The Nash equilibrium: A perspective

Abstract

In 1950, John Nash contributed a remarkable one-page PNAS article that defined and characterized a notion of equilibrium for nperson games. This notion, now called the ''Nash equilibrium,'' has been widely applied and adapted in economics and other behavioral sciences. Indeed, game theory, with the Nash equilibrium as its centerpiece, is becoming the most prominent unifying theory of social science. In this perspective, we summarize the historical context and subsequent impact of Nash's contribution.I n a brief 1950 communication to PNAS (1), John Forbes Nash formulated the notion of equilibrium that bears his name and that has revolutionized economics and parts of other sciences. Nash, a young mathematics graduate student at Princeton, was a part of the Camelot of game theory centered around von Neumann and Morgenstern. They had written Theory of Games and Economic Behavior (2) to expand economic analysis to allow economists to model the ''rules of the game'' that influence particular environments and to extend the scope of economic theory to include strategic small-group situations in which each person must try to anticipate others' actions. von Neumann and Morgenstern's definition of equilibrium for ''noncooperative'' games was largely confined to the special case of ''two-person zero-sum'' games, in which one person's gain is another's loss, so the payoffs always sum to zero (3). Nash proposed a notion of equilibrium that applied to a much wider class of games without restrictions on the payoff structure or number of players (1,4,5). von Neumann's reaction was polite but not enthusiastic. † Nevertheless, the Nash equilibrium, as it has become known, helped produce a revolution in the use of game theory in economics, and it was the contribution for which Nash was cited by the Nobel Prize committee at the time of his award, 44 years later. Equilibrium Points in n-Person GamesThe first part of the 1950 PNAS paper introduces the model of a game with n participants, or ''players,'' who must each select a course of action, or ''strategy'':One may define a concept of an nperson game in which each player has a finite set of pure strategies and in which a definite set of payments to the n players corresponds to each n-tuple of pure strategies, one strategy being taken for each player.(The notion of a strategy is quite general, and it includes ''mixed'' strategies that are probability distributions over decisions, e.g., an inspector who audits on a random basis or a poker player who sometimes bluffs. Another interpretation of a mixed-strategy is that of a population of randomly matched individuals in the role of each player of the game, some proportion of whom make each of a number of available choices. The idea of the Nash equilibrium is that a set of strategies, one for each player, would be stable if nobody has a unilateral incentive to deviate from their own strategy:Any n-tuple of strategies, one for each player, may be regarded as a point in the product space obtained by multiplying the n strategy spaces of the ...

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