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.

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