GAME THEORY

branch of mathematics used to analyze competitive situations whose outcomes depend not only on one's own choices, and perhaps chance, but also on the choices made by other parties, or players. Since the outcome of a game is dependent on what all players do, each player tries to anticipate the probable choices of other players in order to determine his own best choice. How these interdependent strategic calculations may reasonably be made is the subject of the theory. Modern game theory was created practically at one stroke by the publication in 1944 of Theory of Games and Economic Behavior by the mathematician John von Neumann and the economist Oskar Morgenstern. This book stimulated rapid, worldwide development of the mathematical theory and its applications to economics, politics, military science, operations research, business, law, sports, biology, and other fields. It has had a major influence, widening and refining common discourse on strategic thinking. The theory has several major divisions, the following being the most important: branch of applied mathematics fashioned to analyze certain situations in which there is an interplay between parties that may have similar, opposed, or mixed interests. In a typical game, decision-making players, who each have their own goals, try to outsmart one another by anticipating each other's decisions; the game is finally resolved as a consequence of the players' decisions. A solution to a game prescribes the decisions the players should make and describes the game's appropriate outcome. Game theory serves as a guide for players and as a tool for predicting the outcome of a game. Although game theory may be used to analyze ordinary parlour games, its range of application is much wider. In fact, game theory was originally designed by the Hungarian-born American mathematician John von Neumann and his colleague Oskar Morgenstern, a German-born American economist, to solve problems in economics. In their book The Theory of Games and Economic Behavior, published in 1944, von Neumann and Morgenstern asserted that the mathematics developed for the physical sciences, which describes the workings of a disinterested nature, was a poor model for economics. They observed that economics is much like a game in which the players anticipate one another's moves and that it therefore requires a new kind of mathematics, which they appropriately named game theory. Game theory may be applied in situations in which decision makers must take into account the reasoning of other decision makers. By stressing strategic aspectsaspects controlled by the participants rather than by pure chancethe method both supplements and goes beyond the classical theory of probability. It has been used, for example, to determine the formation of political coalitions or business conglomerates, the optimum price at which to sell products or services, the power of a voter or a bloc of voters, the selection of a jury, the best site for a manufacturing plant, and even the behaviour of certain species in the struggle for survival. It would be surprising if any one theory could address such a wide range of games, and, in fact, there is no single game theory. A number of theories exist, each applicable to a different kind of situation and each with its own kind of solution (or solutions). This article discusses some of the simpler games and theories as well as the basic principles involved in game theory. Additional techniques and concepts that may be used in solving decision problems are treated in optimization. For information pertaining to the classical theory of probability, see the articles mathematics, history of; and probability theory. Additional reading The seminal work in game theory is John Von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, 3rd ed. (1953, reprinted 1980). Game theory as a whole is covered in the following books, listed in order of increasing difficulty: Anatol Rapoport, Fights, Games, and Debates (1960, reprinted 1967); Morton D. Davis, Game Theory: A Nontechnical Introduction, rev. ed. (1983); R. Duncan Luce and Howard Raiffa, Games and Decisions: Introduction and Critical Survey (1957, reprinted 1967); and Guillermo Owen, Game Theory, 2nd ed. (1982). Applications of game theory are presented in Nesmith C. Ankeny, Poker Strategy: Winning with Game Theory (1981, reprinted 1982); Robert Axelrod, The Evolution of Cooperation (1984), concerned with evolution and ecology; Steven J. Brams, Game Theory and Politics (1975); and Henry Hamburger, Games as Models of Social Phenomena (1979). Useful essays and journal articles include Robert J. Aumann and Michael Maschler, The Bargaining Set for Cooperative Games, in M. Dresher, L.S. Shapley, and A.W. Tucker (eds.), Advances in Game Theory (1964), pp. 443476, the basis of the AumannMaschler solution concept; Daniel Kahneman and Amos Tversky, The Psychology of Preferences, Scientific American, 246(1):160173 (January 1982), a discussion of the validity of the assumptions that underlie utility theory; L.S. Shapley, A Value for N-Person Games, in H.W. Kuhn and A.W. Tucker (eds.), Contributions to the Theory of Games, vol. 2 (1953), pp. 307317, the foundation of the Shapley value; John Maynard Smith, The Evolution of Behavior, Scientific American, 239(3):176192 (September 1978); and Philip D. Straffin, Jr., The Bandwagon Curve, American Journal of Political Science, 21(4):695709 (November 1977), which explains bandwagon effect on the basis of the Shapley value. Morton D. Davis