An application of mathematical logic to decision-making in games and, by extension, to economics, politics, military conflicts, and biology. The goal of game theory is to find the optimal strategy for one player to use when his opponent also plays optimally. A strategy may incorporate randomness, in which case it is referred to as a mixed strategy.
History of game theoryEarly ideas of game theory can be found in writings throughout history as diverse as the Bible and works by René Descartes, Sun Tzu (author of the 2,400-year-old The Art of War), and Charles Darwin. The basis of modern game theory is an outgrowth of several books that deal with related subjects such as economics and probability. These include Augustin Cournot's Researches into the Mathematical Principles of the Theory of Wealth (1838), which gives an intuitive explanation of what would eventually be formalized by John Nash as Nash equilibrium; Francis Edgeworth's Mathematical Psychics, which explored the notion of competitive equilibria in a two-type (or two-person) economy; and Emile Borel's Algebre et calcul des probabilites (1927), which gave the first insight into mixed strategies.1
Game theory finally came of age through the efforts of two European immigrants to the United States working at the Institute of Advanced Studies in Princeton. Around 1940, the idea of the utility function was taken up by John von Neumann, who had been forced to flee his native Hungary when the Nazis invaded, and the economist Oskar Morgenstern (1902-1976), who had left Austria because he loathed the National Socialists. In Princeton the two immigrants worked together on what they initially thought would be a short paper on the theory of games, but that kept growing until it finally appeared in 1944 as an opus of 600 pages with the title Theory of Games and Economic Behavior.2
Types of gamesIn singular games (e.g., solitaire) the player's strategy is determined solely by the rules. In dual games (e.g., chess, football) one side's strategy must take into account the possible strategies of the other. Dual games are usually zero-sum: one side's gain exactly equals the other's loss. In practical situations, however, they may be non-zero-sum, as where two conflicting nations negotiate a truce that benefits both. Two major strategies are available to players of dual games: the minimax, in which a player evaluates his probable maximum loss and attempts to minimize it; and the maximin, in which a player evaluates his probably minimum gain and attempts to maximize it. Von Neumann showed in his minimax theorem (first stated in 1928) that, since, statistically, minimax and maximin strategies negate each other, most dual games are not worth playing in that their outcome is determined solely by the rules.
In plural or n-person games (e.g., poker), an individual's gain does not necessarily imply another's loss; and more complex considerations must affect each player's choice of action. More over, the outcome may be affected by the formation of coalitions, possibly reducing the n-person game to a dual one.
Related category GAMES AND PUZZLES
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