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 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.

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 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). tive action has proceeded is to look for Nash’s 1950 PNAS paper not only Nash equilibrium would not necessarily be a good predictor of behavior. Each the social institutions that have been formulated the definition of equilibrium invented to change games from prison- but also announced the proof of exis- of the two players in that game had to choose one of two decisions, which, for er’s dilemmas to games in which cooper- tence that he obtained using Kakutani’s ation is sustainable as an equilibrium; (11) fixed point theorem. This technique expositional purposes, we will call ‘‘cooperate’’ or ‘‘defect.’’ The game speci- see e.g., Elinor Ostrom’s 1998 presiden- of proof subsequently became standard tial address to the American Political in economics, e.g., the notion of a com- fies the payoffs for each player for each of the four possible outcomes: (cooper- Science Association (25). For example, petitive equilibrium as a vector of antici- ate, cooperate), (cooperate, defect), (de- just as firms selling similar products may pated prices resulting in production and fect, cooperate), and (defect, defect). undercut each other’s price until price is consumption decisions that generate the The payoffs used were such that each driven down to cost, it is possible for a same vector of prices. In a personal player’s best counter to either of the series of actions and reactions to force communication to one of the authors, other’s choices was to defect, but both players in a game into a situation that is Nash remarked, ‘‘I know that S. Kaku- players would earn more if they both relatively bad for all concerned, which tani’s generalized fixed point theorem cooperated than if they both chose their provides strong incentives for restric- was actually inspired to improve on equilibrium decision and defected. tions on unilateral actions. This kind of ‘‘unraveling’’ is encountered in some some arguments made by von Neumann Nash’s thesis advisor, Albert Tucker, labor markets in which employers may in an economic context in the 1930s.’’ was preparing a talk on recent develop- try to gain an advantage by making Nash shared the 1994 Nobel Prize ments in game theory to be given to the early offers. In the market for federal with John Harsanyi and Reinhard Stanford Psychology Department when appellate court clerks, for example, posi- Selten. Harsanyi was cited for extending he saw the Dresher and Flood payoff tions began to be arranged earlier and the Nash equilibrium to the larger class numbers on a blackboard at the Rand earlier, as some judges tried to hire of games called games of incomplete Corporation. Tucker then devised the clerks just before their competitors. This information, in which players need not famous story of the dilemma faced by continued until offers (for jobs that be assumed to know other players’ pref- two prisoners who are each given incen- would begin only on graduation from erences and feasible choices (12). Selten tives by the prosecutor to confess, even law school) were being made to law stu- was cited for his work on equilibrium though both would be better off if nei- dents 2 years in advance, only on the refinements, which takes the point of ther confesses than if they both do (16, basis of first-year law school grades (see view that the requirements of the Nash 17). In the initial experiment (18) and in ref. 26). This situation was widely equilibrium are necessary conditions for innumerable experiments that followed, viewed as unsatisfactory, because it advice to perfectly rational players but players often succeed, at least to some forced both judges and law students to degree, in cooperating with one another are not sufficient conditions, and there make decisions far in advance, on the and avoiding equilibrium play (19).‡ may be superfluous equilibria that can basis of too little information. The most be removed from consideration by ap- Social scientists across many disci- recent of many attempts to reform this propriate refinements that focus atten- plines have found prisoner’s dilemmas market took the form of a year-long tion on a nonempty subset of Nash helpful in thinking about phenomena moratorium on the hiring of clerks by equilibria (13, 14). The Nash equilib- ranging from ecological degradation appellate judges, which ended the day rium has been extended, refined, and (20) to arms races. What the Nash equi- after Labor Day 2003, with only third- generalized in other directions as well. librium makes clear, even in a game like year law students to be hired. It is still One noteworthy generalization of mixed the prisoner’s dilemma in which it may too early to know whether this relatively strategy equilibrium is ‘‘correlated equi- not be an accurate point predictor, is mild intervention will finally solve the librium’’ (15), which considers not only that the cooperative outcome, because it unraveling of the law clerk market. But independently randomized strategies for a moratorium by itself does not change each player but also jointly randomized ‡H. Raiffa independently conducted experimens with a Pris- the rules of the game sufficiently to al- strategies that may allow coordination oner’s Dilemma game in 1950, but he did not publish them ter the dilemma-like properties of the among groups of players. (see ref. 19). equilibrium, and so we predict that fur-

