It wasn’t A Beautiful Mind—the book or the movie—that made John Forbes Nash, Jr., famous. It was his work in game theory, a theory that models strategic interactions between people as games. Before Nash, the only games theorists could get a handle on were artificial ones with no real-world applications. Nash’s insights enabled economists to expand the use of game theory to interesting practical problems.Game theory

Ed Nosal

Peter Rupert

English

Research Papers in Economics

lowest payoff the player could ever
receive, no matter how his opponent
plays. It turns out—and this is the linch-
pin to von Neumann’s proof—this 
winning strategy also minimizes the
highest payoff that his opponent can 
possibly receive. Hence, the strategy that
maximizes the guaranteed payoff is a
winning one because if a player chooses
to play something else, he will be made
worse off. Because von Neumann was
able to show that two-person zero-sum
games are always solvable, they became
an important vehicle for modeling strate-
gic interactions for many years.
Without doubt, von Neumann’s solution
to two-person zero-sum games was a
major building block in the theory of
games. However, one must recognize
that in most strategic social encounters,
it is generally not the case that whatever
one person gains, the other person loses.
A basic problem with the two-person
zero-sum game is that it is incapable of
describing interesting or realistic social
encounters. In most social encounters,
players can typically take actions that are
mutually beneficial, mutually destruc-
tive, or beneficial to one player and
harmful to the other.
April 15, 2002
Federal Reserve Bank of Cleveland
A Beautiful Theory
by Ed Nosal and Peter Rupert
ISSN 0428-1276
It wasn’t A Beautiful Mind—the book
or the movie—that made John
Forbes Nash, Jr., famous. It was his
work in game theory, a theory that
models strategic interactions between
people as games. Before Nash, the
only games theorists could get a 
handle on were artificial ones with 
no real-world applications. Nash’s
insights enabled economists to
expand the use of game theory to
interesting practical problems.
Long before the book, the movie, the
Golden Globe, and the Academy Award,
John Forbes Nash, Jr., the central char-
acter in A Beautiful Mind, was a legend
of epic proportions in economics. For
his contributions to economics, Nash
was awarded the 1994 Nobel Prize. This
Commentary explains why Nash’s work
had such a profound influence on the
way economists both think about and 
do economics. His influence has been
felt in virtually every field of study 
in economics.
■ Background
Nash was awarded a Nobel Prize for his
contributions to game theory. Game the-
ory is used to analyze problems that fea-
ture “strategic interaction” between indi-
viduals or “players.” Strategic interaction
refers to a situation in which the actions
of one party affect—directly or indi-
rectly—the well-being of other parties.
For example, an airline company that
lowers the price of its tickets will affect
not only its own profitability, but also the
profitability of its competitors, since a
lower price will draw consumers away
from the competition. This example
points to the competitive nature of strate-
gic interaction. But many strategic
encounters also involve a cooperative
element. To continue with the example,
airline carriers may compete aggres-
sively for certain routes but may form
alliances—cooperate—for other routes.
Game theory turns out be a useful tool
for understanding economic behavior
precisely because most social encounters
entail an element of strategic interaction.
Our example points out that a complete
theory of games should contain elements
of both competition and cooperation. 
Nash, of course, was not the first person
to recognize the importance of strategic
interaction between social actors: The
first inhabitants of our planet no doubt
realized that! But, in a series of articles
between 1950 and 1953, Nash laid the
groundwork that helped economists and
other social scientists think fruitfully
about and model strategic aspects of
social engagement. In fact, some of the
most important and pathbreaking work
in economics undertaken in the latter
half of the twentieth century simply
could not have taken place without the
benefit of Nash’s insights. 
■ Pre-1950 Game Theory
To put Nash’s contributions in perspec-
tive, one must appreciate the state of
game theory before he entered the scene.
In 1928 the brilliant mathematician John
von Neumann published the first impor-
tant paper on game theory. In that paper,
von Neumann recognizes that social
interaction typically involves elements
of both competition and cooperation.
However, he also realizes that any model
that embeds these elements will not
always be solvable. Given this limita-
tion, a major success of von Neumann’s
1928 paper was that it completely solved
a game that featured competition and
excluded the possibility of cooperation.
This game of pure strategic competition
is known as a “two-person zero-sum”
game because there are only two players,
and whatever one player gains, the other
player loses. 
When we say that von Neumann was
able to “solve” a game, what we mean is
that he was able to characterize each
player’s “winning strategy” or, in other
words, how each player intended to play
the game. Von Neumann was able to
