The Nash Equilibrium: a Perspective

PERSPECTIVE The Nash equilibrium: A perspective Charles A. Holt* and Alvin E. Roth Department of Economics, University of Virginia, Charlottesville, VA 22904-4182; and Department of Economics and Harvard Business School, Harvard University, Cambridge, MA 02138 Edited by Vernon L. Smith, George Mason University, Fairfax, VA, and approved January 28, 2004 (received for review January 7, 2004) In 1950, John Nash contributed a remarkable one-page PNAS article that deﬁned and characterized a notion of equilibrium for n- person games. This notion, now called the ‘‘Nash equilibrium,’’ has been widely applied and adapted in economics and other behav- ioral 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. n a brief 1950 communication to The notion of a strategy is quite gen- advice is an equilibrium, however, this PNAS (1), John Forbes Nash for- eral, and it includes ‘‘mixed’’ strategies will not be the case, because the advice mulated the notion of equilibrium that are probability distributions over to each player is the best response to that bears his name and that has decisions, e.g., an inspector who audits the advice given to the other players. I This point of view is sometimes also revolutionized economics and parts of on a random basis or a poker player other sciences. Nash, a young mathemat- who sometimes bluffs. Another interpre- used to derive predictions of what play- ics graduate student at Princeton, was a tation of a mixed-strategy is that of a ers would do, if they can be approxi- part of the Camelot of game theory cen- population of randomly matched indi- mated as ‘‘perfectly rational’’ players tered around von Neumann and Mor- viduals in the role of each player of the who can all make whatever calculations genstern. They had written Theory of game, some proportion of whom make are necessary and so are in the posi- Games and Economic Behavior (2) to each of a number of available choices. tion of deriving the relevant advice for expand economic analysis to allow econ- The idea of the Nash equilibrium is that themselves. omists to model the ‘‘rules of the game’’ a set of strategies, one for each player, When the goal is prediction rather that influence particular environments would be stable if nobody has a unilat- than prescription, a Nash equilibrium and to extend the scope of economic eral incentive to deviate from their own can also be interpreted as a potential theory to include strategic small-group strategy: stable point of a dynamic adjustment situations in which each person must try process in which individuals adjust their Any n-tuple of strategies, one for to anticipate others’ actions. von Neu- behavior to that of the other players in each player, may be regarded as a mann and Morgenstern’s definition of the game, searching for strategy choices point in the product space obtained that will give them better results. This equilibrium for ‘‘noncooperative’’ games by multiplying the n strategy spaces was largely confined to the special case point of view has been productive in of the players. One such n-tuple biology also: when mixed strategies are of ‘‘two-person zero-sum’’ games, in counters another if the strategy of interpreted as the proportion of a popu- which one person’s gain is another’s each player in the countering n-tuple lation choosing each of a set of strate- loss, so the payoffs always sum to zero yields the highest obtainable expecta- gies, game payoffs are interpreted as the (3). Nash proposed a notion of equilib- tion for its player against the n Ϫ 1 change in inclusive fitness that results rium that applied to a much wider class strategies of the other players in the from the play of the game, and the dy- of games without restrictions on the countered n-tuple. A self-countering namics are interpreted as population payoff structure or number of players n-tuple is called an equilibrium point. dynamics (6, 7). No presumptions of (1, 4, 5). von Neumann’s reaction was † (ref. 1, p. 49) rationality are made in this case, of polite but not enthusiastic. Neverthe- course, but only of simple self-interested That is, a Nash equilibrium is a set of less, the Nash equilibrium, as it has be- dynamics. This evolutionary approach strategies, one for each of the n players come known, helped produce a revolu- has also been attractive to economists of a game, that has the property that tion in the use of game theory in (e.g., ref. 8). economics, and it was the contribution each player’s choice is his best response Ϫ A third interpretation is that a Nash for which Nash was cited by the Nobel to the choices of the n 1 other players. equilibrium is a self-enforcing agree- Prize committee at the time of his It would survive an announcement test: ment, that is, an (implicit or explicit) award, 44 years later. if all players announced their strategies agreement that, once reached by the simultaneously, nobody would want to players, does not need any external Equilibrium Points in n-Person Games reconsider. The Nash equilibrium has means of enforcement, because it is in The first part of the 1950 PNAS paper found many uses in economics, partly the self interest of each player to follow introduces the model of a game with n because it can be usefully interpreted in participants, or ‘‘players,’’ who must a number of ways. each select a course of action, or When the goal is to give advice to all This Perspective is published as part of a series highlighting ‘‘strategy’’: of the players in a game (i.e., to advise landmark papers published in PNAS. Read more about each player what strategy to choose), this classic PNAS article online at www.pnas.org͞misc͞ classics.shtml. One may define a concept of an n- any advice that was not an equilibrium person game in which each player This paper was submitted directly (Track II) to the PNAS would have the unsettling property that ofﬁce. has a finite set of pure strategies and there would always be some player for *To whom correspondence should be addressed. E-mail: in which a definite set of payments to whom the advice was bad, in the sense [email protected]. the n players corresponds to each that, if all other players followed the †In a personal communication with one of the authors, Nash n-tuple of pure strategies, one strat- parts of the advice directed to them, it notes that von Neumann was a ‘‘European gentleman’’ but egy being taken for each player. would be better for some player to do was not an enthusiastic supporter of Nash’s approach. (ref. 1, p. 48) differently than he was advised. If the © 2004 by The National Academy of Sciences of the USA www.pnas.org͞cgi͞doi͞10.1073͞pnas.0308738101 PNAS ͉ March 23, 2004 ͉ vol. 101 ͉ no. 12 ͉ 3999–4002 Downloaded by guest on September 26, 2021 the agreement if the others do. Viewed Equilibrium and Social Dilemmas is not an equilibrium, is going to be un- in this way, the Nash equilibrium has The Nash equilibrium is useful not just stable in ways that can make coopera- helped to clarify a distinction sometimes when it is itself an accurate predictor of tion difficult to maintain. This observa- still made between ‘‘cooperative’’ and how people will behave in a game but tion has been confirmed in many ‘‘noncooperative’’ games, with coopera- also when it is not, because then it iden- subsequent experiments on this and tive games being those in which agree- tifies situations in which there is a ten- more general ‘‘social dilemmas’’ (see, ments can be enforced (e.g., through the sion between individual incentives and e.g., refs. 21–23). You can put yourself courts), and noncooperative games be- other motivations. A class of problems into a social dilemma game by going to ing those in which no such enforcement that have received a good deal of study the link: http://veconlab.econ.virginia. mechanism exists, so that only equilib- from this point of view is the family of edu/tddemo.htm and playing against rium agreements are sustainable. One ‘‘social dilemmas,’’ in which there is a decisions retrieved from a database. trend in modern game theory, often re- socially desirable action that is not a This Traveler’s Dilemma game is some- ferred to as the ‘‘Nash program,’’ is to Nash equilibrium. Indeed, one of the what more complex than a prisoner’s erase this distinction by including any first responses to Nash’s definition of dilemma, in that the best decision is not relevant enforcement mechanisms in the equilibrium gave rise to one of the best independent of your beliefs about what strategy might be selected by the other model of the game, so that all games known models in the social sciences, the player (24). can be modeled as noncooperative. Prisoners’ Dilemma. This model began Nash took initial steps in this direction life as a simple experiment conducted in Design of Markets and Social Institutions January 1950 at the Rand Corporation in his early and influential model of bar- One of the ways in which research on gaining as a cooperative game (9) and by mathematicians Melvin Dresher and Merrill Flood, to demonstrate that the dilemmas and other problems of collec- then as a noncooperative game (10).

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