CHAPTER
21
---.-----­
BASIC CONCEPTS
Strategy and Game Theory
In Chapter 20 we examined some of the problems that arise in modeling markets
in which there are only a few firms. Perhaps the most difficult of these problems
concerned questions of strategy-that is, with few firms each firm must, to some
extent, be concerned with what its rivals will do. As we saw, making profit­
maximizing decisions requires that each firm make some conjectures about its
competitors' behavior. Under perfect competition such strategic thinking was
unnecessary because the prevailing market price was assumed to convey all the
external information that was relevant to the firm. With relatively few firms the
situation may be more complicated since price-taking behavior is less likely.
One of the primary tools that economists use to study strategic choices is
game theory. This subject was originally developed during the 1920s and grew
rapidly during World War II in response to the need to develop formal ways of
thinking about military strategy.! Today the theory has applicability to problems
as diverse as the development of optimal strategies for five-card stud poker and
the analysis of antimissile defenses. In this chapter we will provide a brief intro­
duction to game theory with a primary focus on its use in explaining pricing
and entry behavior in oligopolistic markets. A few other applications will also
be mentioned.
Game theory models seek to portray complex strategic situations in a highly
simplified and stylized setting. Much like the previous models in this book, game
theory models abstract from most of the personal and institutional details of a
problem in order to arrive at a representation of the situation that is mathemati­
cally tractable. This ability to get to the "heart" of the problem is the greatest
strength of this type of modeling.
1 Much of the pioneering work in game theory was done by the mathematician John von Neumann.
The main reference is J. von Neumann and O. Morgenstern, The Theory of Games and Economic
Behal'ior (Princeton, N.].: Princeton University Press, 1944).
619
620
PART VI • Models of Imperfect Competition
Any situation in which individuals must make strategic choices and in which
the final outcome will depend on what each person chooses to do can be viewed
as a game. All games have three basic elements: (1) players; (2) strategies; and
(3) payoffs. Games may be cooperative, in which players can make binding
agreements, or noncooperative, where such agreements are not possible. Here
we will be concerned primarily with noncooperative games. The basic elements
listed below are included in such games.
Players Each decision-maker in a game is called a "player." These players may be indi­
viduals (as in poker games), firms (as in oligopoly markets), or entire nations (as
in military conflicts). All players are characterized as having the ability to choose
from among a set of possible actions they might take. 2 Usually, the number of
players is fixed throughout the "play" of a game, and games are often charac­
terized by the number of players (that is, two-player, three-player, or n-player
games). In this chapter we will primarily study two-player games and will denote
these players (usually firms) by A and B. One of the important assumptions
usually made in game theory (as in most of economics) is that the specific iden­
tity of the players is irrelevant. There are no "good guys" or "bad guys" in a
game, and players are not assumed to have any special abilities or shortcomings.
Each player is simply assumed to choose the course of action that yields the
most favorable outcome.
Strategies Each course of action open to a player in a game is called a "strategy." Depend­
ing on the game being examined, a strategy may be a very simple action (take
another card in blackjack) or a very complex one (build a laser-based antimissile
defense), but each strategy is assumed to be a well-defined, specific course of
action. J Usually, the number of strategies available to each player will be finite;
many aspects of game theory can be illustrated for situations in which each
player has only two strategies available. 4 In noncooperative games, players can­
not reach binding agreements with each other about what strategies they will
play-each player is uncertain about what the other will do.
Payoffs The final returns to the players of a game at its conclusion are called "payoffs."
Payoffs are usually measured in levels of utility obtained by the players although
frequently monetary payoffs (say, profits for firms) are used instead. In general,
'Sometimes one of the players in a game is taken to be "nature." For this player, actions are not
"chosen" but rather occur with certain possibilities. For example, the weather may affect the out·
comes of a game, but it is not "chosen" by natute. Rather, particular weather outcomes arc assumed
to occur with various probabilities. Games against nature can be analyzed using the methods devel­
oped in Chapter 10.
\In games involVing a sequence of actions (for example, most board games such as chess), a speci·
fication of strategies may involve several decision points (each move in chess). Assuming perfect
knowledge of how the game is played, such complex patterns can often be expressed as choices
among a large but finite set of pure strategies, each of which specifies a complete course of action
until the game is completed. See our discussion of "extensive" and "normal" forms and R. D. Luee
and H. Raiffa, Games and Decisions (New York: John Wiley & Sons, 1957), chap. 3.
• Players may also adopt "mixed" strategies by choosmg to play their pure strategies randomly (say,
by flipping a coin). We will analyze this possibility only briefly in footnotes.
CHAPTER 21 • Strategy and Game Theory
621
it is assumed that players can rank the payoffs of a game ordinally from most
preferred to least preferred and will seek the highest ranked payoff attainable.
Payoffs incorporate all aspects associated with outcomes of a game; these in­
clude both explicit monetary payoffs and implicit feelings by the players about
the outcomes such as whether they are embarrassed or gain self-esteem. Players
prefer payoffs that offer more utility to those that offer less.
EQUILIBRIUM
CONCEPTS IN GAME THEORY In our examination of the theory of markets, we developed the concept of equi­
librium in which both suppliers and demanders were content with the market
outcome. Given the equilibrium price and quantity, no market participant has
an incentive to change his or her behavior. The question therefore arises whether
there are similar equilibrium concepts in game theory models. Are there strategic
choices that, once made, provide no incentives for the players to alter their be­
havior further? Do these equilibria then offer believable explanations of market
outcomes?
Although there are several ways to formalize equilibrium concepts in games,
the most frequently used approach was originally proposed by Coumot (see
Chapter 20) in the nineteenth century and generalized in the early 1950s by
J. Nash. s Under Nash's procedure a pair of strategies, say, (a*, b*), is defined to
be an equilibrium if a* represents player A's best strategy when B plays b" and
b * represents B's best strategy when A plays a". Even if one of the players
reveals the (equilibrium) strategy he or she will use, the other player cannot
benefit from knowing this. For nonequilibrium strategies, as we shall see, this is
not the case. If one player knows what the other's strategy will be, he or she can
often benefit from that knowledge and, in the process, take actions that reduce
the payoff received by the player who has revealed his or her strategy.
Not every game has a Nash equilibrium set of strategies. And, in some cases,
a game may have multiple equilibria, some of which are more plausible than
others. Some Nash equilibria may not be especially desirable for the players in
a game. And, in some cases, other equilibrium concepts may be more reasonable
than those proposed by Nash. Hence, there is a rather complex relationship
between game theory equilibria and more traditional market equilibrium con­
cepts. Still, we have an initial working definition of equilibrium with which to
start our study of game theory:
DEFINITION Nash Equilibrium Strategies A pair of strategies (a", b *) represents an equilibrium solution to a two-player
game if a* is an optimal strategy for A against b* and b* is an optimal strategy for B against a*.6
'John Nash, "Equilibrium Points in n-person Games," Proceedings of the Natiolfal Academy of
Sciences 36 (1950): 48-49. 6 Although this definition is stated only for two-player games, the generalization to If-persons is straightforward, though notationally cumbersome. 622
PART VI • Models of Imperfect Competition
Figure 21.1
The Advertising Game in Extensive Form
In this game A chooses a low (L) or a high (H) advertising budget, then 8 makes a
similar choice. The oval surrounding 8's nodes indicates that they share the same (lack
of) information-8 does not know what strategy A has chosen. Payoffs (with A's first)
are listed at the right.