prove that each player’s winning strategy
provided the highest guaranteed payoff.
By guaranteed payoff we mean the To study such behavior, that is, to cap-
ture the competitive and cooperative
elements that are observed when people
interact, non-zero-sum games are
required. In these kinds of games, how-
ever, von Neumann’s “linchpin” is
destroyed: The strategy that maximizes
the guaranteed payoff to one player no
longer minimizes the highest payoff that
his opponent can receive. This implies
that if a player chooses to play some-
thing other than the strategy that maxi-
mizes the guaranteed payoff, he can
actually make himself better off. So
having players choose strategies that
maximize their guaranteed payoffs is
not a solution to these games. What
strategies should players use? 
■ Nash 1950–53
Between 1950 and 1953, Nash published
four remarkable papers that effectively
redirected research efforts in game 
theory in a fruitful direction. Two papers
focused on competitive or “noncoopera-
tive” non-zero-sum games; another
paper focused on cooperative two-person
games; and the final paper examined the
relationship between cooperative and
noncooperative games and suggested
how elements of competition and 
cooperation could be incorporated into 
a general game.
Noncooperative Game Theory and
the Nash Equilibrium (1950,1951)
Strategies that maximize guaranteed
payoffs do not solve two-person non-
zero-sum games. What, then, is the
solution to a non-zero-sum game? This
is where Nash comes in. His insight was
to have each player imagine how his
opponent intended to play and to have
players use this information in helping
them formulate their own strategies.
Nash envisioned that a player would
take his “best response” to other play-
ers’intended strategies. A solution or
Nash equilibrium to a game is when all
players’intended strategies are their
best responses. Nash used the term
“noncooperative” to describe his
approach: Although in non-zero-sum
games there may be gains from cooper-
ation, Nash assumed it is  “impossible
for players to communicate or collabo-
rate in any way.”
Nash was able to show that for any non-
zero-sum game that has a finite number
of players and actions, an equilibrium to
the game always exists. This is a very
powerful result. It implies that
researchers could now write down a
game that captured realistic aspects of
social interaction—because the game is
non-zero sum—and could always be
assured that the game had a solution. It is
rather comforting to know that the Nash
equilibrium for a two-person zero-sum
game corresponds to von Neumann’s
solution. In this way, Nash’s result
extended and generalized von Neu-
mann’s result to non-zero-sum games
with potentially many players.
There are a huge number of social phe-
nomena that can best be thought of as
noncooperative games. For example,
firms compete for consumers in a variety
of ways: They can compete through the
prices they charge, through the quality of
their services, and so on. Firms may
have an incentive to cooperate with one
another: But it is illegal for firms to form
alliances when these kinds of formal
agreements ultimately hurt consumers.
So, in the context of trying to understand
how firms interact with one another, one
can view them as playing a noncoopera-
tive game with each other. This is not to
say that cooperation is unimportant. 
Tariff wars provide a good example. In
competing for consumers, countries have
been known to use a tariff to raise the
price of imports, thereby making the
home good relatively cheaper. In
response, the other country imposes tar-
iffs. In the end, both countries are gener-
ally worse off. Organizations such as the
World Trade Organization can be inter-
preted as a coordinating or cooperative
mechanism that helps prevent such
destructive competitive behavior. Simi-
larly, the United Nations can be viewed as
a coordinating mechanism that helps pre-
vent other kinds of destructive behavior
between countries. Although in some con-
texts it is appropriate, or at least not mis-
leading, to ignore cooperation, in other
contexts of social engagement it is not.
The Bargaining Problem and the
Nash Bargaining Solution (1950) 
Many strategic situations require explicit
cooperation. Consider, for example, the
case of a union and a firm bargaining
over a contract. It may be that the union
and firm are opposed on many issues, but
cooperation is required for both parties to
agree to sign the contract. How exactly
would the union and firm agree to split
the benefits associated with their relation-
ship? Nash provided the first solution to
this kind of bargaining problem. 
Nash looked at the problem of two play-
ers having to agree to split a “pie.” Both
people want as much of the pie as they
can get, but if the players fail to agree on
a split, the pie disappears. Nash’s
approach to solving this problem was
revolutionary. Anything short of revolu-
tionary was doomed to fail: For years,
many great economists and game theo-
rists tried to solve the two-person bar-
gaining problem without success.
Instead of attempting to determine the
solution to an explicit bargaining game,
Nash suggested a number of reasonable
properties that any solution to a bargain-
ing problem should satisfy. For exam-
ple, two of his properties are: Players
should never leave any “pie on the
