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.

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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 tick-tack-toe 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 ® 2009. © 1993-2008 Microsoft Corporation. All rights reserved.