7,5
5,4
A
6.4
6,3
AN ILLUSTRATIVE
ADVERTISING
GAME
As a way of illustrating the game-theoretic approach to strategic modeling, we
will examine a simple example in which two firms (A and B) must decide how
much to spend on advertising. Each firm may adopt either a "high" (H) budget
or a "low" (L) budget, and we wish to examine possible equilibrium choices in
this situation. It should be stressed at the outset that this game is not especially
realistic-it is intended for pedagogic purposes only.
The Game in
Extensive Form
Figure 21.1 illustrates the specific details of the advertising game. In this game
"tree," the action proceeds from left to right, and each "node" represents a
decision point for the firm indicated there. The first move in this game belongs
to firm A: it must choose its level of advertising expenditures, H or L. Because
firm B's decisions occur to the right of A's, the tree indicates that firm B makes
its decision after firm A. At this stage, two versions of the game are possible
depending on whether B is assumed to know what choice A has made. First we
will look at the case where B does not have this information. The larger oval
surrounding B's two decision nodes indicates that both nodes share the same
(lack of) information. Firm B must choose H or L without knowing what A has
done. Later we will examine the case where B does have this information.
I
CHAPTER 21 • Strategy and Game Theory
Table 21.1
623
The Advertising Game in Normal Form
B's Strategies
L
A's Strategies
H
L
H
The numbers at the end of each tree branch indicate payoffs, here measured
in (thousands or millions of) dollars of profits. Each pair of payoffs lists A's
profits first. For example, the payoffs in Figure 21.1 show that if firm A chooses
H and firm B chooses L, profits will be 6 for A and 4 for B. Other payoffs are
interpreted similarly.
The Game in Normal Form Although the game tree in Figure 21.1 offers a useful visual presentation of the
complete structure of a game, sometimes it is more convenient to describe games
in tabular (or "normal") form. Table 21.1 provides such a presentation for the
advertising game. In the table, firm A's strategies (H or L) are shown at the left,
and B's strategies are shown across the top. Payoffs (again with firm A's coming
first) corresponding to the various strategic choices are shown in the body of the
table. The reader should check that Figure 21.1 and Table 21.1 convey the same
information about this game.
Dominant and Nash Equilibria Table 21.1 makes clear that adoption of a low advertising budget is a dominant
strategy for firm B. No matter what A does, the L strategy provides greater
profits to firm B than does the H strategy. Of course, since the structure of the
game is assumed to be known to both players, firm A will recognize that B has
such a dominant strategy and will opt for the strategy that does the best against
it-that is, firm A will also choose L. Considerations of strategy dominance,
therefore, suggest that the A: L, B:L strategy choice will be made and that the
resulting payoffs will be 7 (to A) and 5 (to B).
The A:L, B:L strategy choice also obeys the Nash criterion for equilibrium.
If A knows that B will play L, its best choice is L. Similarly, if B knows A will
play L, its best choice is also L (indeed, since L is a dominant strategy for B,
this is its best choice no matter what A does). The A:L, B:L choice, therefore,
meets the symmetry required by the Nash criterion.
To see why the other strategy pairs in Table 21.1 do not meet the Nash cri­
terion, let us consider them one at a time. If the players announce A:H, B:L,
this provides A with a chance to better its position-if firm A knows B will opt
for L, it can make greater profits by choosing L. The choice A:H, B:L is there­
fore not a Nash equilibrium. Neither of the two outcomes in which B chooses
H meets the Nash criterion either since, as we have already pointed out, no
matter what A does, B can improve its profits by choosing L instead. Since L
624
PART VI • Models of Imperfect Competition
strictly dominates H for firm B, no outcome in which B plays H can be a Nash
equilibrium.
.
Nature of
Nash Equilibria
Although the advertising game illustrated in Figure 21.1 contains a single Nash
equilibrium, that is not a general property of all two-person games. Example
21.1 illustrates a simple game ("Rock, Scissors, Paper") in which no Nash equi­
librium exists and. another game ("Battle of the Sexes") that contains two Nash
equilibria. Hence the Nash approach may not always identify a well-defined
solution to a game situation. 7 The Nash approach also may not yield an espe­
cially desirable equilibrium nor one that might be expected to persist if a game
were played repeatedly. Some of these issues are illustrated by the Prisoner's
Dilemma game that we take up in the next section.
Example 21.1
Sample Nash Equilibria
Table 21.2 illustrates two familiar games that reflect differing possibilities for
Nash equilibria. Part (a) of the table depicts the children's finger game "Rock,
Scissors, Paper." The zero payoffs along the diagonal show that if players adopt
the same strategy, no payments are made. In other cases the payoffs indicate a
$1 payment from loser to winner under the usual hierarchy (Rock breaks Scis­
sors, Scissors cut Paper, Paper covers Rock). As anyone who has played this
game knows, there is no equilibrium. Any strategy pair is unstable because it
offers at least one of the players an incentive to adopt another strategy. For
example, (A: Scissors, B: Scissors) provides an incentive for either A or B to
choose Rock. Similarly (A: Paper, B: Rock) obviously encourages B to choose
Scissors. The irregular cycling behavior exhibited in the play of this game clearly
indicates the absence of a Nash equilibrium.
Battle of the Sexes In the "Battle of the Sexes" game, a husband (A) and wife
(B) are planning a vacation. A prefers mountain locations, B prefers the sea­
side. Both players prefer a vacation spent together to one spent apart. The pay­
offs in part (b) of Table 21.2 reflect these preferences. Here both of the joint
vacations represent Nash equilibria. With (A: Mountain, B: Mountain) neither
player can gain by taking advantage of knowing the other's strategy. Similar
comments apply to (A: Seaside, B: Seaside). Hence this is a game with two
Nash equilibria.
Query: Are any of the strategies in either of these games dominant? Why aren't
separate vacations Nash equilibria in the Battle of the Sexes?
•
7Nash equilibria can be shown always to exist in certain types of games. For example, in zero-sum
games (where the payoffs sum to zero or any other constant), a Nash equilibrium always exists in
mixed strategies (strategies that consist of various pure strategies played with certain probabilities).
See, for example, Luce and Raiffa, Games and Decisions, appendices 2-5.
1
1
CHAPTER 21
• Strategy and Game Theory
625
I
Table 21.2
Two Simple Games
(al Rock, Scissors, Paper-No Nash Equilibria
B's Strategies
Scissors
Rock
A's Strategies
Rock
Scissors
Paper
o
o
1
1
1
o
-]
0
- 1
Paper
1
1
o
1
o
(b) Battle of the Sexes-Two Nash Equilibria
B's Strategies
Mountain
Seaside
A's Strategies
Mountain
Seaside
THE PRISONER'S
DILEMMA
The Prisoner's Dilemma game was first discussed by A. W. Tucker in the 19405.
