The Nash equilibrium: A 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.

I n a brief 1950 communication to PNAS (1), John Forbes Nash for- mulated the notion of equilibrium that bears his name and that has

revolutionized economics and parts of other sciences. Nash, a young mathemat- ics graduate student at Princeton, was a part of the Camelot of game theory cen- tered around von Neumann and Mor- genstern. They had written Theory of Games and Economic Behavior (2) to expand economic analysis to allow econ- omists 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 Neu- mann 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 equilib- rium 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.† Neverthe- less, the Nash equilibrium, as it has be- come known, helped produce a revolu- tion 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 Games The 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 n- person 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 strat- egy being taken for each player.

(ref. 1, p. 48)

The notion of a strategy is quite gen- eral, 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 interpre- tation of a mixed-strategy is that of a population of randomly matched indi- viduals 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 unilat- eral 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 players. One such n-tuple counters another if the strategy of each player in the countering n-tuple yields the highest obtainable expecta- tion for its player against the n � 1 strategies of the other players in the countered n-tuple. A self-countering n-tuple is called an equilibrium point.

(ref. 1, p. 49)

That is, a Nash equilibrium is a set of strategies, one for each of the n players of a game, that has the property that each player’s choice is his best response to the choices of the n�1 other players. It would survive an announcement test: if all players announced their strategies simultaneously, nobody would want to reconsider. The Nash equilibrium has found many uses in economics, partly because it can be usefully interpreted in a number of ways.

When the goal is to give advice to all of the players in a game (i.e., to advise each player what strategy to choose), any advice that was not an equilibrium would have the unsettling property that there would always be some player for whom the advice was bad, in the sense that, if all other players followed the parts of the advice directed to them, it would be better for some player to do differently than he was advised. If the

advice is an equilibrium, however, this will not be the case, because the advice to each player is the best response to the advice given to the other players. This point of view is sometimes also used to derive predictions of what play- ers would do, if they can be approxi- mated as ‘‘perfectly rational’’ players who can all make whatever calculations are necessary and so are in the posi- tion of deriving the relevant advice for themselves.

When the goal is prediction rather than prescription, a Nash equilibrium can also be interpreted as a potential stable point of a dynamic adjustment process in which individuals adjust their behavior to that of the other players in the game, searching for strategy choices that will give them better results. This point of view has been productive in biology also: when mixed strategies are interpreted as the proportion of a popu- lation choosing each of a set of strate- gies, game payoffs are interpreted as the change in inclusive fitness that results from the play of the game, and the dy- namics are interpreted as population dynamics (6, 7). No presumptions of rationality are made in this case, of course, but only of simple self-interested dynamics. This evolutionary approach has also been attractive to economists (e.g., ref. 8).

A third interpretation is that a Nash equilibrium is a self-enforcing agree- ment, that is, an (implicit or explicit) agreement that, once reached by the players, does not need any external means of enforcement, because it is in the self interest of each player to follow

This Perspective is published as part of a series highlighting landmark papers published in PNAS. Read more about this classic PNAS article online at www.pnas.org�misc� classics.shtml.

This paper was submitted directly (Track II) to the PNAS office.

*To whom correspondence should be addressed. E-mail: cah2k@cms.mail.virginia.edu.

†In a personal communication with one of the authors, Nash notes that von Neumann was a ‘‘European gentleman’’ but was not an enthusiastic supporter of Nash’s approach.

© 2004 by The National Academy of Sciences of the USA

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the agreement if the others do. Viewed in this way, the Nash equilibrium has helped to clarify a distinction sometimes still made between ‘‘cooperative’’ and ‘‘noncooperative’’ games, with coopera- tive games being those in which agree- ments can be enforced (e.g., through the courts), and noncooperative games be- ing those in which no such enforcement mechanism exists, so that only equilib- rium agreements are sustainable. One trend in modern game theory, often re- ferred to as the ‘‘Nash program,’’ is to erase this distinction by including any relevant enforcement mechanisms in the model of the game, so that all games can be modeled as noncooperative. Nash took initial steps in this direction in his early and influential model of bar- gaining as a cooperative game (9) and then as a noncooperative game (10).

Nash’s 1950 PNAS paper not only formulated the definition of equilibrium but also announced the proof of exis- tence that he obtained using Kakutani’s (11) fixed point theorem. This technique of proof subsequently became standard in economics, e.g., the notion of a com- petitive equilibrium as a vector of antici- pated prices resulting in production and consumption decisions that generate the same vector of prices. In a personal communication to one of the authors, Nash remarked, ‘‘I know that S. Kaku- tani’s generalized fixed point theorem was actually inspired to improve on some arguments made by von Neumann in an economic context in the 1930s.’’

