JOHN NASH:
GENIUS, NOBEL AND SCHIZOPHRENIA

For many years, students and scholars in Princeton have seen a ghostly, silent figure shuffling around the corridors of the math and physics building wearing purple sneakers and occasionally writing numerology treatises on the blackboards. They called him the "Phantom of Fine Hall". The Phantom was John Nash, one of the most brilliant mathematicians of his generation, who had spiraled into schizophrenia in the 1950s.

Nash was a mathematical genius whose 27-page dissertation, "Non-Cooperative Games," written in 1950 when he was 21, would be honored with the Nobel Prize in Economics in 1994.

His most important work had been in game theory, which by the 1980s was underpinning a large part of economics. When the Nobel Prize committee began debating a prize for game theory, Nash's name inevitably came up--only to be dismissed, since the prize clearly could not go to a madman. But in 1994 Nash, in remission from schizophrenia, shared the Nobel Prize in economics for work done some 45 years previously.

When the 21-year old John Nash wrote his 27-page dissertation outlining his "Nash Equilibrium" for strategic non-cooperative games, the impact was enormous. On the formal side, his existence proof was one of the first applications of Kakutani's fixed-point theorem later employed with so much gusto by Neo-Walrasians everywhere; on the conceptual side, he spawned much of the literature on non-cooperative game theory which has since grown at a prodigious rate - threatening, some claim, to overwhelm much of economics itself.

When the young Nash had applied to graduate school at Princeton in 1948, his old Carnegie Tech professor, R.J. Duffin, wrote only one line on his letter of recommendation: "This man is a genius".

It was at Princeton that Nash encountered the theory of games, then recently launched by John von Neumann and Oskar Morgenstern. However, they had only managed to solve non-cooperative games in the case of "pure rivalries" (i.e. zero-sum). The young Nash turned to rivalries with mutual gain.

His trick was the use of best-response functions and a recent theorem that had just emerged - Kakutani's fixed point-theorem. His main result, the "Nash Equilibrium", was published in 1950 in the Proceedings of the National Academy of Sciences. He followed this up with a paper which introduced yet another solution concept - this time for two-person cooperative games - the "Nash Bargaining Solution" (NBS) in 1950.

A 1951 paper attached his name to yet another side of economics - this time, the "Nash Programme", reflecting his methodological call for the reduction of all cooperative games into a non-cooperative framework.

His contributions to mathematics were no less remarkable. As an undergraduate, he had inadvertently (and independently) proved Brouwer's fixed point theorem. Later on, he went on to break one of Riemann's most perplexing mathematical conundrums. (This was the problem to prove the isometric embeddability of abstract Riemannian manifolds in flat - or "Euclidean" spaces.) From then on, Nash provided breakthrough after breakthrough in mathematics.

In 1958, on the threshold of his career, Nash got struck by paranoid schizophrenia. He lost his job at M.I.T. in 1959 (he had been tenured there in 1958 - at the age of 29) and was virtually incapicated by the disease for the next two decades or so. He roamed about Europe and America, finally, returning to Princeton where he became a sad, ghostly character on the campus - "the Phantom of Fine Hall".

The disease began to evaporate in the early 1970s and Nash began to gradually return to his work in mathematics. However, Nash himself associated his madness with his living on an "ultralogical" plane, "breathing air too rare" for most mortals, and if being "cured" meant he could no longer do any original work at that level, then, Nash argued, a remission might not be worthwhile in the end.

Nash shared the Nobel prize in 1994 with John C. Harsanyi and Reinhard Selten - for what he claims was his "most trivial work". The press release of the Nobel Foundation decribes his work as follows:

"Games as the Foundation for Understanding Complex Economic Issues"

Game theory emanates from studies of games such as chess or poker. Everyone knows that in these games, players have to think ahead - devise a strategy based on expected countermoves from the other player(s). Such strategic interaction also characterizes many economic situations, and game theory has therefore proved to be very useful in economic analysis.

The foundations for using game theory in economics were introduced in a monumental study by John von Neumann and Oskar Morgenstern entitled Theory of Games and Economic Behavior (1944). Today, 50 years later, game theory has become a dominant tool for analyzing economic issues. In particular, non-cooperative game theory, i.e., the branch of game theory which excludes binding agreements, has had great impact on economic research. The principal aspect of this theory is the concept of equilibrium, which is used to make predictions about the outcome of strategic interaction. John F. Nash, Reinhard Selten and John C. Harsanyi are three researchers who have made eminent contributions to this type of equilibrium analysis.

* John F. Nash introduced the distinction between cooperative games, in which binding agreements can be made, and non-cooperative games, where binding agreements are not feasible. Nash developed an equilibrium concept for non-cooperative games that later came to be called Nash equilibrium.

Read Part 2: Nash's Autobiographical
Description of Ilness