from The New York Times

John F. Nash Jr., a mathematician who shared a Nobel Prize in 1994 for work that greatly extended the reach and power of modern economic theory and whose decades-long descent into severe mental illness and eventual recovery were the subject of a 2001 film, “A Beautiful Mind,” was killed in a car crash Saturday in New Jersey. He was 86.

Dr. Nash, and his wife, Alicia, 82, were killed when the taxi they were riding in lost control and hit a guard rail and another vehicle, said Sgt. Gregory Williams of the New Jersey State Police.

Sergeant Williams said the taxi was traveling southbound on the New Jersey Turnpike when the driver lost control while attempting to pass another vehicle. Dr. and Ms. Nash were ejected from the vehicle and pronounced dead at the scene. The taxi driver and the driver of the other car were treated for non-life threatening injuries. There are no criminal charges at this time.

Dr. Nash was widely regarded as one of the great mathematicians of the 20th century, known for the originality of his thinking and for his fearlessness in wrestling down problems so difficult few others dared tackle them. A one-sentence letter written in support of his application to Princeton’s doctoral program in math said simply, “This man is a genius.”

His theory of noncooperative games, published in 1950 and known as Nash equilibrium, provided a conceptually simple but powerful mathematical tool for analyzing a wide range of competitive situations, from corporate rivalries to legislative decision making. Dr. Nash’s approach is now pervasive in economics and throughout the social sciences and is applied routinely in other fields, like evolutionary biology.

Harold W. Kuhn, an emeritus professor of mathematics at Princeton and a longtime friend and colleague of Dr. Nash’s who died in 2014, said, “I think honestly that there have been really not that many great ideas in the 20th century in economics and maybe, among the top 10, his equilibrium would be among them.” An economist, Roger Myerson of the University of Chicago, went further, comparing the impact of Nash equilibrium on economics “to that of the discovery of the DNA double helix in the biological sciences.”

Dr. Nash also made contributions to pure mathematics that many mathematicians view as more significant than his Nobel-winning work on game theory, including solving an intractable problem in differential geometry derived from the work of the 19th century mathematician G.F.B. Riemann.

His achievements were the more remarkable, colleagues said, for being contained in a small handful of papers published before he was 30.

“Jane Austen wrote six novels, Bach wrote six partitas,” said Barry Mazur, a professor of mathematics at Harvard who was a freshman at M.I.T. when Dr. Nash taught there. “I think Nash’s pure mathematical contributions are on that level. Very, very few papers he wrote on different subjects, but the ones that had impact had incredible impact.”

Yet to a wider audience, Dr. Nash was probably best known for his life story, a tale of dazzling achievement, devastating loss and almost miraculous redemption. The narrative of Dr. Nash’s brilliant rise, the lost years when his world dissolved in schizophrenia, his return to rationality and the awarding of the Nobel, retold in a biography by Sylvia Nasar and in the Oscar-winning film starring Russell Crowe, captured the public mind and became a symbol of the destructive force of mental illness and the stigma that often hounds those who suffer from it.

John Forbes Nash was born on June 13, 1928, in Bluefield, W. Va. His father, John Sr., was an electrical engineer. His mother, Margaret, was a schoolteacher.

As a child, John Nash may have been a prodigy but he was not a sterling student, Sylvia Nasar noted in a 1994 article in The New York Times. “He read constantly. He played chess. He whistled entire Bach melodies,” Ms. Nasar wrote.

In high school, he stumbled across E.T. Bell’s book, “Men and Mathematics,” and soon demonstrated his own mathematical skill by independently proving Fermat’s classic theorem, an accomplishment he recalled in an autobiographical essay written for the Nobel committee.

Intending to become an engineer like his father, he entered Carnegie Mellon (then called Carnegie Institute of Technology). But he chafed at the regimented courses, and encouraged by professors who recognized his mathematical genius, he switched to mathematics.

Receiving his bachelor’s and master’s degrees at Carnegie, he arrived at Princeton in 1948, a time of great expectations, when American children still dreamed of growing up to be physicists like Einstein or mathematicians like the brilliant, Hungarian-born polymath John von Neumann, both of whom attended the afternoon teas at Fine Hall, the home of the math department.

John Nash, tall and good-looking, quickly became known for his intellectual arrogance, his odd habits — he paced the halls, walked off in the middle of conversations, whistled incessantly — and his fierce ambition, his colleagues have recalled.

He invented a game, known as Nash, that became an obsession in the Fine Hall common room. (The same game, invented independently in Denmark, was later sold by Parker Brothers as Hex.) He also took on a problem left unsolved by Dr. von Neumann and Oskar Morgenstern, the pioneers of game theory, in their now classic book, “Theory of Games and Economic Behavior.”

