Neumann, John von

Neumann, John von (1903-57), Hungarian-American
mathematician, who developed the branch of mathematics known as the game
theory. He was born in Budapest, Hungary, and
educated at Zürich, Switzerland, and at the
universities of Berlin and Budapest. He went to the United States in 1930 to
join the faculty of Princeton University. After 1933 he was associated with the
Institute for Advanced Study in Princeton, New Jersey. He became a U.S. citizen
in 1937 and during World War II served as a consultant on the Los Alamos
atomic-bomb project. In March 1955 he became a member of the U.S. Atomic Energy
Commission.

Von Neumann was one of the world's outstanding mathematicians. He
is noted for his fundamental contributions to the theory of quantum mechanics,
particularly the concept of “rings of operators” (now known as Neumann
algebras), and also for his pioneering work in applied mathematics, mainly in
statistics and numerical analysis. He is also known for the design of
high-speed electronic computers. In 1956 the Atomic Energy Commission granted
him the Enrico Fermi Award for outstanding
contributions to the theory and design of electronic computers.

**Microsoft ® Encarta ® Reference Library 2003.** © 1993-2002 Microsoft Corporation. All rights reserved.

von Neumann, John

born Dec. 3, 1903,
Budapest, Hung.

died Feb. 8, 1957,
Washington, D.C., U.S.

original name Johann Von
Neumann Hungarian-American mathematician who made important contributions in
quantum physics, logic, meteorology, and computer science. His theory of games
had a significant influence upon economics.

Von Neumann studied
chemistry at the University of Berlin and, at Technische
Hochschule in Zürich,
received the diploma in chemical engineering in 1926. The same year, he
received the Ph.D. in mathematics from the University of Budapest, with a
dissertation about set theory. His axiomatization has
left a permanent mark on the subject; and his definition of ordinal numbers,
published when he was 20, has been universally adopted.

Von Neumann was privatdocent (lecturer) at Berlin in 1926–29 and at the
University of Hamburg in 1929–30. During this time he worked mainly on quantum
physics and operator theory. Largely because of his work, quantum physics and
operator theory can be viewed as two aspects of the same subject.

In 1930 von Neumann was
visiting lecturer at Princeton University; he was appointed professor in 1931.
In 1932 he gave a precise formulation and proof of the “ergodic
hypothesis” of statistical mathematics. His book on quantum mechanics, The
Mathematical Foundations of Quantum Mechanics, published in 1932, remains a
standard treatment of the subject. In 1933 he became a professor at the newly
founded Institute for Advanced Study, Princeton, keeping that position for the
rest of his life. Meanwhile, he turned his attention to the challenge made in
1900 by a German mathematician, David Hilbert (q.v.), who proposed 23 basic
theoretical problems for20th-century mathematical research. Von Neumann solved
a special case of Hilbert's fifth problem, the case of compact groups.

In the second half of
the 1930s the main part of von Neumann's publications, written partly in
collaboration with F.J. Murray, was on “rings of operators” (now called Neumann
algebras). Of all his work, these concepts will quite probably be remembered
the longest. Currently it is one of the most powerful tools in the study of
quantum physics. An important outgrowth of rings of operators is “continuous
geometry.” Von Neumann saw that what really determines the character of the
dimensional structure of a space is the group of rotations that the structure
allows. The groups of rotations associated with rings of operators make
possible the description of space with continuously varying dimensions.

About 20 of von
Neumann's 150 papers are in physics; the rest are distributed more or less
evenly among pure mathematics (mainly set theory, logic, topological group,
measure theory, ergodic theory, operator theory, and
continuous geometry) and applied mathematics (statistics, numerical analysis,
shock waves, flow problems, hydrodynamics, aerodynamics, ballistics, problems
of detonation, meteorology, and two nonclassical
aspects of applied mathematics, games and computers). His publications show a
break from pure to applied research around 1940.

During World War II, he
was much in demand as a consultant to the armed forces and to civilian
agencies. His two main contributions were his espousal of the implosion method
for bringing nuclear fuel to explosion and his participation in the development
of the hydrogen bomb.