The title stems from the following game situation. Two people are arrested for
a crime. The district attorney has little evidence in the case and is anxious to
extract a confession. She separates the suspects and tells each, "If you confess
and your companion doesn't, I can promise you a reduced (six-month) sentence,
whereas on the basis of your confession, your companion will get 10 years. If
you both confess, you will each get a three-year sentence." Each suspect also
knows that if neither of them confesses, the lack of evidence will cause them to
be tried for a lesser crime for which they will receive two-year sentences. The
normal form payoff matrix for this situation is illustrated in Table 21.3. The
"confess" strategy dominates for both A and B. Hence these strategies constitute
a Nash equilibrium and the district attorney's ploy looks successful. However,
an agreement by both not to confess would reduce their prison terms from three
to two years. This "rational" solution is not stable, and each prisoner has an
incentive to squeal on his or her colleague. This then is the dilemma-outcomes
that appear to be optimal are not stable.
Applications
Prisoner's Dilemma-type problems may arise in many real-world market situ­
ations. Table 21.4 contains an illustration of the dilemma in a different advertis­
ing game. Here the twin L strategies are most profitable, but this choice is un­
stable. This situation resembles the situation discussed in Chapter 20 where we
described why some advertising might be regarded as "defensive" in the sense
that a mutual agreement to reduce expenditures would be profitable to both
parties. Such an agreement in the situation of Table 21.4 would be unstable.
Either firm could increase its profits even further by cheating on the agreement.
Similar situations arise in the tendency for airlines to give "bonus mileage"
(there would be larger profits if all firms stopped offering free trips, but such
a solution is unstable) and in the instability of farmers' agreements to restrict
626
PART VI • Models of Imperiect Competition
Table 21.3
The Prisoner's Dilemma
B
Confess
A
Not Confess
Confess
Not Confess
A: 3 years
B: 3 years
A: 10 years
B: 6 months
A: 6 months
B: 10 years
A: 2 years
B: 2 years
Table 21.4 An Advertising Game with a Desirable Outcome
That Is Unstable
B's Strategies
L
L
A's Strategies H
H
3,10
5,5
output (it is just too tempting for an individual farmer to try to sell more milk).
As these examples show, the difficulty of enforcing agreements may be very det­
rimental to the profits of an industry.
Example 21.2
The Stackelberg Equilibrium
In Example 20.2 we developed a numerical illustration of a duopoly market in
which the outcomes depended on the strategic assumptions made by the com­
petitors. Under the Stackelberg version of that model, each firm had two pos­
60) or a "follower" (produce
sible strategies-to be a "leader" (produce q
q
30). Results from employing these strategies can be viewed as a 2 x 2
game, the payoff matrix for which is shown in Table 21.5. Adoption of the
leader-leader strategy is disastrous for the two firms, resulting in zero profits
for each. A follower-follower strategy (the Cournot solution) is much more
profitable for both firms as in the Prisoner's Dilemma, but this is unstable be­
cause it provides each firm with an incentive to cheat and grab the leader posi­
tion. The game is not strictly a Prisoner's Dilemma game either, however, since
the leader-leader solution is not a Nash equilibrium-if firm A knows that
firm B will be a leader, it might as well be a follower and vice versa. Each of
the leader-follower strategy pairs is an equilibrium, but, as before, the formal
problem offers no guidance on which pair will be chosen. Presumably, it will
CHAPTER 21 • Strategy and Game Theory
Table 21.5
627
Payoff Matrix for the Stackelberg Model
B's Strategies Follower (qn :::: 60)
(qn
30)
Leader
Leader
A's Strategies
(q. :::: 60)
Follower
(q. :::: 30)
:O
ITI
B:O
A: $ 900
B: $1,800
=
A:
B:
A:
B:
$1,800
$ 900
$1,600
$1,600
depend on outside factors such as the history of the industry or the personali­
ties of the firms' managers.
Query: How do you think the Stackelberg game would evolve if it were played
many times (see the discussion that follows)?
•
Cooperation and
Repetition
Communication and cooperation between participants can be an important part
of a game. In the Prisoner's Dilemma, for example, the inability to reach a co­
operative agreement not to confess leads to a second-best outcome. If the parties
could cooperate, they might do better. Similarly, in the Stackelberg game, an
agreement to operate as a cartel would raise profits above any of those listed in
Table 21.5.
As an example of how communications alone can affect the outcome of a
game, consider the payoff matrix shown in Table 21.6. In this version of the
advertising game, the adoption of strategy H by firm A has disastrous conse­
quences for firm B, causing a loss of 50 when B plays Land 25 when His
chosen. Without any communication A would choose L (this dominates H) and
B would choose H (which dominates L). Firm A would therefore end up with
+ 15 and B with + 10. However, by recognizing the potency of strategy H, A
may be able to improve its situation. It can threaten to play H unless B plays L.
If this threat is indeed credible (a topic we take up later), A can increase its
profits from 15 to 20.
If games are to be played many times, cooperative behavior may be fostered.
In the Prisoner's Dilemma game, for example, it seems doubtful that the district
attorney's ploy would work if it were used repeatedly. In this case, prisoners
might hear about the method and act accordingly in their interrogations. In
other contexts, firms that are continually exasperated by their inability to obtain
favorable market outcomes may come to perceive the kind of cooperative behav­
ior that is necessary. In antitrust theory, for example, some markets are believed
to be characterized by "tacit collusion" among the participants. Firms act as a
628
PART VI • Models of Imperfect Competition
Table 21.6
A Threat Game in Advertising
B's Strategies
A's Strategies
L
H
L
H
20,5
10, -50
15,10
5, -25
cartel even though they never meet to plot a common strategy. We will explore
the formal aspects of this problem later. Finally, repetition of the threat game
(Table 21.6) offers player A the opportunity to take reprisals on B for failing to
choose L. Imposing severe losses on B for "improper" behavior may be far more
persuasive than simply making abstract threats.
ATWO-PERIOD
ADVERTISING
GAME
These observations suggest that repeated games, perhaps with some types of
communication or cooperation, may involve complex scenarios that better re­
flect real-world markets than do the simple single-period models we have stud­
ied so far. In order to illustrate the formal aspects of such games, we will return
to a reformulated version of the advertising game presented at the beginning of
this chapter. We present the game first in extensive form in order to understand
its temporal aspects. Figure 21.2 repeats that game, but now we assume that
firm B knows which advertising spending level A has chosen. In graphical terms,
the oval around B's nodes has been eliminated in Figure 21.2 to indicate this
additional information. B's strategic choices now must be phrased in a way that
takes the information it has into account. In Table 21.7 we indicate such an
extended delineation of strategies. In all, there are four such strategies covering
the possible informational contingencies. Each strategy is stated as a pair of ac­
tions indicating what B will do depending on its information. The strategy (L, L)
indicates that B chooses L if A chooses L (its first strategy) and L also if A
chooses H (its second strategy). Similarly (H, L) indicates that B chooses H if
A chooses Land B chooses L if A chooses H. Although this table conveys little
more than did the previous normal form for the advertising game (Table 21.1),
explicit consideration of contingent strategy choices does enable us to explore
equilibrium notions for dynamic games in a simplified setting.