table,” and identical players should split
the pie evenly. Nash was able to show
that for the set of properties that he
viewed as being reasonable, there would
be a unique split of the pie and the
actual split could be described by a sim-
ple mathematical equation, which is
now known as the “Nash bargaining
solution.” Nash’s solution to the bar-
gaining problem was an extremely
important development because it 
provided an answer to a longstanding,
unresolved question, and his approach 
to solving the problem was quite novel. 
But, in a way, the novelty of the
approach also carried with it a liability.
Nash’s approach to solving a (two-per-
son) cooperative game is fundamentally
different from the approach he used to
solve a (non-zero-sum) noncooperative
game. The problem, it appears, is that
one kind of “theory” is needed to under-
stand (or solve) noncooperative strategic
interactions and a completely different
kind of theory is needed to solve coop-
erative strategic interactions. But, in
practice, social engagements are gener-
ally not exclusively cooperative or
exclusively noncooperative—but rather
a mixture of both. How, then, should
one think about problems that have both
elements?
The Nash Program (1953)
When people cooperate in a strategic sit-
uation, they first discuss or negotiate
what they would like to receive, empha-
sizing the actions they may take if their
opponent does not agree, and then, typi-
cally, they sign an agreement. Consider,
once again, the situation of a firm and
union bargaining over a new contract.
The firm and union may bargain over
wages, working conditions, and so on.The firm may threaten to lock out work-
ers if the union does not agree to its pro-
posal; or workers can threaten to strike,
providing the firm with no labor until
the contract dispute is resolved. 
Nash demonstrated that one could
“reduce” this cooperative situation to a
noncooperative game. Specifically,
players first announce “threats”—
actions they would take in the event that
no settlement is reached. Then, players
announce their settlement offers: If the
settlement offers are not feasible or are
incompatible, then players will under-
take their threat actions. This is a non-
cooperative game because players do
not sign contracts or make prior agree-
ments. In the case of splitting a pie, set-
tlement offers would not be compatible
if the players’desired shares added up to
more the whole pie. Nash showed that
there is a solution, that is, a “Nash equi-
librium,” to this noncooperative game,
where players announce certain threats,
and settlement offers are compatible.
So, players will end up “splitting a pie”
in a noncooperative manner. What is
remarkable about the solution to this
noncooperative game is that it corre-
sponds to the (Nash bargaining) solution
of a purely cooperative game; that is, it
yields the same split of the pie. This is
an important result because, although
the cooperative and noncooperative
approaches to game theory are quite dif-
ferent, Nash showed they could lead to
the same solution. 
How should one think about the resolu-
tion of strategic encounters that have
elements of both competition and coop-
eration? Nash’s 1953 paper suggests
that a cooperative situation could be
reduced to a noncooperative one. There-
fore, a suitably specified noncooperative
game is able to characterize elements of
competition and cooperation and the
solution to this game is given by the
Nash equilibrium. 
■ Conclusions
In the 1940s, a group of influential 
economists and mathematicians were
convinced that the most fruitful way to
view economic and social interactions
was through the lens of game theory.
However, by the end of the 1940s enthu-
siasm had stalled because a main engine
of game theory, the two-person zero-
sum game, was incapable of describing
relevant problems. In short, Nash
showed us how to think about and solve
the more interesting non-zero-sum
games. He was the first to make a 
distinction between noncooperative
games and cooperative games. Both of
these frameworks are capable of
addressing interesting economics prob-
lems, and Nash showed that solutions
existed for each type. Although non-
cooperative games are fundamentally
different from cooperative games, Nash
demonstrated that the different
approaches could lead to the same solu-
tion. In the end, Nash provided us with
a framework—the noncooperative
game—and a solution—the Nash equi-
librium—that are capable of analyzing
both the cooperative and competitive
aspects of social interaction. There is
virtually no area in economics or the
social sciences that has not benefited
from Nash’s “beautiful theory.”Federal Reserve Bank of Cleveland
Research Department
P.O. Box 6387
Cleveland, OH 44101
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credited. Please send copies of reprinted
material to the editor.
Ed Nosal is an economic advisor at the 
Federal Reserve Bank of Cleveland, and 
Peter Rupert is a senior economic advisor 
at the Bank. The authors thank Martin
Osborne for helpful comments.
The views expressed here are those of the
authors and not necessarily those of the Fed-
eral Reserve Bank of Cleveland, the Board of 
Governors of the Federal Reserve System, or
its staff. 
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A beautiful theory

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