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“The Nash equilibrium: A perspective”

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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 inf luence 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

www.pnas.org�cgi�doi�10.1073�pnas.0308738101 PNAS � March 23, 2004 � vol. 101 � no. 12 � 3999 – 4002

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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 inf luential 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 superf luous 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).

4000 � www.pnas.org�cgi�doi�10.1073�pnas.0308738101 Holt and Roth

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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-
f licts 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

Holt and Roth PNAS � March 23, 2004 � vol. 101 � no. 12 � 4001

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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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4002 � www.pnas.org�cgi�doi�10.1073�pnas.0308738101 Holt and Roth

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