game theory

The most famous of Von Neumann's many contributions to mathematics was embodied in The Theory of Games and Economic Behavior (1944; rev. edn. 1953), which he wrote in collaboration with the American economist Oskar Morgenstern. It was concerned to develop a general theory of the rational behaviour of two or more people in what are basically conflicting or competing situations, although coalitions are also considered. First he developed the theory of two-person, zero-sum games, in which one player's gain is the other player's loss; he then suggested the concept of equilibrium, known as the minimax solution, in which one player minimizes the maximum loss the other player can impose on him. The solution of a game depended on the players using mixed strategies where the actual moves are chosen at random. Later, games for many players were devised, not necessarily zero-sum; here coalitions were possible, and a number of solution concepts were developed. The special case of 'the prisoner's dilemma' was made explicit, where neither player would wish to change his move given the strategy of the other — a pair of strategies that is worse for both players than any alternative. The development of the theory of games since Von Neumann has been considerable and involves decision making (especially under conditions of uncertainty), games against nature, and the very important interpretation of the theory known as linear programming.

(Published 1987)

— Frank George

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