There are three Nash equilibria in this game: (1) A:L, B:L, L; (2) A:L, B:L, H;
and (3) A:H, B:H, L. Each of these strategy pairs meets the criterion of being
optimal for each player given the strategy of the other. Pairs (2) and (3) are
implausible, however, because they incorporate a noncredible threat that firm B
would not carry out if it were in a position to do so. Consider, for example, the
pair A:L, B:L, H. Under this choice B promises to play H if A plays H. A glance
at Figure 21.2 shows that this threat is not credible. If B were presented with
the fact of A having chosen H, it will make profits of 3 if it chooses H, but 4 if
it chooses L. The threat implicit in the L, H strategy is therefore not credible.
CHAPTER 21 • Strategy and Game Theory
Figure 21.2
629
The Advertising Game in Sequential Form
In this form of the advertising game, firm B knows firm A's advertising choice. Strate­
gies for B must be phrased taking this information into account. (See Table 21.7.)
7,5
5.4
A
6,4
6,3
Table 21.7
Contingent Strategies in the Advertising Game
A's Strategies
L
H
L, L
B's Strategies
L,H
H,L
H,H
7,5
6,4
7,5
6,3
5,4
6,3
5,4
6,4
Even though B's strategy L, H is one component of a Nash equilibrium, firm A
should be able to infer the noncredibility of the threat implicit in it.
By eliminating strategies that involve noncredible threats, A can conclude that
B would never play L, H or H, L.8 Proceeding in this way, the advertising game
is reduced to the payoff matrix originally shown in Table 21.1 and, as we dis­
cussed previously, in that case L, L (always playing L) is a dominant strategy
for B. Firm A can recognize this and will opt for strategy L. The Nash equilib­
'The process of eliminating strategies involving noncredible threars is termed "backward induction."
This method of solving games by "folding back the tree" was developed by H. Kuhn. See "Extensive
Games and the Problem of Information," in H. Kuhn and A. Tucker, eds., Contributions to the
Theory of Games (Princeton, N.].: Princeton University Press, 1953), pp. 193-216.
630
PART VI • Models of Imperfect Competition
rium A:L, B: L, L has therefore been shown to be the only one of the three in
Table 21.7 that does not involve noncredible threats. Such an equilibrium is
termed a "perfect equilibrium," which we define more formally as follows:
DEFINITION
Perfect Equilibrium A Nash equilibrium in which the strategy choices of each player do not involve non­
credible threats. That is, no strategy in such an equilibrium requires a player to carry out an action that
would not be in its interest at the time. 9
By using the concepts of strategic dominance, Nash equilibrium, and perfect
equilibrium, we are now in a position to examine a few game-theoretic models
of firm behavior.
MODELS OF
PRICING BEHAVIOR
We begin by illustrating some of the insights that game theory can provide to
the analysis of pricing. As in Chapter 20, most of the interesting results can be
shown for the duopoly case. Later in the chapter we briefly discuss some com­
plications involved in extending game theory models to games involving many
players.
The Bertrand
Equilibrium
Suppose there are two firms (A and B) each producing a homogeneous good at
constant marginal cost, c. The demand for the good is such that all sales go to
the firm with the lowest price and that sales are split evenly if PA = PB • The
available pricing strategies here consist of all prices greater than or equal to c~­
no firm would choose to operate at a loss by choosing a price less than c.
In this case the only Nash equilibrium is PA
PB
c. That is, the Nash
equilibrium is the competitive solution even though there are only two firms. To
see why, suppose firm A chooses a price greater than c. The profit-maximizing
response for firm 8 is to choose a price slightly less than PA and corner the entire
market. But 8's price, if it exceeds c, still cannot be a Nash equilibrium since it
PB
provides A with further incentives for price cutting. Only by choosing PA
c will the two firms in this market have achieved a Nash equilibrium. This
pricing strategy is sometimes referred to as a "Bertrand equilibrium" after the
French economist who discovered it.lO
An alternative definition of perfection focuses on the "subgames" implicit in any extensive game.
A "subgame" is a game that begins at one decision node and includes all future actions stemming
from decisions at this node. For a Nash equilibrium choice of strategies to be a subgame perfect
equilibrium, the strategies specified must constitute a Nash equilibrium in each subgame encoun­
tered during the play. In Figure 21.2 the Nash equilibrium A:L, 8:1-, L is a perfect equilibrium
because once the game reaches 8's decision node, the choice B:L is a Nash equilibrium. The Nash
equilibrium A:L, B:L, 11 is not a perfect equilibrium because the choice B:l1 is not a Nash equilib­
rium for the subgame starting at 8's decision node after A plays H. See R. Selten, "Reexamination
of the Perfectness Concept for Equilibrium Points in Extensive Games," International Journal of
Game Theory (March 1975): 25-55. In this article Selten proposes another defimtion of a "perfect"
equilibrium as a Nash equilibrium that is robust to errors made by the players. Here we adopt the
earlier notion of (subgame) perfection.
IOJ. Bertrand, "Theorie Mathematique de la Richesse Socia Ie," Journal de Savants (1883): 499-508.
9
1
CHAPTER 21 • Strategy and Game Theory
Two-Stage
Price Games
631
The simplicity and definiteness of the Bertrand result depend crucially on the
assumptions underlying the model. If firms do not have equal costs (see prob­
lem 21.3) or if the goods produced by the two firms are not perfect substitutes,
the competitive result no longer holds. Other duopoly models that depart from
the Bertrand result treat price competition as only the final stage of a two-stage
game in which the first stage involves various types of entry or investment con­
siderations for the firms. In Example 20.1 we examined Cournot's example of a
natural spring duopoly in which each spring owner chose how much water to
supply. In the present context we might assume that each firm in a duopoly must
choose a certain capacity output level for which marginal costs are constant up
to that level and infinite thereafter. It seems clear that a two-stage game in which
firms choose capacity first (and then price) is formally identical to the Cournot
analysis. The quantities chosen in the Cournot equilibrium represent a Nash
equilibrium since each firm correctly perceives what the other's output will be
(see Equations 20.16 and 20.17). Once these capacity decisions are made, the
only price that can prevail is that for which total quantity demanded is equal to
the combined capacities of the two firms.
To see why Bertrand-type price competition will result in such a solution,
suppose capacities are given by qA and qB and that
(21.1)
where D
I
is the inverse demand function for the good. A situation in which
(21.2)
is not a Nash equilibrium. With this price, total quantity demanded exceeds
qA + qB so anyone firm could increase its profits by raising price a bit and still
selling qA. Similarly,
(21.3)
is not a Nash equilibrium since now total sales fall short of qA +
. At least
one firm (say, firm A) is selling less than its capacity. By cutting price slightly,
firm A can increase its profits by taking all possible sales up to qA' Of course, B
will respond to a loss of sales by dropping its price a bit too. Hence the only
Nash equilibrium that can prevail is the Cournot result: II
P.
