Draft chapter from An introduction to game theory by Martin J. Osborne. Version: 2002/7/23.
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2
Nash Equilibrium: Theory
2.1 Strategic games 11
2.2 Example: the Prisoner’s Dilemma 12
2.3 Example: Bach or Stravinsky? 16
2.4 Example: Matching Pennies 17
2.5 Example: the Stag Hunt 18
2.6 Nash equilibrium 19
2.7 Examples of Nash equilibrium 24
2.8 Best response functions 33
2.9 Dominated actions 43
2.10 Equilibrium in a single population: symmetric games and
symmetric equilibria 49
Prerequisite: Chapter 1.
2.1 Strategic games
ASTRATEGIC GAME is a model of interacting decision-makers. In recognitionof the interaction, we refer to the decision-makers as players. Each player
has a set of possible actions. The model captures interaction between the players
by allowing each player to be affected by the actions of all players, not only her
own action. Specifically, each player has preferences about the action profile—the
list of all the players’ actions. (See Section 17.4, in the mathematical appendix, for
a discussion of profiles.)
More precisely, a strategic game is defined as follows. (The qualification “with
ordinal preferences” distinguishes this notion of a strategic game from a more
general notion studied in Chapter 4.)
I DEFINITION 11.1 (Strategic game with ordinal preferences) A strategic game (with
ordinal preferences) consists of
• a set of players
• for each player, a set of actions
• for each player, preferences over the set of action profiles.
11
12 Chapter 2. Nash Equilibrium: Theory
A very wide range of situations may be modeled as strategic games. For exam-
ple, the players may be firms, the actions prices, and the preferences a reflection of
the firms’ profits. Or the players may be candidates for political office, the actions
campaign expenditures, and the preferences a reflection of the candidates’ proba-
bilities of winning. Or the players may be animals fighting over some prey, the ac-
tions concession times, and the preferences a reflection of whether an animal wins
or loses. In this chapter I describe some simple games designed to capture funda-
mental conflicts present in a variety of situations. The next chapter is devoted to
more detailed applications to specific phenomena.
As in the model of rational choice by a single decision-maker (Section 1.2), it is
frequently convenient to specify the players’ preferences by giving payoff functions
that represent them. Bear in mind that these payoffs have only ordinal significance.
If a player’s payoffs to the action profiles a, b, and c are 1, 2, and 10, for example,
the only conclusion we can draw is that the player prefers c to b and b to a; the
numbers do not imply that the player’s preference between c and b is stronger
than her preference between a and b.
Time is absent from the model. The idea is that each player chooses her ac-
tion once and for all, and the players choose their actions “simultaneously” in the
sense that no player is informed, when she chooses her action, of the action chosen
by any other player. (For this reason, a strategic game is sometimes referred to
as a “simultaneous move game”.) Nevertheless, an action may involve activities
that extend over time, and may take into account an unlimited number of contin-
gencies. An action might specify, for example, “if company X’s stock falls below
$10, buy 100 shares; otherwise, do not buy any shares”. (For this reason, an action
is sometimes called a “strategy”.) However, the fact that time is absent from the
model means that when analyzing a situation as a strategic game, we abstract from
the complications that may arise if a player is allowed to change her plan as events
unfold: we assume that actions are chosen once and for all.
2.2 Example: the Prisoner’s Dilemma
One of the most well-known strategic games is the Prisoner’s Dilemma. Its name
comes from a story involving suspects in a crime; its importance comes from the
huge variety of situations in which the participants face incentives similar to those
faced by the suspects in the story.
EXAMPLE 12.1 (Prisoner’s Dilemma) Two suspects in a major crime are held in sep-
arate cells. There is enough evidence to convict each of them of a minor offense,
but not enough evidence to convict either of them of the major crime unless one of
them acts as an informer against the other (finks). If they both stay quiet, each will
be convicted of the minor offense and spend one year in prison. If one and only
one of them finks, she will be freed and used as a witness against the other, who
will spend four years in prison. If they both fink, each will spend three years in
prison.
2.2 Example: the Prisoner’s Dilemma 13
This situation may be modeled as a strategic game:
Players The two suspects.
Actions Each player’s set of actions is {Quiet, Fink}.
