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 the 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 a 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

The payoff, or outcome, refers to what happens at the end of a game. In such games as chess or draughts, the 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.

C Extensive and Normal Form

One of the most important distinctions in characterizing different forms of games is that made 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, including 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 games such as noughts and crosses, droughts and games employing playing cards such as 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 a player’s 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. Draughts 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 between different varieties of games, depending on the number of players and the circumstances of play.

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 pack and the dealing 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 draughts or two-team games such as a bridge. Two-person games have been extensively analysed by game theorists. A major difficulty in extending the results of two-person theory to n-person games is predicting the possible interactions among various players. In most two-person 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, that is, 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 1944 von Neumann and Oskar Morgenstern showed 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, two-person zero-sum games constitute a major part of the 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 the steadily growing interest in the subject. Von Neumann and Morgenstern indicated the immediate utility of their work on the mathematical game theory by linking it 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, 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. 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 examples of non-zero-sum games, as 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 over-simplification 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”.

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