(21.4)
In general, this price will fall short of the monopoly price but will exceed mar­
ginal cost (as was the case in Example 20.1).12 Results of this two-stage game
are therefore indistinguishable from those arising from the Cournot model of
the previous chapter.
The contrast between the Bertrand and Cournot games is striking-the for­
mer predicts competitive outcomes in a duopoly situation whereas the latter
"For completeness, it should also be noted that no situation in which PA ,.£ PB can be an equilibrium
since the low-price firm has an incentive to raise price and the high-price firm wishes to cut price.
12Equation 20.10 also suggests another source of inefficiency in the Cournot solution-except in
special cases, marginal costs will not be the same among firms. See problem 21.4.
632
PART VI • Models of Imperfect Competition
predicts monopoly-like inefficiencies. This suggests that actual behavior in du­
opoly markets may exhibit a wide variety of outcomes depending on the precise
way in which competition occurs. The principal lesson of the two-stage Cournot
game is that, even with Bertrand price competition, decisions made prior to this
final stage of a game can have an important impact on market behavior. This
lesson will be reflected again in some of the game theory models of entry we
describe later in this chapter.
Tacit Collusion
Our analysis of the Prisoner's Dilemma concluded that, if the game were played
several times, the participants might devise ways to adopt more cooperative
strategic choices. The same query might be raised about the Bertrand game­
would repetition of this game offer some mechanism for the players to attain
supracompetitive profits by pursuing a monopoly pricing policy? One possibil­
ity, discussed in Chapter 20, would be for the players to establish a cartel and
explicitly set price or output targets. As we showed, such explicit agreements
are subject to a number of difficulties in enforcement. Here we adopt a non­
cooperative approach to the collusion question by exploring models of "tacit"
collusion. That is, we use game theory concepts to see whether there exist equi­
librium strategies that, though not explicitly coordinated, would allow firms to
achieve monopoly profits.
Our initial result from the Bertrand model poses a significant stumbling block
to achieving tacit collusion. Since the single-period Nash equilibrium in this
model results in PA
PB = c, we need to ask whether this situation would
change if the game were repeated during many periods. With any finite number
of repetitions, it seems clear that the Bertrand result remains unchanged. Any
strategy in which firm A, say, chooses PA > c during the final period (T) offers
firm B the possibility of earning profits by setting PA > PB > c. The threat of
charging Pi\. > c in period T is therefore not credible. Since a similar argument
applies to any period prior to T, we can conclude that the only perfect equilib­
rium is one in which firms charge the competitive price in every period. The
strict assumptions of the Bertrand model make tacit collusion impossible over
any finite period.
If firms are viewed as having an infinite time horizon, however, matters change
significantly. In this case there is no "final" period so collusive strategies may
exist that are not undermined by the logic of the Bertrand result. One such
possibility is for firms to adopt "trigger" strategies in which each firm (again,
say, firm A) sets PA = PM (where PM is the monopoly price) in every period for
which firm B adopts a similar price, but chooses PA = c if firm B has cheated
in the previous period.
To determine whether these trigger strategies constitute a perfect equilibrium,
we must ask whether they constitute a Nash equilibrium in every period. Sup­
pose the firms have colluded for a time and firm A thinks about cheating in this
period. Knowing that firm B will choose PB = PM, it can set its price slightly
below PM and, in this period, obtain the entire market for itself. It will thereby
earn (almost) the entire monopoly profits (7TM) in this period. But, by doing this,
firm A will lose its share of monopoly profits (7TMI2) forever after because its
CHAPTER 21 • Strategy and Game Theory
633
treachery will trigger firm B's retaliatory strategy. Since the present value (see
Chapter 24) of these lost profits is given by
7TM
2
(21.5)
r
(where r is the per-period interest rate), cheating will be unprofitable if
7TM
7TM
1
(21.6)
< T'~'
This condition holds for values of r less than !. We can therefore conclude that
the trigger strategies constitute a perfect equilibrium for sufficiently low inter­
est rates. The collusion implicit in these strategies is noncooperative. The firms
never actually have to meet in seedy hotel rooms to adopt strategies that yield
monopoly profits. 13
Example 21.3
Tacit Collusion in Steel Bars
Suppose only two firms produce steel bars suitable for jailhouse windows. Bars
are produced at a constant average and marginal cost of $10, and the demand
for bars is given by
Q = 5,000
lOOP.
(21.7)
Under Bertrand competition, each firm will charge a price of $10 and a total
of 4,000 bars will be sold. Since the monopoly price in this market is $30, each
firm has a dear incentive to consider collusive strategies. With the monopoly
price, total profits are $40,000 (each firm's share of total profits is $20,000)
so anyone firm will consider a next-period price cut only if
1
$40,000 > $20,000 -.
r
(21.8)
If we consider the pricing period in this model to be one year and a reason­
able value of r to be 0.20, the present value of each firm's future profit share is
$100,000 so there is dearly little incentive to cheat on price. Alternatively, each
firm might be willing to incur costs (say, by monitoring the other's price or by
developing a "reputation" for reliability) of up to $60,000 in present value to
maintain the agreement.
Tacit Collusion with More Firms Viability of a trigger price strategy may de­
pend importantly on the number of firms. With eight producers of steel bars,
the gain from cheating on a collusive agreement is still $40,000 (assuming the
II It seems clear from Equation 21.6 that other supracompetitive prices might also yield perfect
equilibria depending on the value of r. For a discussion, see J. Friedman, Oligopoly and the Theory
of Games (Amsterdam: North-Holland Publishing Co., 1977).
634
PART VI • Models of Imperfect Competition
cheater can corner the entire market). The present value of a continuing agree­
ment is only $25,000 (= $40,000 -0- 8 . 1/.2) so the trigger price strategy is
not viable for anyone firm. Even with three or four firms or less responsive
demand conditions, the gain from cheating may exceed the costs required to
make tacit collusion work. Hence the commonsense idea that tacit collusion
is easier with fewer firms is supported by this model.
Ouery: How does the interest rate determine the maximum number of firms
that can successfully collude in this problem? What is the maximum if r = 0.2?
How about the case when r = 0.1? Explain your results intuitively.
•
Generalizations
and Limitations
The contrast between the competitive results of the Bertrand model and the
monopoly results of the (infinite time period) collusive model suggests that the
viability of tacit collusion in game theory models is very sensitive to the particu­
lar assumptions made. Two assumptions in our simple model of tacit collusion
are especially important: (1) that firm B can easily detect whether firm A has
cheated; and (2) that firm B responds to cheating by adopting a harsh response
that not only punishes firm A, but also condemns firm B to zero profits forever.
In more general models of tacit collusion, these assumptions can be relaxed by,
for example, allowing for the possibility that it may be difficult for firm B to
recognize cheating by A. Some models examine alternative types of punishment
B might inflict on A-for example, B could cut price in some other market in
which A also sells. Other categories of models explore the consequences of in­
troducing differentiated products into models of tacit collusion or of incorpo­
rating other reasons why the demand for a firm's product may not respond
instantly to price changes by its rival. As might be imagined, results of such
modeling efforts are quite varied. 14 In all such models, the notions of Nash and
perfect equilibria continue to play an important role in identifying whether tacit
collusion can arise from strategic choices that appear to be viable.