Nash shared the 1994 Nobel Prize with John Harsanyi and Reinhard Selten. Harsanyi was cited for extending the Nash equilibrium to the larger class of games called games of incomplete information, in which players need not be assumed to know other players’ pref- erences and feasible choices (12). Selten was cited for his work on equilibrium refinements, which takes the point of view that the requirements of the Nash equilibrium are necessary conditions for advice to perfectly rational players but are not sufficient conditions, and there may be superfluous equilibria that can be removed from consideration by ap- propriate refinements that focus atten- tion on a nonempty subset of Nash equilibria (13, 14). The Nash equilib- rium has been extended, refined, and generalized in other directions as well. One noteworthy generalization of mixed strategy equilibrium is ‘‘correlated equi- librium’’ (15), which considers not only independently randomized strategies for each player but also jointly randomized strategies that may allow coordination among groups of players.

Equilibrium and Social Dilemmas The Nash equilibrium is useful not just when it is itself an accurate predictor of how people will behave in a game but also when it is not, because then it iden- tifies situations in which there is a ten- sion between individual incentives and other motivations. A class of problems that have received a good deal of study from this point of view is the family of ‘‘social dilemmas,’’ in which there is a socially desirable action that is not a Nash equilibrium. Indeed, one of the first responses to Nash’s definition of equilibrium gave rise to one of the best known models in the social sciences, the Prisoners’ Dilemma. This model began life as a simple experiment conducted in January 1950 at the Rand Corporation by mathematicians Melvin Dresher and Merrill Flood, to demonstrate that the Nash equilibrium would not necessarily be a good predictor of behavior. Each of the two players in that game had to choose one of two decisions, which, for expositional purposes, we will call ‘‘co- operate’’ or ‘‘defect.’’ The game speci- fies the payoffs for each player for each of the four possible outcomes: (cooper- ate, cooperate), (cooperate, defect), (de- fect, cooperate), and (defect, defect). The payoffs used were such that each player’s best counter to either of the other’s choices was to defect, but both players would earn more if they both cooperated than if they both chose their equilibrium decision and defected.

Nash’s thesis advisor, Albert Tucker, was preparing a talk on recent develop- ments in game theory to be given to the Stanford Psychology Department when he saw the Dresher and Flood payoff numbers on a blackboard at the Rand Corporation. Tucker then devised the famous story of the dilemma faced by two prisoners who are each given incen- tives by the prosecutor to confess, even though both would be better off if nei- ther confesses than if they both do (16, 17). In the initial experiment (18) and in innumerable experiments that followed, players often succeed, at least to some degree, in cooperating with one another and avoiding equilibrium play (19).‡

Social scientists across many disci- plines have found prisoner’s dilemmas helpful in thinking about phenomena ranging from ecological degradation (20) to arms races. What the Nash equi- librium makes clear, even in a game like the prisoner’s dilemma in which it may not be an accurate point predictor, is that the cooperative outcome, because it

is not an equilibrium, is going to be un- stable in ways that can make coopera- tion difficult to maintain. This observa- tion has been confirmed in many subsequent experiments on this and more general ‘‘social dilemmas’’ (see, e.g., refs. 21–23). You can put yourself into a social dilemma game by going to the link: http://veconlab.econ.virginia. edu/tddemo.htm and playing against decisions retrieved from a database. This Traveler’s Dilemma game is some- what more complex than a prisoner’s dilemma, in that the best decision is not independent of your beliefs about what strategy might be selected by the other player (24).

Design of Markets and Social Institutions One of the ways in which research on dilemmas and other problems of collec- tive action has proceeded is to look for the social institutions that have been invented to change games from prison- er’s dilemmas to games in which cooper- ation is sustainable as an equilibrium; see e.g., Elinor Ostrom’s 1998 presiden- tial address to the American Political Science Association (25). For example, just as firms selling similar products may undercut each other’s price until price is driven down to cost, it is possible for a series of actions and reactions to force players in a game into a situation that is relatively bad for all concerned, which provides strong incentives for restric- tions on unilateral actions. This kind of ‘‘unraveling’’ is encountered in some labor markets in which employers may try to gain an advantage by making early offers. In the market for federal appellate court clerks, for example, posi- tions began to be arranged earlier and earlier, as some judges tried to hire clerks just before their competitors. This continued until offers (for jobs that would begin only on graduation from law school) were being made to law stu- dents 2 years in advance, only on the basis of first-year law school grades (see ref. 26). This situation was widely viewed as unsatisfactory, because it forced both judges and law students to make decisions far in advance, on the basis of too little information. The most recent of many attempts to reform this market took the form of a year-long moratorium on the hiring of clerks by appellate judges, which ended the day after Labor Day 2003, with only third- year law students to be hired. It is still too early to know whether this relatively mild intervention will finally solve the unraveling of the law clerk market. But a moratorium by itself does not change the rules of the game sufficiently to al- ter the dilemma-like properties of the equilibrium, and so we predict that fur-

