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.