ENTRY, EXIT,
AND STRATEGY
Our treatment of entry and exit in competitive and noncompetitive markets in
previous chapters left little room for strategic considerations. A potential entrant
was viewed as being concerned only with the relationship between prevailing
market price and its own (average or marginal) costs. We assumed that making
that comparison involved no special problems. Similarly, we assumed firms will
promptly leave a market they find to be unprofitable. Upon closer inspection,
however, the entry and exit issue can become considerably more complex. The
fundamental problem is that a firm wishing to enter or leave a market must
make some conjecture about how its action will affect market price in subse­
quent periods. Making such conjectures obviously requires the firm to consider
I'See J. Tirole, The Theory of Industrial Organization (Cambridge, Mass.: MIT Press, 1988),
chap. 6.
CHAPTER 21 • Strategy and Game Theory
635
what its rivals will do. What appears to be a relatively straightforward decision
comparing price and cost may therefore involve a number of possible strategic
ploys, especially when a firm's information about its rivals is imperfect.
Sunk Costs and
Commitment
Many game-theoretic models of the entry process stress the importance of a
firm's commitment to a specific market. If the nature of production requires
firms to make specific capital investments in order to operate in a market and if
these cannot easily be shifted to other uses, a firm that makes such an investment
has committed itself to being a market participant. Expenditures on such invest­
ments are called sunk costs, defined more formally as follows:
DEFINITION Sunk Costs Sunk costs are one-time investments that must be made in order to enter a market. Such
investments allow the firm to produce in the market but have no residual value if the firm exits the market.
Investments in sunk costs might include expenditures such as unique types of
equipment (for example, a newsprint-making machine) or job-specific training
for workers (developing the skills to use the newsprint machine). Sunk costs
have many characteristics similar to what we have called "fixed costs" in that
both these costs are incurred even if no output is produced. Rather than being
incurred periodically as are many fixed costs (heating the factory), however,
sunk costs are incurred only once in connection with the entry process. 15 When
the firm makes such an investment, it has committed itself to the market, and
that may have important consequences for its strategic behavior.
Sunk Costs, First­
Mover Advantages,
and Entry Deterrence
Although at first glance it might seem that incurring sunk costs by making the
commitment to serve a market puts a firm at a disadvantage, in most models
that is not the case. Rather, one firm can often stake out a claim to a market by
making a commitment to serve it and in the process limit the kinds of actions its
rivals find profitable. Many game theory models, therefore, stress the advantage
of moving first.
As an example, consider again the Stackelberg leadership game in Table 21.5.
Suppose we treat the output decision as reflecting commitments of the firms to
a particular level of productive capacity, which they will maintain in future
periods. With simultaneous moves, either of the follower-leader pairs in the pay­
off matrix represents a possible Nash equilibrium. However, if one firm (say,
firm A) has the opportunity to move first in this game, it will choose to be a
15
Mathematically, the notion of sunk costs can be integrated into the per-period total cost function as
TCt = S + Ft + cqt,
where $ is the per-period amortization of sunk costs (for example, the interest paid for funds used
to finance specific capital investments), F is per-period fixed costs, c is marginal cost, and q, is per­
$ + Ft , but if the production period is long enough, some or all of
period output. If q, = 0, TC,
F, may also be avoidable. No portion of S is avoidable, however.
636
PART VI • Models of Imperfect Competition
leader (q A = 60) and thereby limit firm B's options. Adoption of a relatively
large initial capacity by firm A gives it an advantage-there is simply not much
"room" left in the market for firm B. Given firm A's advantage in moving first,
B's most profitable decision is to be a follower.
Other situations in which a first mover might have an advantage include in­
vesting in research and development or pursuing product differentiation strate­
gies. In international trade theory, for example, it is sometimes claimed that
protection or subsidization of a domestic industry may allow it to enter an In­
dustry first, thereby gaining strategic advantage. Similarly, pursuit of "brand
proliferation" strategies by existing toothpaste or breakfast cereal companies
may make it more difficult for those who come later to develop a sufficiently
different product to warrant a place in the market. The success of such first­
mover strategies is by no means assured, however. Careful modeling of the stra­
tegic situation is required in order to identify whether moving first does offer
any real advantages.
In some cases, first-mover advantages may be large enough to deter all entry
by rivals. Intuitively, it seems plausible that the first mover could make the stra­
tegic choice to have a very large capacity and thereby discourage all other firms
from entering the market. The economic rationality of such a decision is not
clear-cut, however. In the Stackelberg model introduced in Chapter 20, for ex­
ample, the only sure way for one spring owner to deter all entry is to satisfy the
total market demand at the firm's marginal and average cost-that is, one firm
would have to offer q
120 at a price of zero to have a fully successful entry
deterrence strategy. Obviously, such a choice results in zero profits for the in­
cumbent firm and would not represent profit maximization. Instead, it would be
better for that firm to accept some entry by following the Stackelberg leadership
strategy.
With economies of scale in production, the possibility for profitable entry
deterrence is increased. If the firm that is to move first can adopt a large enough
scale of operation, it may be able to limit the scale of the potential entrant. The
potential entrant will therefore experience such high average costs that there
would be no advantage to its entering the market. Example 21.4 illustrates this
possibility in the case of Cournot's natural springs. Whether this example is of
general validity depends, among other factors, on whether the market is con­
testable. If other firms with large scales of operations elsewhere can take advan­
tage of prices in excess of marginal cost to practice hit-and-run entry of the type
described in Chapter 20, the entry deterrence strategy will not succeed.
Example 21.4
Entry Deterrence in Natural Springs
If the natural spring owners in our previous examples experience economies of
scale in production, entry deterrence becomes a profitable strategy for the first
firm to choose its quantity. The simplest way to incorporate economies of scale
into the Cournot model is to assume each spring owner must pay a fixed cost
of operations. If that fixed cost is given by $784 (a carefully chosen number!),
CHAPTER 21 • Strategy and Game Theory
637
it is clear that the Nash equilibrium leader-follower strategies remain profitable
for both firms (see Table 21.5). When firm A moves first and adopts the lead­
er's role, however, B's profits are rather small (900 - 784 = 116), and this
suggests that firm A could push B completely out of the market simply by be­
ing a bit more aggressive.
Since B's reaction function (Equation 20.18) is unaffected by considerations
of fixed costs, firm A knows that
120 - qA
2
(21.9)
and that market price is given by
P
(21.10)
Hence A knows that B's profits are
'TrB = PqB
(21.11)
784,
which, when B is a follower (that is, when B moves second) depends only on
qA. Substituting Equation 21.9 into 21.11 yields
'FrB
=
C
20
;
qA
r-
784.
