The American mathematician John Nash, who has died aged 86 in a car crash along with his wife, made his public mark as the subject of the 2001 film A Beautiful Mind.
In his discipline, he gave his name to the Nash equilibrium – a position in a situation of competition or conflict in which both sides have selected a strategy which neither side can independently change without ending up in a less desirable position.
He earned his early reputation, and his 1994 Nobel prize in economics, by proving mathematically that there is at least one Nash equilibrium lying in wait to trap us in every situation of competition or conflict where the parties are unwilling or unable to communicate.
Nash arrived at Princeton University in 1948 to study for a postgraduate degree in mathematics, bearing a one-line recommendation from his previous professor which simply read: "This man is a genius." Game theory This he proved by publishing what is surely the shortest paper ever to win its author a Nobel prize. Called "Equilibrium Points in N-Person Games", it contained just 317 words. It was to prove a major contribution to game theory, a field which explores the roots of many problems in conflict and the failure of co-operation.
Nash himself, however, did not regard his contribution particularly highly, continuing to work on basic problems in mathematics.
As an undergraduate he had produced an independent proof of Brouwer’s fixed-point theorem – the theorem that tells us that, no matter how much we stir a cup of coffee, there will always be one small bit that is just where it was before we started stirring.
The mental illness that incapacitated Nash and eventually prompted the film A Beautiful Mind began to make itself felt in 1959. He could be quite funny about it in later years, and told a New Scientist interviewer in 2004: "Mathematicians are comparatively sane as a group. It is the people who study logic that are not so sane." Paranoid schizophrenia But the reality was far from funny. His diagnosed paranoid schizophrenia led to the breakup of his marriage – although he remarried his wife, Alicia, (who died with him in the car crash) in 2001 – the loss of his job, and nearly the loss of his Nobel prize, when members of the selection committee were worried that he might never be able to carry on serious research again.
But carry on he did. In between bouts of hospitalisation, with gradual recovery despite many severe episodes until the early 1990s, Nash continued to make contributions to many areas of mathematics, especially in the solution of partial differential equations, basic tools for engineers and for people developing computer models of economies and societies.
John Forbes Nash was born in Bluefield, West Virginia. His father, John, was an electrical engineer, and his mother, Margaret Virginia (née Martin), had been a teacher. He attended what is now Carnegie Mellon University, initially majoring in chemical engineering but switching to chemistry and then mathematics, graduating in 1948 with bachelor’s and master’s degrees, after which he accepted a doctoral fellowship at Princeton.
On completion of his thesis – just 27 pages long – he was hired by Massachusetts Institute of Technology (MIT). In 1957 he married Alicia Lopez-Harrison de Lardé. He was also a consultant for the Rand corporation from 1950 to 1954, but lost his position, even though all charges were dropped, after he had been arrested for indecent exposure in a police entrapment operation in Santa Monica, California.
Nash’s illness led to his resignation from his position at MIT in 1959, and Alicia had him admitted to hospital. Divorce The stress of Nash’s illness took its toll, and he and his wife were divorced in 1963. After various periods of hospitalisation he was finally discharged in 1970, but continued to struggle with paranoid delusions over the next two decades.
In the New Scientist interview, he said: "I was a long way into mental illness before I heard any voices. Ultimately I realised I am generating these voices in my own mind: this is dreaming, this is not communication. This is coming from an internal source, not from the cosmos. And simply to understand that is to escape from the thing in principle. After understanding that, the voices died out."
Of his portrayal in film he said: "It's not me, but Russell Crowe plays the role well."
He is survived by his son, John Nash, also a mathematician, by another son from a previous relationship, John Stier, and by a sister, Martha.