash through her professional income.

In 1998, Sylvia Nasar completed an extensive (461 pages) biography of Nash titled “A Beautiful Mind – The life of mathematical genius and Nobel Laureate John Nash.” It was nominated for a Pulitzer Prize. Subsequently it was made into a movie, “A Beautiful Mind.” The picture received four Academy Awards, including best picture.

The director of the film, Ron Howard, said, “[The movie] captures the spirit of [Nash’s] journey, and I think that it is authentic in what it conveys to a large extent. Certain aspects of it are dealt with symbolically. How do you understand what goes on inside a person’s mind when under stress, when mentally ill, when operating at the highest levels of achievement. The script tries to offer insight, but it’s impossible to be entirely accurate.”

Of his portrayal by an actor in the film, Nash said: “It’s not me, but Russell Crowe plays the part well.”

John Forbes Nash, mathematician, was born June 13, 1928; he died May 23, 2015.

– John D. C. Little, MIT

Author’s note: I was a doctoral student in physics at MIT from 1950 through 1954, overlapping Nash’s first few years at MIT. In the late stages of my Ph.D. thesis, I informally audited a course in real analysis for a few weeks. It happened to be taught by Nash. People ask me what he was like. I found him to be competent but not especially inspiring.

As I remember, he seemed somewhat abstracted, as if focused on something else. The only idiosyncrasy I recall is that, if Nash were near the window, he would occasionally absent-mindedly fiddle with the venetian blind cord, while he answered questions. Although I did not know it at the time, in this period, he was working on two brilliant papers that were published in 1954 and 1956.