4000 ͉ www.pnas.org͞cgi͞doi͞10.1073͞pnas.0308738101 Holt and Roth ther changes will be needed if the famil- terms of the supply and demand of the discussed and then showing the Nash iar problems are to be avoided in the goods to be sold, with no way to discuss predictions subsequently. long term. the rules of the game that make one An alternative solution to this prison- kind of auction different from another Modeling Learning and er’s dilemma problem of timing in such or make auctions different from other Stochastic Equilibrium markets is to arrange an organized kinds of markets (such as stock markets Experimentation has helped move game clearinghouse in which both employers or shopping malls). Today, that discus- theorists to focus on approaches that are and job candidates participate. Such sion is most often carried forward by able to predict how people actually be- clearinghouses arose, for example, as a analyzing the Nash equilibria of the auc- have when the perfect foresight and per- result of situations in which medical stu- tion rules. fect rationality assumptions of classical dents were being hired well over 1 year game theory are not satisfied. The ex- before graduation; this happened in the Experimental Economics perimental literature is full of examples U.S. in the 1940s and in the U.K. in the It is worth mentioning that Nash both both of games in which observed behav- 1960s. Today, graduates of American commented on and participated in early ior quickly converges to equilibrium be- medical schools (and others applying for experiments in economics (see ref. 35). havior and games in which equilibrium residencies at American hospitals) sub- It was a natural progression to move is a persistently poor predictor. This has mit rank order lists of preferred posi- from a fascination with mathematical helped to reinforce the trend, already tions to a clearinghouse called the Na- models of strategic behavior to the ob- apparent in the theoretical literature, to tional Resident Matching Program. servation of decisions made by people extend the static, often deterministic, Roth (27, 28) has studied these U.S. and who are playing for real money payoffs formulation of equilibrium and consider U.K. matching markets. It turns out that under controlled conditions in the labo- dynamic and stochastic models. one important factor in whether such a ratory. Indeed, as game theory started Experiments make clear that players labor market clearinghouse succeeds or its move to the forefront of economic often do not conform to equilibrium fails is whether the clearinghouse is de- theory, it generated scores of testable behavior when they first experience a signed so that it is a Nash equilibrium predictions and helped lay the ground- game, even if it is a game in which be- for applicants and employers to partici- work for the introduction of experimen- havior quickly converges to equilibrium pate in a way that produces a matching tal methods into economics (36, 37). as the players gain experience. One re- of workers to jobs that is stable, in the The increasing use of experimental action to this has been to develop mod- sense that no employer and applicant methods in economics and the growing els of learning that converge to equilib- who are not matched to one another interaction between economics and psy- rium in the limit, starting from would both prefer to be (29, 30). chology was itself recognized by the nonequilibrium behavior (see, e.g., refs. Participation in such a clearinghouse 2002 Economics Nobel Prize that was 40 and 41). Another has been the begin- is even more straightforward if the awarded to a psychologist, Daniel Kah- ning of attempts to develop models of clearinghouse is constructed so that it is neman (see, e.g., ref. 38) and an econo- learning that can predict observed be- a Nash equilibrium for applicants to mist, Vernon Smith (see, e.g., ref. 39). havior in simple experimental games simply put down their true preferences, Before Smith’s experiments, it was (see, e.g., ref. 42). In particular, learning regardless of how likely they think they widely believed that the competitive models are useful for explaining pat- are to receive each of the jobs for which predictions of supply͞demand intersec- terns of adjustment, e.g., whether prices they have applied, or how other appli- tions required very large numbers of converge from above or below, as well cants are ranking those jobs. For match- well-informed traders. Smith showed as the ultimate steady-state distributions ing markets like these entry level labor that competitive efficient outcomes (e.g., refs. 43 and 44). markets, this kind of equilibrium is pos- could be observed with surprisingly Some games are played only once, sible for an appropriately designed small numbers of traders, each with no e.g., the exact strategic environments in clearinghouse. Thus the current version direct knowledge of the others’ costs or many military, legal, and political con- of the National Resident Matching Pro- values. An important developing area of flicts are unique to the particular time gram algorithm, designed by Roth and game theory is to explain these and and place. In that case, there is no his- Peranson (31), has the property that other experimental results in the context tory that can be used to form precise applicants can confidently be advised it of well-specified dynamic models of the