(21.12)
Consequently, firm A can ensure nonpositive profits for firm B by choosing
(21.13 )
With qA = 64, firm A becomes the only supplier of natural spring water. Since
market price is $56 (= 120 - 64) in this case, firm A's profits are
(56, 64) - 784
2,800,
(21.14)
a significant improvement over the leader-follower outcome. The ability to
move first coupled with the fixed costs assumed here therefore makes entry
deterrence a successful strategy in this case.
Query: Why is the time pattern of play in this game crucial to the entry deter­
rence result? How does the result here contrast with our analysis of a contest­
able monopoly in Example 20.4?
•
Limit Pricing
So far our discussion of strategic considerations in entry decisions has focused
on issues of sunk costs and output commitments. Prices were assumed to be
determined through auction or Bertrand processes only after such commitments
were made. A somewhat different approach to the entry deterrence question
concerns the possibility of an incumbent monopoly accomplishing this goal
through its pricing policy alone. That is, are there situations where a monopoly
might purposely choose a low ("limit") price policy with the goal of deterring
entry into its market?
638
PART VI • Models of Imperfect Competition
In most simple cases, the limit pricing strategy does not seem to yield maxi­
mum profits nor to be sustainable over time. If an incumbent monopoly opts for
a price of PL < PM (where PM is the profit-maximizing price), it is obviously
hurting its current-period profits. But this limit price will deter entry in the f~­
ture only if PL falls short of the average cost of any potential entrant. If the
monopoly and its potential entrant have the same costs (and if capacity choices
do not play the role they did in the previous example), the only limit price that
IS sustainable in the presence of potential entry is PL = AC, adoption of which
would obviously defeat the purpose of being a monopoly since profits would be
zero. Hence the basic monopoly model offers little room for limit price behavior­
either there are barriers to entry that allow the monopoly to sustain PM, or there
are no such barriers, in which case competitive pricing prevails.
Believable models of limit pricing behavior must therefore depart from tra­
ditional assumptions. The most important set of such models are those involving
incomplete information. If an incumbent monopoly knows more about a par­
ticular market situation than does a potential entrant, it may be able to take
advantage of its superior knowledge to deter entry. As an example, consider the
game tree illustrated in Figure 21.3. Here firm A, the incumbent monopolist,
may have either "high" or "low" production costs as a result of past decisions.
Firm A does not actually choose its costs currently but, since these costs are not
known to B~ we must allow for the two possibilities. Clearly, the profitability of
B~s entry into the market depends on A's costs-with high costs B's entry is
profitable (7TH
3) whereas if A has low costs, entry is unprofitable (7TB = -1).
What is B to do? One possibility would be for B to use whatever information it
does have to develop a subjective probability estimate of A's true cost struc­
ture. 16 That is, B must assign probability estimates to the states of nature "low
cost" and "high cost." If B assumes there is a probability p that A has high cost
and (1
p) that it has low cost, entry will yield positive expected profits (see
Chapter 9) provided
E(7TB)
=
p(3)
+
(1 -
pH -1) > 0,
(21.15)
which holds for
p>!.
(21.16)
The particularly intriguing aspects of this game concern whether A can influ­
ence B's probability assessment. Clearly, regardless of its true costs, firm A is
better off if B adopts the no-entry strategy, and one way to assure that is for A
to make B believe that p < t. As an extreme case, if A can convince B that it is
certainly a low-cost producer (p = 0), B will clearly be deterred from entry even
if the true cost situation is otherwise. For example, if A chooses a low price
policy when it serves the market as a monopoly, this may signal (see Chapter 10)
16Games that require one player to devise probability estimates for the other's situation are termed
"Bayesian games of incomplete information" after the statistician Thomas Bayes who pioneered in
the development of the mathematics of subjective probabilities. For further details on such games,
see Tirole, The Theory of Industrial Organization, pp. 432-453.
CHAPTER 21 • Strategy and Game Theory
Figure 21.3
639
An Entry Game
Firm A has either a "high" or a "low" cost structure that cannot be observed by B. If B
assigns a subjective probability (p) to the possibility that A is high cost, it will enter
providing p
Firm A may try to influence B's probability estimate.
*'
1,3
B
A
4,0
3,-1
6,0
to B that A's costs are low and thereby deter entry. Such strategy might be
profitable for A even though it would require it to sacrifice some profits if its
costs are actually high. This provides a possible rationale for low limit pricing
as an entry deterrence strategy.
Unfortunately, as we said in Chapter 10, examination of the possibilities for
signaling equilibria in situations of asymmetric information raises many com­
plexities. Since firm B knows that A may create false signals and firm A knows
that B will be wary of its signals, a number of solutions to this game seem
possible. The viability of limit pricing as a strategy for achieving entry deterrence
depends crucially on the types of informational assumptions that are made. P
Predatory Pricing
Tools used to study limit pricing can also shed light on the possibility for "preda­
tory" pricing. Ever since the formation of the Standard Oil monopoly in the late
nineteenth century, part of the mythology of American business has been that
John D. Rockefeller was able to drive his competitors out of business by charg­
ing ruinously low (predatory) prices. Although both the economic logic and the
"For an examination of some of these issues, see P. Milgrom and J. Roberts, "Limit Pricing and
Entry under Conditions of Incomplete Information: An Equilibrium Analysis," Econometrica
(March 1982): 443-460.
640
PART VI • Models of Impertect Competition
empirical facts behind this version of the Standard Oil story have generally been
discounted,18 the possibility of encouraging exit through predation continues to
provide interesting opportunities for theoretical modeling.
The structure of many models of predatory behavior is similar to that used in
limit pricing models-that is, the models stress asymmetric information. An
incumbent firm wishes to encourage its rival to exit the market so it takes ac­
tions intended to affect the rival's view of the future profitability of market par­
ticipation. The incumbent may, for example, adopt a low price policy in an
attempt to signal to its rival that its costs are low-even if they are not. Or the
incumbent may adopt extensive advertising or product differentiation activities
with the intention of convincing its rival that it has economies of scale in under­
taking such activities. Once the rival is convinced that the incumbent firm pos­
sesses such advantages, it may recalculate the expected profitability of its pro­
duction decisions and decide to exit the market. Of course, as in the limit pricing
models, such successful predatory strategies are not a foregone conclusion. Their
viability depends crucially on the nature of the informational asymmetries in the
market.
n-PLAYER
GAME THEORY
All of the game theory examples we have developed so far in this chapter involve
only two players. Although this limitation is useful for illustrating some of the
strategic issues that arise in the play of a game (or the operation of a duopoly
market), it also tends to obscure some important questions. In this final section,
therefore, we briefly examine more general n-player games.
Coalitions
The most important element added to game theory when one moves beyond two
players is the possibility for the formation of subsets of players who agree on
coordinated strategies. Although the possibility for forming such coalitions ex­
ists in two-player games (the two firms in a duopoly could form a carte!), the
number of possible coalitions expands rapidly as games with larger numbers of
players are considered. In some games, simply listing the number of potential
coalitions and the payoffs they might receive can be a major analytical task.
