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 27page dissertation,
"NonCooperative 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 uponly 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 21year old John Nash wrote his 27page dissertation
outlining his "Nash Equilibrium" for strategic noncooperative
games, the impact was enormous. On the formal side, his existence
proof was one of the first applications of Kakutani's fixedpoint
theorem later employed with so much gusto by NeoWalrasians
everywhere; on the conceptual side, he spawned much of the
literature on noncooperative 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 noncooperative games
in the case of "pure rivalries" (i.e. zerosum). The young
Nash turned to rivalries with mutual gain.
His trick was the use of bestresponse functions and a recent
theorem that had just emerged  Kakutani's fixed pointtheorem.
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 twoperson 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 noncooperative
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, noncooperative
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 noncooperative
games, where binding agreements are not feasible. Nash developed
an equilibrium concept for noncooperative games that later
came to be called Nash equilibrium.
Read Part 2: Nash's
Autobiographical
Description of
Ilness
