John von Neumann and Game Theory: a Brief Study of

“Game Theory, is the 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.”


“Game Theory, is the 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 amusements such as checkers, tic-tac-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, starting in 1912, wrote several papers on games of chance and theories of play (“la theorie du jeu”). In 1928 the Hungarian-American mathematician John von Neumann, published an article entitled “Zur Theorie der Gesellschaftspiele” (“Theory of Parlour Games”)proving the “minimax theorem” which established the mathematical framework for all subsequent theoretical developments.

The Minimax Theorem

This theorem says that there is always a rational solution to a precisely defined conflict between two people whose interests are completely opposite. It is a rational solution in that both parties can convince themselves that they cannot expect to do any better, given the nature of the conflict.

The U.S. Embrace

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.



Basic Concepts

Game

A particular sort of conflict in which ‘n’ amount 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.

Move

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.

Payoff

Also known as outcome, this is what happens at the end of a game. In some games this is as simple as declaring a winner or loser; in other games it can be an amount of money or points.

Perfect Information

A game is said to have perfect information if all moves are known to each of the players involved. Chess is a game with perfect information, and poker or bridge are games in which players have only partial information at their disposal.

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.

Applications

John Von Neumann, working with the Austrian economist Oskar Morgenstern at Princeton, linked Game Theory with economic behaviour : models can be developed for markets of various commodities with differing numbers of buyers and sellers, fluctuating values of supply and demand, and seasonal and cyclical variations. (See “Theory of Games and Economic Behaviour” – Von Neumann and Morgenstern)

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. For example, consider the problem of splitting one cake between two children. No matter how carefully a parent may try to divide the cake, at least one of the children will feel that they have been allocated a smaller piece. The solution is to let one child divide the cake and the other child can then choose a piece. In this way, greed ensures fair division because the first child cannot object because she divided the cake herself, and the second child was given the choice of pieces. This example is a simple illustration of the “minimax” principle upon which Game Theory is based.

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.

John von Neumann: 1903-1957

John von Neumann was born on December 3, 1903 in Budapest, Hungary, and educated at Zurich and at the universities of Berlin and Budapest. He died in Washington D.C., 1957.

He was a maths prodigy in Budapest; as a child, he could divide two eight-digit numbers in his head, he entertained family guests by memorizing columns from phone books, then reciting names, addresses and phone numbers perfectly. Earning a doctorate at twenty-two, at twenty-three he became the youngest person to lecture at the University of Berlin.

In 1930 he went to the United States to join the faculty of Princeton University and at the age of thirty, along with Albert Einstein, he was appointed one of the first professors of the Institute for Advanced Study, in Princeton, New Jersey. His contribution to the development of the electronic digital computer was so important, that almost all such machines are now referred to as von Neumann processors.

Through the 1930’s and early 1940’s, Von Neumann worked on game theory, hoping it would form the basis for a future exact science of economics. In 1937 he was accepted as a U.S. citizen and during World War II he served as a consultant on the Los Alamos atomic-bomb project.

In the late 1940’s, John von Neumann began to develop a theory of automata. He envisaged a systematic theory which would be mathematical and logical in form, and which would contribute in an essential way to our understanding of natural systems (natural automata) as well as to our understanding of both analogue and digital computers (artificial automata).

Von Neumann was involved in such diverse fields as ordnance, submarine warfare, bombing objectives, nuclear weapons (including the hydrogen bomb), military strategy, weather prediction, intercontinental ballistic missiles, high-speed digital computers, and computing methods. 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, and he built in 1952 the first computer using a flexible stored program, the MANIAC I.

In 1954, he was appointed to the U.S. Atomic Energy Commission. He received many awards and honours during his lifetime.

In 1956 the Atomic Energy Commission granted him the Enrico Fermi Award for outstanding contributions to the theory and design of electronic computers.

Some of Von Neumann’s works…

“The General and Logical Theory of Automata” (1951)

“Theory and Organisation of Complicated Automata” (1949)

“Probabilistic Logic and the Synthesis of Reliable Organisms from Unreliable Components” (1952)

“The Theory of Automata: Construction, Reproduction, Homogeneity” (1952)

“The Computer and the Brain” (1958)

Von Neumann was especially interested in complicated automata, such as the human nervous system and the tremendously large computers he foresaw for the future.

The two problems in automata theory that von Neumann concentrated on are both intimately related to complexity: the problems of reliability and self- reproduction.

More about Von Neumann…

Von Neumann and Automata Theory

Von Neumann and Information Theory

Von Neumann and Biology

References

“Prisoner’s Dilemma : John von Neumann, Game Theory and the Puzzle of the Bomb” – William Poundstone

“Artificial Life : The Quest for a New Creation” – Steven Levy

Author: thee_InVection_report

News Service: TheExperiement Network

URL: http://ei.cs.vt.edu/~history/VonNeumann.html

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