Game Theory (mathematics)

I

Introduction

Game Theory (mathematics), 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 noughts and crosses, chess, 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 1930s established the mathematical framework for all subsequent theoretical developments. During World War II military strategists in such areas as logistics, submarine warfare, and air defence 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 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 to 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, refers to what happens at the end of a game. In such games as chess or draughts, payoff may be as simple as declaring a winner or a loser. In poker or other gambling situations the payoff is usually money; the amount being predetermined by antes and bets amassed during the course of play.