‡H. Raiffa independently conducted experimens with a Pris- oner’s Dilemma game in 1950, but he did not publish them (see ref. 19).

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ther changes will be needed if the famil- iar problems are to be avoided in the long term.

An alternative solution to this prison- er’s dilemma problem of timing in such markets is to arrange an organized clearinghouse in which both employers and job candidates participate. Such clearinghouses arose, for example, as a result of situations in which medical stu- dents were being hired well over 1 year before graduation; this happened in the U.S. in the 1940s and in the U.K. in the 1960s. Today, graduates of American medical schools (and others applying for residencies at American hospitals) sub- mit rank order lists of preferred posi- tions to a clearinghouse called the Na- tional Resident Matching Program. Roth (27, 28) has studied these U.S. and U.K. matching markets. It turns out that one important factor in whether such a labor market clearinghouse succeeds or fails is whether the clearinghouse is de- signed so that it is a Nash equilibrium for applicants and employers to partici- pate in a way that produces a matching of workers to jobs that is stable, in the sense that no employer and applicant who are not matched to one another would both prefer to be (29, 30).

Participation in such a clearinghouse is even more straightforward if the clearinghouse is constructed so that it is a Nash equilibrium for applicants to simply put down their true preferences, regardless of how likely they think they are to receive each of the jobs for which they have applied, or how other appli- cants are ranking those jobs. For match- ing markets like these entry level labor markets, this kind of equilibrium is pos- sible for an appropriately designed clearinghouse. Thus the current version of the National Resident Matching Pro- gram algorithm, designed by Roth and Peranson (31), has the property that applicants can confidently be advised it is in their best interest to submit rank order lists of residencies that correspond to their true preferences. That game theorists have started to play a role in designing such clearinghouses and other markets is an indication of how game theory has grown from a conceptual to a practical tool.

Auctions are another kind of market in which it is becoming increasingly common for game theorists to be asked for design advice (see, e.g., refs. 32 and 33). And the economic theory of auc- tions [for which a Nobel Prize was given to William Vickrey in 1996, in large part for his seminal 1961 paper (34)] is a perfect example of how game theory and the Nash equilibrium have changed economics. Before game theory, econo- mists often analyzed markets simply in

terms of the supply and demand of the goods to be sold, with no way to discuss the rules of the game that make one kind of auction different from another or make auctions different from other kinds of markets (such as stock markets or shopping malls). Today, that discus- sion is most often carried forward by analyzing the Nash equilibria of the auc- tion rules.

Experimental Economics It is worth mentioning that Nash both commented on and participated in early experiments in economics (see ref. 35). It was a natural progression to move from a fascination with mathematical models of strategic behavior to the ob- servation of decisions made by people who are playing for real money payoffs under controlled conditions in the labo- ratory. Indeed, as game theory started its move to the forefront of economic theory, it generated scores of testable predictions and helped lay the ground- work for the introduction of experimen- tal methods into economics (36, 37).

The increasing use of experimental methods in economics and the growing interaction between economics and psy- chology was itself recognized by the 2002 Economics Nobel Prize that was awarded to a psychologist, Daniel Kah- neman (see, e.g., ref. 38) and an econo- mist, Vernon Smith (see, e.g., ref. 39). Before Smith’s experiments, it was widely believed that the competitive predictions of supply�demand intersec- tions required very large numbers of well-informed traders. Smith showed that competitive efficient outcomes could be observed with surprisingly small numbers of traders, each with no direct knowledge of the others’ costs or values. An important developing area of game theory is to explain these and other experimental results in the context of well-specified dynamic models of the interaction of strategic traders.

Another emerging connection be- tween game theory and experimentation is the increased use of experimental methods in teaching. A well designed classroom experiment shows students that the seemingly abstract equilibrium models can have surprising predictive power. The Internet makes it much eas- ier to run complex games with large groups of students. For example, �30 different types of games, auctions, and markets can be set up and run from a site, (http:��veconlab.econ.virginia.edu� admin.htm) that also provides sample data displays from classroom experi- ments. Most of the data displays and dynamically generated data graphs have options for hiding the relevant Nash predictions when the results are being

discussed and then showing the Nash predictions subsequently.