The mathematical
cornerstone of von Neumann's theory of games is the “minimax
theorem,” which he stated in 1928; its elaboration and applications are in the
book he wrote jointly with Oskar Morgenstern in 1944,
Theory of Games and Economic Behavior. The minimax theorem says that for a large class of two-person
games, there is no point in playing. Either player may consider, for each
possible strategy of play, the maximum loss that he can expect to sustain with
that strategy and then choose as his “optimal” strategy the one that minimizes
the maximum loss. If a player follows this reasoning, then he can be
statistically sure of not losing more than that value called the minimax value. Since (this is the assertion of the theorem)
that minimax value is the negative of the one,
similarly defined, that his opponent can guarantee for himself, the long-run
outcome is completely determined by the rules.

In computer theory, von
Neumann did much of the pioneering work in logical design, in the problem of
obtaining reliable answers from a machine with unreliable components, the
function of “memory,” machine imitation of “randomness,” and the problem of
constructing automata that can reproduce their own kind. One of the most
striking ideas, to the study of which he proposed to apply computer techniques,
was to dye the polar ice caps so as to decrease the amount of energy they would
reflect—the result could warm the Earth enough to make the climate of Iceland
approximate that of Hawaii.

The “axiomatic method”
is sometimes mentioned as the secret of von Neumann's success. In his hands it
was not pedantry but perception; he got to the root of the matter by
concentrating on the basic properties (axioms) from which all else follows. His
insights were illuminating and his statements precise.

Game Theory

I INTRODUCTION

Game Theory, mathematical analysis of any
situation involving a conflict of interest, with the intent of indicating the
optimal choices that, under given conditions, will lead to a desired outcome. Although game theory has roots in the study of such well-known
amusements as checkers, tick-tack-toe, and poker—hence the name—it also
involves much more serious conflicts of interest arising in such fields as
sociology, economics, and political and military science.

Aspects of game theory were first explored
by the French mathematician Émile Borel,
who wrote several papers on games of chance and theories of play. The
acknowledged father of game theory, however, is the Hungarian-American
mathematician John von Neumann, who in a series of papers in the 1920s and '30s
established the mathematical framework for all subsequent theoretical
developments. During World War II military strategists in such areas as
logistics, submarine warfare, and air defense drew on
ideas that were directly related to game theory. Game theory thereafter
developed within the context of the social sciences. Despite such empirically
related interests, however, it is essentially a product of mathematicians.

II BASIC CONCEPTS

In game theory, the term *game* means a
particular sort of conflict in which *n* of individuals or groups (known
as players) participate. A list of rules stipulates the conditions under which
the game begins, the possible legal “moves” at each stage of play, the total
number of moves constituting the entirety of the game, and the terms of the
outcome at the end of play.

A Move

In game theory, a move is the way in which the
game progresses from one stage to another, beginning with an initial state of
the game through the final move. Moves may alternate between players in a
specified fashion or may occur simultaneously. Moves are made either by
personal choice or by chance; in the latter case an object such as a die,
instruction card, or number wheel determines a given move, the probabilities of
which are calculable.

B Payoff

*Payoff,* or *outcome,* is
a game-theory term referring to what happens at the end of a game. In such
games as chess or checkers, payoff may be as simple as declaring a winner or a
loser. In poker or other gambling situations the payoff is usually money; its
amount is predetermined by antes and bets amassed during the course of play, by
percentages or by other fixed amounts calculated on the odds of winning, and so
on.

C Extensive and Normal Form

One of the most important distinctions made in
characterizing different forms of games is that between extensive and normal. A
game is said to be in extensive form if it is characterized by a set of rules
that determines the possible moves at each step, indicating which player is to
move, the probabilities at each point if a move is to be made by a chance
determination, and the set of outcomes assigning a particular payoff or result
to each possible conclusion of the game. The assumption is also made that each
player has a set of preferences at each move in anticipation of possible
outcomes that will maximize the player's own payoff or minimize losses. A game
in extensive form contains not only a list of rules governing the activity of
each player, but also the preference patterns of each player. Common parlor games such as checkers and ticktacktoe and games
employing playing cards such as “go fish” and gin rummy are all examples.

Because of the enormous numbers of strategies
involved in even the simplest extensive games, game theorists have developed
so-called normalized forms of games for which computations can be carried out
completely. A game is said to be in normal form if the list of all expected
outcomes or payoffs to each player for every possible combination of strategies
is given for any sequence of choices in the game. This kind of theoretical game
could be played by any neutral observer and does not depend on player choice of
strategy.

