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game theory
A mathematical formalism used to study human games, economics, 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.
Early 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
| Glossary of game theory | |
|---|---|
| categorical game | A game in which a tie is impossible |
| finite game | A game in which each player has a finite number of moves and a finite number of choices at each move |
| futile game | A game that allows a tie when played properly by both players |
| impartial game | A game in which the possible moves are the same for each player in any position |
| mixed strategy | A collection of moves together with a corresponding set of weights which are followed probabilistically in the playing of a game |
| partisan game | A game for which each player has a different set of moves in any position |
| payoff matrix | An m × n matrix that gives the possible outcome of a two-person zeros-sum game when player A has m possible moves and player B has n moves |
| strategy | A set of moves that a player plans to follow while playing a game |
| zero-sum game | A game in which players make payments only to each other. One player's loss is the other player's gain, so the total amount of "money" available is constant |
References
- Borel, Emile. "Algebre et calcul des probabilites," Comptes Rendus Academie des Sciences, Vol. 184, 1927.
- 230. Neumann, J. von and Morgenstern, O. Theory of Games and Economic Behavior. New York: Wiley, 1964.
Related categories
• GAMES AND PUZZLES
• MATHEMATICS
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