John F. Nash Jr, the subject of Sylvia Nasar's compelling biography, has had a magnified version of a fairly common life pattern. It was common in that the loss of an economically, socially and personally productive life to mental illness happens fairly often. It was magnified in that Nash possessed extraordinary intellectual abilities and, partly because of them, had unusual support, which may have played a role in an unexpected degree of recovery, if only after a long incapacity. Nash was born in West Virginia in 1928, his father a business executive, both parents with intellectual backgrounds rather atypical for that state. He acquired early the sense of social status that frequently marked his subsequent relations. His undergraduate education was at Carnegie-Mellon University, where he recognised his bent for mathematics. While there, he took one required course in economics, which stimulated his thinking and led to his great contribution to that field.

He then went to study for his PhD at Princeton University, one of the great centres of mathematics in the United States. His outstanding ability was easily seen but he was always a controversial figure. He was seen to be difficult. In spite of his ability, he was not offered a position at Princeton on getting his doctorate or indeed at some of the other great centres. He was appointed at the Massachusetts Institute of Technology, which had a very good and rising department, ornamented by the great Norbert Wiener.

His dissertation topic was itself defiant of convention. It was in game theory, a field that serious mathematicians have generally disdained, even though it was started by the highly respected John von Neumann. Game theory is the mathematical study of social interactions among rational individuals. The basic problem is the circularity of rational behaviour. The environment in which each individual makes his choices is created by the decisions of others. Indeed, the Austrian economist Oskar Morgenstern and the US sociologist Robert K. Merton had expressed the view in the 1930s that economic and social forecasting was impossible. Each entrepreneur or other decision-maker would base his or her current investments on forecasts of the choices made by others; but these actions were in turn dependent on the forecasts made about the original individual's choices.

Von Neumann's 1928 paper began the modern study of game theory. As the somewhat whimsical term indicates, von Neumann considered social games (eg chess or card games) as paradigms for social interaction in general. In 1944, von Neumann and Morgenstern wrote a magisterial book, Theory of Games and Economic Behavior, which attracted immediate attention among economists. The specific solution concepts of the book have not, in fact, been used significantly, but the idea of studying social and economic interactions as games has been highly fruitful.

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The direct analysis of interactions has not been absent from economics. Already in 1838, Antoine-Auguste Cournot had studied markets in which there are few sellers from a viewpoint identical with one of Nash's contributions. In 1881, Francis Y. Edgeworth had studied bargaining among small numbers. Indeed, it is their concepts that are used today.

Nash (without being aware of Cournot's work) re-invented his concept (called by him "equilibrium" and now invariably referred to as "Nash equilibrium"), gave it a general formulation and established its consistency. He also contributed to Edgeworth's approach to give a coherent account of the outcome of a bargaining process, different from Edgeworth's account (known in the current literature as the "core"). The second, which was the outcome of his undergraduate economics course, was the more original, but both have been important.

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Nash's thesis was apparently not regarded well by von Neumann. The book's account of their meeting is not sharp enough for me to figure out what von Neumann disapproved of; perhaps there is no fuller statement now available.

Nash did not stay with game theory, though he did make one important contribution to bargaining theory a few years later. His initial career steps were indicative of the future. He was always taken seriously but did not obtain the highest honours (such as the Fields medal, the mathematician's analogue to Nobel prizes). He was considered in 1958, when he was only 29 and so had more opportunities ahead; the winner, Rene Thom, was certainly highly qualified.

Unlike many mathematicians or other scholars, Nash did not have a definite research programme to which he was committed. What he sought were individual problems, problems whose solution would enhance his reputation. He did write four papers that were each of outstanding importance. He tackled what is recognised to be the greatest unsolved problem of mathematics, a conjecture made by the 19th-century mathematician Bernhard Riemann. A talk on his progress was the first revelation of his illness.

Nash was always somewhat eccentric, but not to an extent that would seem unusual by the standards of mathematicians or indeed of academics generally. It was only around 1958 that clear signs of mental illness occurred. A great feature of the book is the extremely detailed account of the individual steps by which his paranoid schizophrenia gradually developed and was recognised. The very different treatments of different institutions, his relations with other mathematicians and his elaborate accounts of conspiracies and occult signs make for a gripping story, and one that is very well told.

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The care with which Nasar has sought out witnesses and records pays off in the rich understanding she and we achieve. Nash's relations with his wife, who continued to take care of him, after their divorce, and with his son, who succumbed to the same illness, are narrated. By contrast, the long period, some 20 years, of gradual recovery in the relatively benign environment of Princeton is passed over quickly.

Nasar makes the important point that mental illness is common. Other characters go into mental institutions or are otherwise incapacitated many times; usually, their recovery is quicker than in Nash's case.

In the end, it was the economists who gave Nash his highest recognition. Game theory has become an important tool of economic analysis. The majority of the committee that awards the Bank of Sweden's Nobel Memorial Prize in Economics were convinced of the need to recognise the field, but they were afraid of presenting it to one who might create a public scandal. In the end, they were reassured that Nash had changed, and so they could associate him with two other scholars who had developed his work further. This gave rise to some furious internal controversy. Somehow, despite the secrecy that is supposed to surround the prize, Nasar has obtained a fairly complete account of the disputes. Some of the participants have told me her account is not thoroughly accurate, but I would guess the more spectacular aspects are.

As a gripping narrative, as an account of mental illness and as a study of a very interesting scholar, I think this book should find many readers.

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Kenneth Arrow, Nobel laureate, is emeritus professor of economics and operations research, Stanford University, USA.