Dr. John F. Nash Jr., a Nobel Prize-winning mathematician who revolutionized the mathematical field of game theory, and whose descent into and recovery from mental illness inspired the film “A Beautiful Mind,” died in an accident while riding in a taxicab on the New Jersey Turnpike on May 23. He was 86. His wife, Alicia, 82, also died.

Born in Bluefield, West Virginia, Nash first studied in Pittsburgh before moving to Princeton. His recommendation letter contained just one line: “This man is a genius.”

In 1994, Dr. Nash received the Nobel Prize in economics. 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. Even this week, Nash received the Abel Prize, another top honour in the field of mathematics.

Nash married Alicia Larde in 1957, but he developed severe schizophrenia soon after, and Alicia had him committed for psychiatric care several times. The couple divorced in 1962.

But the two stayed close, and his condition had begun to improve by the 1980s. They remarried in 2001.

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 screen three years later. The film, although criticized by some viewers for presenting a romanticized version of the mathematician’s life, won four Oscars, including one for best picture. Portrayed by actor Russell Crowe, Dr. Nash became an international celebrity — perhaps the most famous mathematician in recent memory.

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.

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.

Dr. Nash was described as having insights before he could hammer out the proofs of their accuracy, the ideas 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.

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.

He thought that the New York Times was publishing messages from extraterrestrials and that he could understand them.

At one point, he concluded 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.”

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.”