predictions about others’ decisions. is in their best interest to submit rank interaction of strategic traders. Therefore, learning must occur by intro- order lists of residencies that correspond Another emerging connection be- spection, or thinking about what the to their true preferences. That game tween game theory and experimentation other person might do, what they think theorists have started to play a role in is the increased use of experimental you might do, etc. Such introspection is designing such clearinghouses and other methods in teaching. A well designed likely to be quite imprecise, especially markets is an indication of how game classroom experiment shows students when thinking about others’ beliefs or theory has grown from a conceptual to that the seemingly abstract equilibrium their beliefs about your beliefs. There a practical tool. models can have surprising predictive has been some recent progress in for- Auctions are another kind of market power. The Internet makes it much eas- mulating models of noisy introspection, in which it is becoming increasingly ier to run complex games with large which can then be used to predict and common for game theorists to be asked groups of students. For example, Ͼ30 explain ‘‘non-Nash’’ behavior in experi- for design advice (see, e.g., refs. 32 and different types of games, auctions, and ments using games played only once 33). And the economic theory of auc- markets can be set up and run from a (24, 45). tions [for which a Nobel Prize was given site, (http:͞͞veconlab.econ.virginia.edu͞ If a game is repeated, e.g., with ran- to William Vickrey in 1996, in large part admin.htm) that also provides sample dom matchings from a population of for his seminal 1961 paper (34)] is a data displays from classroom experi- players, some noise may persist even perfect example of how game theory ments. Most of the data displays and after average tendencies have stabilized. and the Nash equilibrium have changed dynamically generated data graphs have The ‘‘quantal response equilibrium’’ is economics. Before game theory, econo- options for hiding the relevant Nash based on the idea that players’ responses mists often analyzed markets simply in predictions when the results are being to differences in expected payoffs are

Holt and Roth PNAS ͉ March 23, 2004 ͉ vol. 101 ͉ no. 12 ͉ 4001 sharper when such differences are large bargaining experiments is that people situations as competitive equilibrium is and are more random when such differ- are often as concerned with fairness is- used in large markets. Students in eco- ences are small (see ref. 46 for an exis- sues as they are with their own payoffs nomics classes today probably hear John tence proof and ref. 47 for application (see, e.g., ref. 48). The incorporation of Nash’s name as much as or more than to bidding in an auction). This notion of fairness and other notions of nonselfish that of any economist. equilibrium is a generalization of the preferences into standard models often In the half century after the publica- Nash equilibrium in the sense that the brings economic game theory into con- tion of Nash’s PNAS paper, game the- quantal response predictions converge tact with evolutionary explanations ory moved into center stage in eco- to a Nash equilibrium as the noise is of human behavior (see, e.g., refs. 49 nomic theory. Game theory has also diminished. But the effect of nonnegli- and 50). become part of a lively scientific conver- gible noise is not merely to spread deci- sation with experimental and other em- sions around Nash predictions; strategic Nash’s Contributions in Perspective pirical scientists and, increasingly, the interactions cause feedbacks in some In the last 20 years, the notion of a source of practical advice on the design games that magnify and distort the ef- Nash equilibrium has become a required of markets and other economic environ- fects of noise. This approach has been part of the tool kit for economists and ments. Looking ahead, if game theory’s used to explain data from some labora- other social and behavioral scientists, so next 50 years are to be as productive, tory experiments in which observed be- well known that it does not need explicit the challenges facing game theorists in- havior deviates from a unique Nash citation, any more than one needs to clude learning to incorporate more var- equilibrium and ends up on the oppo- cite Adam Smith when discussing com- ied and realistic models of individual site side of the set of feasible decisions petitive equilibrium. There have been behavior into the study of strategic be- (24, 43). modifications, generalizations, and re- havior and learning to better use analyt- Still another approach seeks to recon- finements, but the basic equilibrium ical, experimental, and computational cile experimental evidence and equilib- analysis is the place to begin (and some- tools in concert to deal with complex rium predictions by considering how times end) the analysis of strategic inter- strategic environments. those predictions would differ if system- actions, not only in economics but also atic regularities in participants’ prefer- in law, politics, etc. The Nash equilib- This work was funded in part by National ences were modeled. One lesson that rium is probably invoked as often in Science Foundation Infrastructure Grant SES consistently emerges from small-group small-group (and not-so-small-group) 0094800.

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