Game theory

Game theory is a method of exploring the predicted and actual behavior of individuals in interactions with formalized incentive structures. Seemingly different types of interactions can be characterized as having similar incentive structures, thus being examples of a particular "game."

Though touched on by earlier mathematical results, modern game theory became a prominent branch of mathematics in the 1940s, especially after the 1944 publication of The Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. Game theory is closely related to economics in that it seeks to find rational strategies in situations where the outcome depends not only on one's own strategy and "market conditions", but upon the strategies chosen by other players with possibly different or overlapping goals. It also finds wider application in fields such as political science and military strategy.

The results can be applied to simple games of entertainment or to more significant aspects of life and society. An example of the latter is the prisoner's dilemma as popularized by mathematician Albert W. Tucker, which has many implications about the nature of human cooperation. Biologists have used game theory to understand and predict certain outcomes of evolution, such as the concept of evolutionarily stable strategy introduced by John Maynard Smith in his essay Game Theory and the Evolution of Fighting. See also Maynard Smith's book Evolution and the Theory of Games.

Game theory classifies games into many categories that determine which particular methods can be applied to solving them (and indeed how one defines "solved" for a particular category). Some common categories are:

Zero-sum games are those in which the total benefit to all players in the game must add to zero (or more informally put, that each player benefits only at the expense of another). Chess and Poker are zero-sum games, because one wins exactly the amount one's opponents lose. Business and politics, for example, are nonzero-sum games because some outcomes are good for all players or bad for all players. It is easier, however, to analyze a zero-sum game, and it turns out to be possible to transform any game into a zero-sum game by adding an additional dummy player often called "the board," whose losses compensate the players' net winnings.

Cooperative games are those in which the players may freely communicate among themselves before making game decisions and may make bargains to influence those decisions.

Complete information games are those in which each player has the same game-relevant information as every other player. Chess is a complete-information game, while Poker is not.

John Nash developed an "Optimum" strategy for games where no such optimum was previously defined, known as Nash equilibrium. This concept was further refined by Reinhart Selten. These men were awarded the Swedish Bank Prize in 1994 for their work on game theory, along with John Harsanyi who developed the analysis of games of incomplete information.

Other branches of mathematics, in particular probability and statistics, are commonly used in conjuction with game theory to analyse many games.