Modeling Learning and Stochastic Equilibrium Experimentation has helped move game theorists to focus on approaches that are able to predict how people actually be- have when the perfect foresight and per- fect rationality assumptions of classical game theory are not satisfied. The ex- perimental literature is full of examples both of games in which observed behav- ior quickly converges to equilibrium be- havior and games in which equilibrium is a persistently poor predictor. This has helped to reinforce the trend, already apparent in the theoretical literature, to extend the static, often deterministic, formulation of equilibrium and consider dynamic and stochastic models.

Experiments make clear that players often do not conform to equilibrium behavior when they first experience a game, even if it is a game in which be- havior quickly converges to equilibrium as the players gain experience. One re- action to this has been to develop mod- els of learning that converge to equilib- rium in the limit, starting from nonequilibrium behavior (see, e.g., refs. 40 and 41). Another has been the begin- ning of attempts to develop models of learning that can predict observed be- havior in simple experimental games (see, e.g., ref. 42). In particular, learning models are useful for explaining pat- terns of adjustment, e.g., whether prices converge from above or below, as well as the ultimate steady-state distributions (e.g., refs. 43 and 44).

Some games are played only once, e.g., the exact strategic environments in many military, legal, and political con- flicts are unique to the particular time and place. In that case, there is no his- tory that can be used to form precise predictions about others’ decisions. Therefore, learning must occur by intro- spection, or thinking about what the other person might do, what they think you might do, etc. Such introspection is likely to be quite imprecise, especially when thinking about others’ beliefs or their beliefs about your beliefs. There has been some recent progress in for- mulating models of noisy introspection, which can then be used to predict and explain ‘‘non-Nash’’ behavior in experi- ments using games played only once (24, 45).

If a game is repeated, e.g., with ran- dom matchings from a population of players, some noise may persist even after average tendencies have stabilized. The ‘‘quantal response equilibrium’’ is based on the idea that players’ responses to differences in expected payoffs are

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sharper when such differences are large and are more random when such differ- ences are small (see ref. 46 for an exis- tence proof and ref. 47 for application to bidding in an auction). This notion of equilibrium is a generalization of the Nash equilibrium in the sense that the quantal response predictions converge to a Nash equilibrium as the noise is diminished. But the effect of nonnegli- gible noise is not merely to spread deci- sions around Nash predictions; strategic interactions cause feedbacks in some games that magnify and distort the ef- fects of noise. This approach has been used to explain data from some labora- tory experiments in which observed be- havior deviates from a unique Nash equilibrium and ends up on the oppo- site side of the set of feasible decisions (24, 43).

Still another approach seeks to recon- cile experimental evidence and equilib- rium predictions by considering how those predictions would differ if system- atic regularities in participants’ prefer- ences were modeled. One lesson that consistently emerges from small-group

bargaining experiments is that people are often as concerned with fairness is- sues as they are with their own payoffs (see, e.g., ref. 48). The incorporation of fairness and other notions of nonselfish preferences into standard models often brings economic game theory into con- tact with evolutionary explanations of human behavior (see, e.g., refs. 49 and 50).

Nash’s Contributions in Perspective In the last 20 years, the notion of a Nash equilibrium has become a required part of the tool kit for economists and other social and behavioral scientists, so well known that it does not need explicit citation, any more than one needs to cite Adam Smith when discussing com- petitive equilibrium. There have been modifications, generalizations, and re- finements, but the basic equilibrium analysis is the place to begin (and some- times end) the analysis of strategic inter- actions, not only in economics but also in law, politics, etc. The Nash equilib- rium is probably invoked as often in small-group (and not-so-small-group)

situations as competitive equilibrium is used in large markets. Students in eco- nomics classes today probably hear John Nash’s name as much as or more than that of any economist.

In the half century after the publica- tion of Nash’s PNAS paper, game the- ory moved into center stage in eco- nomic theory. Game theory has also become part of a lively scientific conver- sation with experimental and other em- pirical scientists and, increasingly, the source of practical advice on the design of markets and other economic environ- ments. Looking ahead, if game theory’s next 50 years are to be as productive, the challenges facing game theorists in- clude learning to incorporate more var- ied and realistic models of individual behavior into the study of strategic be- havior and learning to better use analyt- ical, experimental, and computational tools in concert to deal with complex strategic environments.

This work was funded in part by National Science Foundation Infrastructure Grant SES 0094800.

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