JohnvonNeumann
Game theory
Von Neumann founded the field of
game theory as a mathematical discipline. Von Neumann proved his
minimax theorem in 1928. This theorem establishes that in
zero-sum games with
perfect information (i.e., in which players know at each time all moves that have taken place so far), there exists a pair of strategies for both players that allows each to minimize his maximum losses (hence the name minimax). When examining every possible strategy, a player must consider all the possible responses of his adversary. The player then plays out the strategy which will result in the minimization of his maximum loss.
Such strategies, which minimize the maximum loss for each player, are called optimal. Von Neumann showed that their minimaxes are equal (in absolute value) and contrary (in sign). Another result he proved during his German period was the nonexistence of a static equilibrium. An equilibrium can only exist in an expanding economy.
Paul Samuelson edited an anniversary volume dedicated to this short German paper in 1972 and stated in the introduction that von Neumann was the only mathematician ever to make a significant contribution to economic theory.
Von Neumann improved and extended the minimax theorem to include games involving imperfect information and games with more than two players, publishing this result in his 1944
Theory of Games and Economic Behavior (written with
Oskar Morgenstern). The public interest in this work was such that
The New York Times ran a front-page story. In this book, von Neumann declared that economic theory needed to use
functional analytic methods, especially
convex sets and
topological fixed point theorem, rather than the traditional
differential calculus, because the
maximum–operator did not preserve differentiable functions.
Independently,
Leonid Kantorovich's functional analytic work on mathematical economics also focused attention on optimization theory, non-differentiability, and
vector lattices. Von Neumann's functional-analytic techniques—the use of
duality pairings of
real vector spaces to represent prices and quantities, the use of
supporting and
separating hyperplanes and convex set, and fixed-point theory—have been the primary tools of mathematical economics ever since.
[31] Von Neumann was also the inventor of the method of proof, used in game theory, known as
backward induction (which he first published in 1944 in the book co-authored with Morgenstern,
Theory of Games and Economic Behaviour).
US$ 50.18 billion (2012)[3]