Dr. von Neumann and Dr. Morgenstern, an economist at Princeton, addressed only so-called zero-sum games, in which one player’s gain is another’s loss. But most real world interactions are more complicated, where players’ interests are not directly opposed, and there are opportunities for mutual gain. Dr. Nash’s solution, contained in a 27-page doctoral thesis he wrote when he was 21, provided a way of analyzing how each player could maximize his benefits, assuming that the other players would also act to maximize their self-interest.

This deceptively simple extension of game theory paved the way for economic theory to be applied to a wide variety of other situations besides the marketplace.

“It was a very natural discovery,” Dr. Kuhn said. “A variety of people would have come to the same results at the same time, but John did it and he did it on his own.”

After receiving his doctorate at Princeton, Dr. Nash served as a consultant for the RAND Corporation and as an instructor at M.I.T. and still had a penchant for attacking problems that no one else could solve. On a dare, he developed an entirely original approach to a longstanding problem in differential geometry, showing that abstract geometric spaces called Riemannian manifolds could be squished into arbitrarily small pieces of Euclidean space.

As his career flourished and his reputation grew, however, Dr. Nash’s personal life became increasingly complex. A turbulent romance with a nurse in Boston, Eleanor Stier, resulted in the birth of a son, John David Stier, in 1953. Dr. Nash also had a series of relationships with men, and while at RAND in the summer of 1954, he was arrested in a men’s bathroom for indecent exposure, according to Ms. Nasar’s biography. And doubts about his accomplishments gnawed at him: two of mathematics’ highest honors, the Putnam prize and the Fields medal, had eluded him.

In 1957, after two years of on-and-off courtship, he married Alicia Larde, an M.I.T. physics major from an aristocratic Central American family. But early in 1959, with Alicia pregnant with their son, John, Dr. Nash began to unravel. His brilliance turned malignant, leading him into a landscape of paranoia and delusion, and in April, he was hospitalized at McLean Hospital, outside Boston, sharing the psychiatric ward with, among others, the poet Ezra Pound.

It was the first step of a steep decline. There were more hospitalizations. He underwent electroshock therapy and fled for a while to Europe, sending cryptic postcards to colleagues and family members. For many years he roamed the Princeton campus, a lonely figure scribbling unintelligible formulas on the same blackboards in Fine Hall where he had once demonstrated startling mathematical feats.

Though game theory was gaining in prominence, and his work cited ever more frequently and taught widely in economics courses around the world, Dr. Nash had vanished from the professional world.

“He hadn’t published a scientific paper since 1958,” Ms. Nasar wrote in the 1994 Times article. “He hadn’t held an academic post since 1959. Many people had heard, incorrectly, that he had had a lobotomy. Others, mainly those outside of Princeton, simply assumed that he was dead.”

Indeed, Dr. Myerson recalled in a telephone interview that one scholar who wrote to Dr. Nash in the 1980s to ask permission to reprint an article received the letter back with one sentence scrawled across it: “You may use my article as if I were dead.”

Still, Dr. Nash was fortunate in having family members, colleagues and friends, in Princeton and elsewhere, who protected him, got him occasional money and work, and in general helped him survive. Alicia Nash divorced him in 1963, but continued to stand by him, taking him into her house to live in 1970. (The couple married a second time in 2001.)

By the early 1990s, when the Nobel committee began investigating the possibility of awarding Dr. Nash its memorial prize in economics, his illness had quieted. He later said that he simply decided that he was going to return to rationality. “I emerged from irrational thinking, ultimately, without medicine other than the natural hormonal changes of aging,” he wrote in an email to Dr. Kuhn in 1996.

Colleagues, including Dr. Kuhn, helped persuade the Nobel committee that Dr. Nash was well enough to accept the prize — he shared it with two economists, John C. Harsanyi of the University of California at Berkeley, and Reinhard Selten, of the Rheinische Friedrich-Wilhelms University in Bonn, Germany — and they defended him when some questioned giving the prize to a man who had suffered from a serious mental disorder.

The Nobel, the publicity that attended it, and the making of the film were “a watershed in his life,” Dr. Kuhn said of Dr. Nash. “It changed him from a homeless unknown person who was wandering around Princeton to a celebrity, and financially, it put him on a much better basis.”

Dr. Nash continued to work, traveling and speaking at conferences and attempting, among other things, to formulate a new theory of cooperative games. Friends described him as charming and diffident, a bit socially awkward, a little quiet, with scant trace of the arrogance of his youth.

“You don’t find many mathematicians approaching things this way now, bare-handedly attacking a problem,” the way Dr. Nash did, said Dr. Mazur.