Preferences Suspect 1’s ordering of the action profiles, from best to worst, is
(Fink, Quiet) (she finks and suspect 2 remains quiet, so she is freed), (Quiet,
Quiet) (she gets one year in prison), (Fink, Fink) (she gets three years in prison),
(Quiet, Fink) (she gets four years in prison). Suspect 2’s ordering is (Quiet, Fink),
(Quiet, Quiet), (Fink, Fink), (Fink, Quiet).
We can represent the game compactly in a table. First choose payoff functions
that represent the suspects’ preference orderings. For suspect 1 we need a function
u1 for which
u1(Fink, Quiet) > u1(Quiet, Quiet) > u1(Fink, Fink) > u1(Quiet, Fink).
A simple specification is u1(Fink, Quiet) = 3, u1(Quiet, Quiet) = 2, u1(Fink, Fink) =
1, and u1(Quiet, Fink) = 0. For suspect 2 we can similarly choose the function
u2 for which u2(Quiet, Fink) = 3, u2(Quiet, Quiet) = 2, u2(Fink, Fink) = 1, and
u2(Fink, Quiet) = 0. Using these representations, the game is illustrated in Fig-
ure 13.1. In this figure the two rows correspond to the two possible actions of
player 1, the two columns correspond to the two possible actions of player 2, and
the numbers in each box are the players’ payoffs to the action profile to which the
box corresponds, with player 1’s payoff listed first.
Suspect 1
Suspect 2
Quiet Fink
Quiet 2, 2 0, 3
Fink 3, 0 1, 1
Figure 13.1 The Prisoner’s Dilemma (Example 12.1).
The Prisoner’s Dilemma models a situation in which there are gains from coop-
eration (each player prefers that both players choose Quiet than they both choose
Fink) but each player has an incentive to “free ride” (choose Fink) whatever the
other player does. The game is important not because we are interested in under-
standing the incentives for prisoners to confess, but because many other situations
have similar structures. Whenever each of two players has two actions, say C
(corresponding to Quiet) and D (corresponding to Fink), player 1 prefers (D, C) to
(C, C) to (D, D) to (C, D), and player 2 prefers (C, D) to (C, C) to (D, D) to (D, C),
the Prisoner’s Dilemma models the situation that the players face. Some examples
follow.
2.2.1 Working on a joint project
You are working with a friend on a joint project. Each of you can either work hard
or goof off. If your friend works hard then you prefer to goof off (the outcome of
14 Chapter 2. Nash Equilibrium: Theory
the project would be better if you worked hard too, but the increment in its value
to you is not worth the extra effort). You prefer the outcome of your both working
hard to the outcome of your both goofing off (in which case nothing gets accom-
plished), and the worst outcome for you is that you work hard and your friend
goofs off (you hate to be “exploited”). If your friend has the same preferences then
the game that models the situation you face is given in Figure 14.1, which, as you
can see, differs from the Prisoner’s Dilemma only in the names of the actions.
Work hard Goof off
Work hard 2, 2 0, 3
Goof off 3, 0 1, 1
Figure 14.1 Working on a joint project.
I am not claiming that a situation in which two people pursue a joint project
necessarily has the structure of the Prisoner’s Dilemma, only that the players’ pref-
erences in such a situation may be the same as in the Prisoner’s Dilemma! If, for
example, each person prefers to work hard than to goof off when the other person
works hard, then the Prisoner’s Dilemma does not model the situation: the players’
preferences are different from those given in Figure 14.1.
? EXERCISE 14.1 (Working on a joint project) Formulate a strategic game that models
a situation in which two people work on a joint project in the case that their pref-
erences are the same as those in the game in Figure 14.1 except that each person
prefers to work hard than to goof off when the other person works hard. Present
your game in a table like the one in Figure 14.1.
2.2.2 Duopoly
In a simple model of a duopoly, two firms produce the same good, for which each
firm charges either a low price or a high price. Each firm wants to achieve the
highest possible profit. If both firms choose High then each earns a profit of $1000.
If one firm chooses High and the other chooses Low then the firm choosing High
obtains no customers and makes a loss of $200, whereas the firm choosing Low
earns a profit of $1200 (its unit profit is low, but its volume is high). If both firms
choose Low then each earns a profit of $600. Each firm cares only about its profit,
so we can represent its preferences by the profit it obtains, yielding the game in
Figure 14.2.
High Low
High 1000, 1000 −200, 1200
Low 1200,−200 600, 600
Figure 14.2 A simple model of a price-setting duopoly.