D Perfect Information

A game is said to have perfect information
if all moves are known to each of the players involved. Checkers and chess are
two examples of games with perfect information; poker and bridge are games in
which players have only partial information at their disposal.

E Strategy

A strategy is a list of the optimal
choices for each player at every stage of a given game. A strategy, taking into
account all possible moves, is a plan that cannot be upset, regardless of what
may occur in the game.

III KINDS OF GAMES

Game theory distinguishes different varieties of games,
depending on the number of players and the circumstances of play in the game
itself.

A One-Person Games

Games such as solitaire are one-person, or singular,
games in which no real conflict of interest exists; the only interest involved
is that of the single player. In solitaire only the chance structure of the
shuffled deck and the deal of cards come into play. Single-person games,
although they may be complex and interesting from a probabilistic view, are not
rewarding from a game-theory perspective, for no adversary is making
independent strategic choices with which another must contend.

B Two-Person Games

Two-person, or dual, games include the largest
category of familiar games such as chess, backgammon, and checkers or two-team
games such as bridge. (More complex conflicts—*n*-person, or plural,
games—include poker, Monopoly, Parcheesi, and any game in which multiple
players or teams are involved.) Two-person games have been extensively analyzed
by game theorists. A major difficulty that exists, however, in extending the
results of two-person theory to *n*-person games is predicting the
interaction possible among various players. In most two-party games the choices
and expected payoffs at the end of the game are generally well-known, but when
three or more players are involved, many interesting but complicating
opportunities arise for coalitions, cooperation, and collusion.

C Zero-Sum Games

A game is said to be a zero-sum game
if the total amount of payoffs at the end of the game is zero. Thus, in a
zero-sum game the total amount won is exactly equal to the amount lost. In
economic contexts, zero-sum games are equivalent to saying that no production
or destruction of goods takes place within the “game economy” in question. Von
Neumann and Oskar Morgenstern showed in 1944 that any
*n*-person non-zero-sum game can be reduced to an *n* + 1 zero-sum
game, and that such *n* + 1 person games can be generalized from the
special case of the two-person zero-sum game. Consequently, such games
constitute a major part of mathematical game theory. One of the most important
theorems in this field establishes that the various aspects of maximal-minimal
strategy apply to all two-person zero-sum games. Known as the minimax theorem, it was first proven by von Neumann in
1928; others later succeeded in proving the theorem with a variety of methods
in more general terms.

IV APPLICATIONS

Applications of game theory are wide-ranging and
account for steadily growing interest in the subject. Von Neumann and
Morgenstern indicated the immediate utility of their work on mathematical game
theory by linking it with economic behavior. Models
can be developed, in fact, for markets of various commodities with differing
numbers of buyers and sellers, fluctuating values of supply and demand, and
seasonal and cyclical variations, as well as significant structural differences
in the economies concerned. Here game theory is especially relevant to the
analysis of conflicts of interest in maximizing profits and promoting the
widest distribution of goods and services. Equitable division of property and
of inheritance is another area of legal and economic concern that can be
studied with the techniques of game theory.

In the social sciences, *n*-person game theory
has interesting uses in studying, for example, the distribution of power in
legislative procedures. This problem can be interpreted as a three-person game
at the congressional level involving vetoes of the president and votes of
representatives and senators, analyzed in terms of successful or failed
coalitions to pass a given bill. Problems of majority rule and individual
decision making are also amenable to such study.

Sociologists have developed an entire branch of game
theory devoted to the study of issues involving group decision making.
Epidemiologists also make use of game theory, especially with respect to
immunization procedures and methods of testing a vaccine or other medication.
Military strategists turn to game theory to study conflicts of interest
resolved through “battles” where the outcome or payoff of a given war game is
either victory or defeat. Usually, such games are not examples of zero-sum
games, for what one player loses in terms of lives and injuries
is not won by the victor. Some uses of game theory in analyses of
political and military events have been criticized as a dehumanizing and
potentially dangerous oversimplification of necessarily complicating factors.
Analysis of economic situations is also usually more complicated than zero-sum
games because of the production of goods and services within the play of a
given “game.”

Contributed By: Joseph Warren Dauben

**Microsoft ® Encarta ® Reference Library 2003.** © 1993-2002 Microsoft Corporation. All rights reserved.