My old student, Dr. Andrew Blais, asks in an e-mail whether
I might write a tutorial on the work of Nobel Laureate John Nash, who died
recently with his wife in a tragic automobile accident. I replied that I did not know enough to write
such a tutorial, but there might be one or two folks out there who would like
to know something about Nash’s contributions to the mathematical discipline
known as Game Theory. Herewith a brief
explanation. In order to keep this
short, I shall have to refer interested readers to my second blog, entitled Formal Methods in Political Philosophy,
for which I wrote a book length primer on a number of branches of formal
analysis, including Game Theory.
Game Theory was invented by the great Hungarian-American
mathematician John von Neumann, who set forth the theory in a classic book, The Theory of Games and Economic Behavior
co-authored by Oskar Morgenstern and published in 1944. The central theorem of the book asserts that
every two person zero sum game with mixed strategies has a solution. Each term in this sentence has a precise
mathematical definition. In particular,
the term “zero sum game” is given an exact definition by von Neumann and
Morgenstern. By the way, almost nobody who
uses the phrase knows what it means, not even professional economists who
should know better.
Assuming that in a two person game, the first person has n
pure strategies [a finite positive integer] and the second player has m pure
strategies, and understanding that a mixed strategy is a probability
distribution over a player’s pure strategies, the triads formed by the first
player’s mixed strategy, the second player’s mixed strategy, and the payoff to
the first player can be represented by points in an n+m-1 dimensional simplex. A solution is a point in that space – called a
saddle point – which has the property that if either player holds firm to his
or her mixed strategy choice, the other player can only do worse by moving away
from that point to an adjacent mixed strategy.
Von Neumann, using L. E. J. Brouwer’s famous “fixed point
theorem,” proved that every such game has a solution. What is more [this is the point of all of
this hand waving], if there are multiple solutions, they all have the same
payoff to each player. Hence, they are
interchangeable.
Powerful as this theorem is, its application is extremely limited,
because the games that meet the stringent definition of being zero sum are rare
indeed. What John Nash did was to extend
von Neumann’s proof by demonstrating that a very much wider set of games have
solutions in the von Neumann sense. However, in this wider set of games, if
there are multiple solutions [multiple saddle points], it is not in general the
case that they all result in the same payoffs to the players. Hence these solution points are local but not
global solutions.
O. K. That is about
as much as I know about John Nash, except that he was played by Russell Crowe
in the movie A Beautiful Mind.
2 comments:
When I was in junior high school (now called 'middle school'?) I read a wonderful introduction to game theory called *The Compleat Stategist". Maybe Dover has republished it?
Von Neumann was one of those people who could sit and listen to a lecture on stuff he didn't know well, absorb it and then explain it to everyone else. He was probably the only person in the room who, when Gödel presented his incompleteness theorems, "got it"--including how the Second Theorem was supposed to go.
I have had a very good life, and I would not have any of it different, but I think I would have given years off that life to be blessed with von Neumann's ability to grasp deep formal structures apparently effortlessly. I have done a lot of math in my life but I always go point to point to point, plodding from premises to conclusion like a turtle crossing a road. To have that sort of mind must be something like what it would have been like to be Mozart.
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