Bearing in mind that what matters are the players’ preferences, not the partic-
2.2 Example: the Prisoner’s Dilemma 15
ular payoff functions that we use to represent them, we see that this game, like the
previous one, differs from the Prisoner’s Dilemma only in the names of the actions.
The action High plays the role of Quiet, and the action Low plays the role of Fink;
firm 1 prefers (Low, High) to (High, High) to (Low, Low) to (High, Low), and firm 2
prefers (High, Low) to (High, High) to (Low, Low) to (Low, High).
As in the previous example, I do not claim that the incentives in a duopoly are
necessarily those in the Prisoner’s Dilemma; different assumptions about the relative
sizes of the profits in the four cases generate a different game. Further, in this case
one of the abstractions incorporated into the model—that each firm has only two
prices to choose between—may not be harmless; if the firms may choose among
many prices then the structure of the interaction may change. (A richer model is
studied in Section 3.2.)
2.2.3 The arms race
Under some assumptions about the countries’ preferences, an arms race can be
modeled as the Prisoner’s Dilemma. (The Prisoner’s Dilemma was first studied in
the early 1950s, when the USA and USSR were involved in a nuclear arms race, so
you might suspect that US nuclear strategy was influenced by game theory; the
evidence suggests that it was not.) Assume that each country can build an arsenal
of nuclear bombs, or can refrain from doing so. Assume also that each country’s
favorite outcome is that it has bombs and the other country does not; the next best
outcome is that neither country has any bombs; the next best outcome is that both
countries have bombs (what matters is relative strength, and bombs are costly to
build); and the worst outcome is that only the other country has bombs. In this
case the situation is modeled by the Prisoner’s Dilemma, in which the action Don’t
build bombs corresponds to Quiet in Figure 13.1 and the action Build bombs corre-
sponds to Fink. However, once again the assumptions about preferences necessary
for the Prisoner’s Dilemma to model the situation may not be satisfied: a country
may prefer not to build bombs if the other country does not, for example (bomb-
building may be very costly), in which case the situation is modeled by a different
game.
2.2.4 Common property
Two farmers are deciding how much to allow their sheep to graze on the village
common. Each farmer prefers that her sheep graze a lot than a little, regardless of
the other farmer’s action, but prefers that both farmers’ sheep graze a little than
both farmers’ sheep graze a lot (in which case the common is ruined for future
use). Under these assumptions the game is the Prisoner’s Dilemma. (A richer model
is studied in Section 3.1.5.)
16 Chapter 2. Nash Equilibrium: Theory
2.2.5 Other situations modeled as the Prisoner’s Dilemma
A huge number of other situations have been modeled as the Prisoner’s Dilemma,
from mating hermaphroditic fish to tariff wars between countries.
? EXERCISE 16.1 (Hermaphroditic fish) Members of some species of hermaphroditic
fish choose, in each mating encounter, whether to play the role of a male or a
female. Each fish has a preferred role, which uses up fewer resources and hence
allows more future mating. A fish obtains a payoff of H if it mates in its preferred
role and L if it mates in the other role, where H > L. (Payoffs are measured in
terms of number of offspring, which fish are evolved to maximize.) Consider an
encounter between two fish whose preferred roles are the same. Each fish has two
possible actions: mate in either role, and insist on its preferred role. If both fish
offer to mate in either role, the roles are assigned randomly, and each fish’s payoff
is 12 (H + L) (the average of H and L). If each fish insists on its preferred role, the
fish do not mate; each goes off in search of another partner, and obtains the payoff
S. The higher the chance of meeting another partner, the larger is S. Formulate this
situation as a strategic game and determine the range of values of S, for any given
values of H and L, for which the game differs from the Prisoner’s Dilemma only in
the names of the actions.
2.3 Example: Bach or Stravinsky?
In the Prisoner’s Dilemma the main issue is whether or not the players will cooperate
(choose Quiet). In the following game the players agree that it is better to cooperate
than not to cooperate, but disagree about the best outcome.
EXAMPLE 16.2 (Bach or Stravinsky?) Two people wish to go out together. Two con-
certs are available: one of music by Bach, and one of music by Stravinsky. One per-
son prefers Bach and the other prefers Stravinsky. If they go to different concerts,
each of them is equally unhappy listening to the music of either composer.
