Mathematician John Nash, a Nobel Prize winner who inspired the movie “A Beautiful Mind,” was killed in a car crash along with his wife on Saturday. (Reuters)

John F. Nash Jr., who revolutionized the mathematical field of game theory, was endowed with a mind that was highly original and deeply troubled. But it became known to most people by Hollywood’s description. His mind was beautiful.

Dr. Nash, a Nobel Prize-winning mathematician whose descent into and recovery from mental illness inspired the Academy Award-winning film “A Beautiful Mind,” died May 23 in a two-car accident on the New Jersey Turnpike. He was 86. His wife, Alicia, who was 82, also died.

According to preliminary findings by the New Jersey State Police, the Nashes were in a taxicab traveling southbound near Monroe when their driver lost control of the vehicle. The taxi driver’s injuries were not considered life-threatening, New Jersey police said. The Nashes lived in Princeton Junction, N.J.

In 1994, when Dr. Nash received the Nobel Prize in economic sciences, the award marked not only an intellectual triumph but also a personal one. More than four decades earlier, as a Princeton University graduate student, he had produced a 27-page thesis on game theory — in essence, the applied mathematical study of decision-making in situations of conflict — that would become one of the most celebrated works in the field.

Before the academic world could fully recognize his achievement, Dr. Nash descended into a condition eventually diagnosed as schizophrenia. For the better part of 20 years, his once supremely rational mind was beset by delusions and hallucinations.

John Nash, left, and his wife Alicia, in 2002. (Laura Rauch/AP)

By the time Dr. Nash emerged from his disturbed state, his ideas had influenced economics, foreign affairs, politics, biology — virtually every sphere of life fueled by competition. But he had been absent from professional life for so long that some scholars assumed he was dead.

“We helped lift him into daylight,” Assar Lindbeck, the former chairman of the committee for the Nobel Prize in economics, told Sylvia Nasar, Dr. Nash’s biographer. “We resurrected him in a way.”

Nasar’s book, titled “A Beautiful Mind,” was published in 1998 and adapted for the big screen three years later. The film, although criticized by some for presenting a romanticized version of the mathematician’s life, won four Oscars, including for best picture. Portrayed by Russell Crowe, Dr. Nash became an international celebrity — perhaps the most famous mathematician in recent memory.

Complexity in competition

Modern game theory was first articulated by mathematician John von Neumann and economist Oskar Morgenstern in the 1944 volume “Theory of Games and Economic Behavior.”

Its objective: to understand and ultimately predict the interactions between rivals in given circumstances. During the Cold War standoff between the United States and the Soviet Union, game theory became increasingly fashionable and immensely useful.

Von Neumann and Morgenstern had assumed the existence of a “zero-sum” game such as checkers, in which one party’s loss was the adversary’s gain. Dr. Nash — who, ironically, was said to have struggled since childhood with social interactions — observed that few human rivalries function in so simple a fashion.

He expanded game theory to include cooperative games (in which binding agreements can be made) and non-cooperative games (in which they cannot), and to allow for the possibility of mutual gain. Such an outcome became known as the Nash equilibrium.

Nash equilibriums, which he described in the hieroglyphics of mathematical symbols, exist everywhere. Two magazines might charge the same price so that each may achieve maximum profit. Two rival nations might agree to arms treaties that limit each of their stockpiles but guarantee both countries a measure of security.

The utility of Dr. Nash’s work had limitations. One is that rivals frequently do not fully know each other’s strategies, as his theories assumed. Another limitation is that in many cases, there is not a single possible outcome for a conflict but rather many potential outcomes. Game theorists John Harsanyi and Reinhard Selten shared with Dr. Nash the 1994 Nobel Prize for contributions in those areas of the field. The prize citation recognized all three men for their “pioneering analysis.”

Dr. Nash was described as having insights before he could hammer out the proofs of their accuracy, the thoughts coming to him more like revelations than like scholarly findings. As early as 1958, Fortune magazine had ranked him among the greatest mathematicians of the era.

“Everyone else would climb a peak by looking for a path somewhere on the mountain,” Nasar quoted a former colleague as saying. “Nash would climb another mountain altogether and from a distant peak would shine a searchlight back on the first peak.”

The emperor of Antarctica

His mental illness came on when he was about 30, during what might have been one of the richest periods of his career. Dr. Nash was working at the time at the Massachusetts Institute of Technology and was studying quantum theory.

