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NASH EQUILIBRIUM

Nash equilibrium is a fundamental concept in the theory

of games and the most widely used method of predicting

the outcome of a strategic interaction in the social sciences.

A game (in strategic or normal form) consists of the

following three elements: a set of players, a set of actions

(or pure-strategies) available to each player, and a payoff

(or utility) function for each player. The payoff functions

represent each player’s preferences over action profiles,

where an action profile is simply a list of actions, one for

each player. A pure-strategy Nash equilibrium is an action

profile with the property that no single player can obtain

a higher payoff by deviating unilaterally from this profile.

This concept can best be understood by looking at

some examples. Consider first a game involving two players,

each of whom has two available actions, which we call

A and B. If the players choose different actions, they each

get a payoff of 0. If they both choose A, they each get 2,

and if they both choose B, they each get 1. This “coordination”

game may be represented as follows, where player

1 chooses a row, player 2 chooses a column, and the resulting

payoffs are listed in parentheses, with the first component

corresponding to player 1’s payoff:

The action profile (B,B) is an equilibrium, since a

unilateral deviation to A by any one player would result in

a lower payoff for the deviating player. Similarly, the

action profile (A,A) is also an equilibrium.

As another example, consider the game “matching

pennies,” which again involves two players, each with two

actions. Each player can choose either heads (H) or tails

(T); player 1 wins a dollar from player 2 if their choices

are the same, and loses a dollar to player 2 if they are not.

This game has no pure-strategy Nash equilibria.

In some cases, instead of simply choosing an action,

players may be able to choose probability distributions

over the set of actions available to them. Such randomizations

over the set of actions are referred to as mixed strategies.

Any profile of mixed strategies induces a probability

distribution over action profiles in the game. Under certain

assumptions, a player’s preferences over all such lotteries

can be represented by a function (called a von

Neumann-Morgenstern utility function) that assigns a real

number to each action profile. One lottery is preferred to

another if and only if it results in a higher expected value

of this utility function, or expected utility. A mixed strategy

Nash-equilibrium is then a mixed strategy profile with

the property that no single player can obtain a higher

value of expected utility by deviating unilaterally from this

profile.

The American mathematician John Nash (1950)

showed that every game in which the set of actions available

to each player is finite has at least one mixed-strategy

equilibrium. In the matching pennies game, there is a

mixed-strategy equilibrium in which each player chooses

heads with probability 1/2. Similarly, in the coordination

game of the above example, there is a third equilibrium in

which each player chooses action A with probability 1/3

and B with probability 2/3. Such multiplicity of equilibria

arises in many economically important games, and has

prompted a large literature on equilibrium refinements

with the purpose of identifying criteria on the basis of

which a single equilibrium might be selected.

Nash equilibria can sometimes correspond to outcomes

that are inefficient, in the sense that there exist

alternative outcomes that are both feasible and preferred

by all players. This is the case, for instance, with the equilibrium

(B,B) in the coordination game above. An even

more striking example arises in the prisoner’s dilemma

game, in which each player can either “cooperate” or

“defect,” and payoffs are as follows:

The unique Nash equilibrium is mutual defection, an

outcome that is worse for both players than mutual cooperation.

Now consider the game that involves a repetition

of the prisoner’s dilemma for n periods, where n is commonly

known to the two players. A pure strategy in this

repeated game is a plan that prescribes which action is to

be taken at each stage, contingent on every possible history

of the game to that point. Clearly the set of pure

strategies is very large. Nevertheless, all Nash equilibria of

this finitely repeated game involve defection at every

stage. When the number of stages n is large, equilibrium

payoffs lie far below the payoffs that could have been

attained under mutual cooperation.

It has sometimes been argued that the Nash prediction

in the finitely repeated prisoner’s dilemma (and in

many other environments) is counterintuitive and at odds

with experimental evidence. However, experimental tests

of the equilibrium hypothesis are typically conducted

with monetary payoffs, which need not reflect the preferences

of subjects over action profiles. In other words, individual

preferences over the distribution of monetary

payoffs may not be exclusively self-interested.

Furthermore, the equilibrium prediction relies on the

hypothesis that these preferences are commonly known to

all subjects, which is also unlikely to hold in practice.

To address this latter concern, the concept of Nash

equilibrium has been generalized to allow for situations in

which players are faced with incomplete information. If

each player is drawn from some set of types, such that the

probability distribution governing the likelihood of each

type is itself commonly known to all players, then we have

a Bayesian game. A pure strategy in this game is a function

that associates with each type a particular action. A BayesNash

equilibrium is then a strategy profile such that no

player can obtain greater expected utility by deviating to a

different strategy, given his or her beliefs about the distribution

of types from which other players are drawn.

Allowing for incomplete information can have dramatic

effects on the predictions of the Nash equilibrium

concept. Consider, for example, the finitely repeated prisoner’s

dilemma, and suppose that each player believes that

there is some possibility, perhaps very small, that his or her

opponent will cooperate in all periods provided that no

defection has yet been observed, and defect otherwise. If

the number of stages n is sufficiently large, it can be

shown that mutual defection in all stages is inconsistent

with equilibrium behavior, and that, in a well-defined

sense, the players will cooperate in most periods. Hence,

in applying the concept of Nash equilibrium to practical

situations, it is important to pay close attention to the

information that individuals have about the preferences,

beliefs, and rationality of those with whom they are strategically

interacting.

Nash Equilibrium

Figure 3

D (3,0) (1,1)

C (2,2) (0,3)

C D

INTERNATIONAL ENCYCLOPEDIA OF THE SOCIAL SCIENCES, 2ND EDITION 541

iess_B3_H-O 4/12/07 4:13 PM Page 541

SEE ALSO Game Theory; Multiple Equilibria;

Noncooperative Games; Prisoner’s Dilemma

(Economics)

BIBLIOGRAPHY

Cournot, A. A. 1838. Recherches sur les principes mathématiques

de la théorie des richesses. Paris: L. Hachette.

Fudenberg, Drew, and Jean Tirole. 1991. Game Theory.

Cambridge, MA: MIT Press.

Harsanyi, John C. 1967–1968. Games with Incomplete

Information Played by Bayesian Players. Management Science

14 (3): 159–182, 320–334, 486–502.

Harsanyi, John C., and Reinhard Selten. 1998. A General Theory

of Equilibrium Selection in Games. Cambridge, MA: MIT

Press.

Kreps, David, Paul Milgrom, John Roberts, and Robert Wilson.

1982. Rational Cooperation in the Finitely Repeated

Prisoner’s Dilemma. Journal of Economic Theory 27: 245–252.

Nash, John F. 1950. Equilibrium Points in N-Person Games.

Proceedings of the National Academy of Sciences 36 (1): 48–49.

Osborne, Martin J., and Ariel Rubinstein. 1994. A Course in

Game Theory. Cambridge, MA: MIT Press.

von Neumann, John, and Oskar Morgenstern. 1944. Theory of

Games and Economic Behavior. Princeton, NJ: Princeton

University Press.

Rajiv Sethi