We may model this situation as the two-player strategic game in Figure 16.1,
in which the person who prefers Bach chooses a row and the person who prefers
Stravinsky chooses a column.
Bach Stravinsky
Bach 2, 1 0, 0
Stravinsky 0, 0 1, 2
Figure 16.1 Bach or Stravinsky? (BoS) (Example 16.2).
This game is also referred to as the “Battle of the Sexes” (though the conflict
it models surely occurs no more frequently between people of the opposite sex
than it does between people of the same sex). I call the game BoS, an acronym
that fits both names. (I assume that each player is indifferent between listening
to Bach and listening to Stravinsky when she is alone only for consistency with
2.4 Example: Matching Pennies 17
the standard specification of the game. As we shall see, the analysis of the game
remains the same in the absence of this indifference.)
Like the Prisoner’s Dilemma, BoS models a wide variety of situations. Consider,
for example, two officials of a political party deciding the stand to take on an issue.
Suppose that they disagree about the best stand, but are both better off if they take
the same stand than if they take different stands; both cases in which they take
different stands, in which case voters do not know what to think, are equally bad.
Then BoS captures the situation they face. Or consider two merging firms that
currently use different computer technologies. As two divisions of a single firm
they will both be better off if they both use the same technology; each firm prefers
that the common technology be the one it used in the past. BoS models the choices
the firms face.
2.4 Example: Matching Pennies
Aspects of both conflict and cooperation are present in both the Prisoner’s Dilemma
and BoS. The next game is purely conflictual.
EXAMPLE 17.1 (Matching Pennies) Two people choose, simultaneously, whether to
show the Head or the Tail of a coin. If they show the same side, person 2 pays
person 1 a dollar; if they show different sides, person 1 pays person 2 a dollar. Each
person cares only about the amount of money she receives, and (naturally!) prefers
to receive more than less. A strategic game that models this situation is shown
in Figure 17.1. (In this representation of the players’ preferences, the payoffs are
equal to the amounts of money involved. We could equally well work with another
representation—for example, 2 could replace each 1, and 1 could replace each −1.)
Head Tail
Head 1,−1 −1, 1
Tail −1, 1 1,−1
Figure 17.1 Matching Pennies (Example 17.1).
In this game the players’ interests are diametrically opposed (such a game is
called “strictly competitive”): player 1 wants to take the same action as the other
player, whereas player 2 wants to take the opposite action.
This game may, for example, model the choices of appearances for new prod-
ucts by an established producer and a new firm in a market of fixed size. Suppose
that each firm can choose one of two different appearances for the product. The
established producer prefers the newcomer’s product to look different from its
own (so that its customers will not be tempted to buy the newcomer’s product),
whereas the newcomer prefers that the products look alike. Or the game could
18 Chapter 2. Nash Equilibrium: Theory
model a relationship between two people in which one person wants to be like the
other, whereas the other wants to be different.
? EXERCISE 18.1 (Games without conflict) Give some examples of two-player strate-
gic games in which each player has two actions and the players have the same pref-
erences, so that there is no conflict between their interests. (Present your games as
tables like the one in Figure 17.1.)
2.5 Example: the Stag Hunt
A sentence in Discourse on the origin and foundations of inequality among men (1755)
by the philosopher Jean-Jacques Rousseau discusses a group of hunters who wish
to catch a stag. They will succeed if they all remain sufficiently attentive, but each
is tempted to desert her post and catch a hare. One interpretation of the sentence is
that the interaction between the hunters may be modeled as the following strategic
game.
EXAMPLE 18.2 (Stag Hunt) Each of a group of hunters has two options: she may
remain attentive to the pursuit of a stag, or catch a hare. If all hunters pursue the
stag, they catch it and share it equally; if any hunter devotes her energy to catching
a hare, the stag escapes, and the hare belongs to the defecting hunter alone. Each
hunter prefers a share of the stag to a hare.
The strategic game that corresponds to this specification is:
Players The hunters.
Actions Each player’s set of actions is {Stag, Hare}.
Preferences For each player, the action profile in which all players choose Stag
(resulting in her obtaining a share of the stag) is ranked highest, followed
by any profile in which she chooses Hare (resulting in her obtaining a hare),
followed by any profile in which she chooses Stag and one or more of the
other players chooses Hare (resulting in her leaving empty-handed).
Like other games with many players, this game cannot easily be presented in a
table like that in Figure 17.1. For the case in which there are
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