As his condition worsened, Dr. Nash suffered delusions, hallucinations and impressions of being hunted. Men wearing red ties, he came to believe, were part of a “crypto-Communist Party.”

He thought that the New York Times was publishing messages from extraterrestrials and that he could understand them. He gave a student an intergalactic driver’s license, Nasar wrote.

At one point, he declined a prestigious appointment to the University of Chicago because he believed that he was in line to become emperor of Antarctica. At another point, he concluded, according to Nasar, that he was a “messianic figure of great but secret importance” and searched numerals — once the object of his brilliance — for hidden messages.

“I felt like I might get a divine revelation by seeing a certain number; a great coincidence could be interpreted as a message from heaven,” Dr. Nash said years later in the PBS “American Experience” documentary “A Brilliant Madness.”

He let his hair grow long. He traveled abroad and attempted to give up his U.S. citizenship, and at various times considered himself a Japanese shogun, the biblical figure Job and a Palestinian refugee, among other identities.

During one of his stays in mental institutions, a former colleague came for a visit.

“How could you, a mathematician devoted to reason and logical proof . . . how could you believe that extraterrestrials are sending you messages?” he asked, according to Nasar.

“Because,” Dr. Nash responded, “the ideas about supernatural beings came to me the same way that my mathematical ideas did. So I took them seriously.”

‘Big Brains’

John Forbes Nash Jr. was born June 13, 1928, in Bluefield, W.Va. His father was an electrical engineer and his mother was an English and Latin teacher.

As a child, John Jr. acquired a nickname: “Big Brains.” His family encouraged education, but he recalled in his Nobel biographical sketch the need to “learn from the world’s knowledge rather than from the knowledge of the immediate community.”

In 1945, he enrolled at what is now Carnegie Mellon University in Pittsburgh and completed his undergraduate work after switching from chemical engineering to chemistry and finally to mathematics. So great was his progress that he received a master’s degree in addition to his bachelor’s degree, both in mathematics, upon his graduation in 1948. He then moved to Princeton University, where, as a second-year student, he wrote the thesis that became the intellectual underpinning of his contributions to game theory.

Dr. Nash was “handsome as a god,” a former classmate told Nasar, but deeply unusual. He rode a bicycle in figure-eights. He joined a group of students that carried on the long tradition at Princeton of playing complex games and even invented a game of his own.

Dr. Nash received his doctorate in 1950, joined the MIT faculty and soon took a research position at the Rand Corp. in California. In that period of his career, he untangled what he described as a “classical unsolved problem” related to differential geometry and to general relativity.

Also during that period, Dr. Nash met Eleanor Stier, a nurse with whom he had a son, John David Stier, in 1953. A year later, Dr. Nash was arrested for indecent exposure at a men’s restroom in Santa Monica, Calif., and was immediately dismissed from Rand. According to Nasar’s biography, he denied that he was gay, showing a picture of Stier and their infant son to Rand officials as evidence.

He then returned to MIT, where he met Alicia Larde, a physics student from El Salvador, and they married in 1957. Shortly thereafter, Alicia became pregnant with their son, John Charles Martin Nash, and Dr. Nash began to show signs of mental instability.

During his illness, Dr. Nash was divorced from his wife, moved in and out of hospitals and endured dangerous treatments including insulin-coma therapy. Alicia Nash later took him into her home and cared for him even though they were no longer married.

He spent much of his time on the Princeton campus, where some recognized him as the genius that he was. Others knew him as the Ghost of Fine Hall, a reference to the building that houses the mathematics department.

In time, and seemingly against all odds, he appeared to overcome the illness that had afflicted him for so long. He insisted that he “willed” his recovery.

“I decided I was going to think rationally,” Dr. Nash told an interviewer.

Dr. Nash and Alicia were remarried in 2001. “We thought it would be a good idea,” she later said. “After all, we’ve been together most of our lives.”

Survivors include his sons, John David Stier of Lynn, Mass., and John Charles Martin Nash of Princeton Junction; and a sister.

Dr. Nash remarked in his Nobel biographical sketch that his return to rational scientific thought was “not entirely a matter of joy as if someone returned from physical disability to good physical health.”

“Without his ‘madness,’ ” Dr. Nash wrote, “Zarathustra would necessarily have been only another of the millions or billions of human individuals who have lived and then been forgotten.”