As in the formation of cartels in oligopolistic markets, the likelihood of form­
ing successful coalitions in n-player games is importantly influenced by organi­
zational costs. These costs involve both information costs associated with deter­
mining coalition strategies and enforcement costs associated with ensuring that
a coalition's chosen strategy is actually followed by its members. If there are
incentives for members to cheat on established coalition strategies, then moni­
toring and enforcement costs may be high. In some cases, such costs may be so
high as to make the establishment of coalitions prohibitively costly. For these
games, then, all n-players operate independently.
IR J. S. McGee and others have pointed out that predatory pricing was a far less profitable strategy
for Rockefeller than simply buying up his rivals at market price (which seems to have been what
occurred). See J. S. McGee, "Predatory Pricing: The Standard Oil (NJ) Case," Journal of Law and
Economics (1958): 137-169; and "Predatory Pricing Revisited," Journal of Law and Economics
(October 1980): 289-330. Recent literature has examined whether predatory pricing can affect a
rival firm's market value.
CHAPTER 21 • Strategy and Game Theory
Game Theory,
General Equilibrium,
and the Core
641
In its most abstract theoretical development, n-player game theory has many
similarities to the type of general economic equilibrium theory described in Chap­
ters 16 and 17. An economy of n individuals can be viewed as an n-player game,
and the coalitions formed may be thought of as firms, local governments, or
any other type of economic organization. Of course, this way of modeling the
economy is quite abstract, and the results are therefore probably not appropri­
ate for any kind of detailed empirical study. Nevertheless, It-player game theory
has been widely used as a conceptual tool for understanding the nature of some
types of economic activity.
Central to these abstract uses of It-player game theory is the concept of the
core of a game. This represents an attempt to generalize the notion of Pareto
optimality to situations where subsets of individuals may form coalitions to im­
prove the welfare of the subset's members. We have already illustrated some
theoretical results related to the core in Chapter 8 when we discussed the Edge­
worth model of exchange. Here we briefly illustrate how that concept has been
adapted to game theory. We begin with a definition: 19
DEFINITION Core of an n-Player Game The core of an n-player game consists of that set of coalitions and strategies for
which no subset of players would find it advantageous to seek improvements through further coalition
activity.
This core concept, therefore, includes the Pareto definition of optimality as a
special case (since every individual in a game can form his or her own one­
person coalition and no unambiguous improvement in welfare is possible). The
concept is more general than the Pareto notion, however, since it also allows for
the balancing of power among multiplayer coalitions. Many games do not have
a core under this definition-as in some of the duopoly models of the previous
chapter, the play of such games represents an endless jockeying of the players
(and coalitions) for favorable outcomes. Bur, for games that have one (or possi­
bly many) allocations in the core, a wide variety of results relevant to economics
has been obtained.
Perhaps the most interesting of these results concern the relationship between
the core of a game and market-type institutions. Several authors have demon­
strated that core equilibria in n-player games can often be given a price system
interpretation. to Other questions that have been examined include the modeling
of public goods and externalities in game-theoretic contexts and ways of intro­
ducing money and financial institutions into economic games. 2l Such applica­
"This definition must necessarily be rather informal. For a complete development, see M. Shublk,
Game Theory in the Social Sciences (Cambridge, Mass.: MIT Press. 1982), chap. 6.
'" For an example, see R. M. Anderson, "An Elementary Core Equivalen~e Theorem," Econometriw (November 1978): 1483-1487. 21 For some references, see Shubik, Game Theory in the Social Sciences. chap. 12. 642
PART VI • Models of Imperfect Competition
tions serve to illustrate, in a conceptual way, how particular laws and institu­
tions may arise to solve problems that are common to practically all economies.
SUMMARY In this chapter we have briefly examined the economic theory of games with
particular reference to the use of that theory to explain strategic behavior in
duopoly markets. Some of the conclusions of this examination include:
• Concepts such as players, strategies, and payoffs are common
to
all games.
• Many games also possess a number of types of equilibrium solutions. With
a Nash equilibrium, each player's strategic choice is optimal given its rival's
choice. In dynamic games only Nash equilibria that involve credible threats
are viable.
• The Prisoner's Dilemma represents a particularly instructive two-person
game. In this game the most preferred outcome is unstable, though in re­
peated games the players may adopt various enforcement strategies.
• Game-theoretic models of duopoly pricing start from the Bertrand result
that the only Nash equilibrium in a simple game is competitive (marginal
cost) pricing. Consideration of possible output commitments and first-mover
strategies may result in noncompetitive results, however. Tacit collusion at
the monopoly price is sustainable in infinite-period games under certain
circumstances.
• Much of the game-theoretic modeling of entry and exit stresses the impor­
tance of the informational environment. In situations of asymmetric infor­
mation, incumbent firms may be able to capitalize on superior information
by adopting strategies that result in entry deterrence.
APPLICA TION
The Game of Chicken
Basic Scenario of the Game
In this chapter we looked at a number of simple two­
person games. There are 78 distinct variants of such
games, many of which provide useful economic insights.
Another example is provided by the game of "Chicken"
for which a typical payoff matrix is shown in Table 21.8.
In this game each player has two strategies: (1) to "co­
operate" or
"not to cooperate." Payoffs to the play­
ers are recorded in the table with 0 being the least favor­
able outcome and 3 being the most favorable for each
player. This game has two Nash equilibria points-the
off-diagonal strategy pairs. Both of these solutions are,
as we shall see, vulnerable to threats, however. It is the
possibility of these threats that gives this game its name
and makes the game empirically interesting.
Use of the title "Chicken" to describe this game derives
from a 19505 "game" played by hot-rod gangs (for the
true flavor of the game, see the great James Dean movie,
Rebel without a Cause). The game's players start at op­
posite ends of a deserted road with their left wheels on
the center line of the road. They then race toward each
other with the one veering off first being branded the
"chicken." In this game then "cooperation" consists of
veering off the center line whereas "noncooperation" re­
fers to a determined strategy of staying on that line. If
both drivers "chicken out," they both live, but gain no
social prestige from their peers. Hence, the twin co­
operative strategy is unstable with each driver having
an incentive to cheat on it (that is, stick to the center
line). If one driver succeeds in cheating in this way, he
CHAPTER 21
Table 21.8
• Strategy and Game Theory
643
The Game of Chicken
B's Strategies
Do Not
Cooperate
Cooperate
A's Strategies
Cooperate
Do Not Cooperate
or she receives considerable prestige (a "3") whereas the
chicken loses face (a "1 "). A twin strategy of nonco­
operation is a disaster, however, since the cars crash and
both drivers die (a "0"). Unlike the Prisoner's Dilemma,
this noncooperative solution is unstable-all the entries
in the table are Pareto superior to the crash solution. But
each driver has an incentive to threaten to take a non­
cooperative strategy in the hope that the opponent will
be cajoled into taking a cooperative strategy. It is in each
player's interest to appear to be committed to a nonco­
operative strategy even if he or she intends to "chicken
out" at some point along the way.
Applications to Oligopoly
Most applications of the game of Chicken to explaining
oligopoly behavior focus on the stability of situations in
which one firm could gain a clear advantage by altering
its behavior. That is, they focus on explaining why "co­
operative" solutions appear to be stable despite the fact
that they do not meet the Nash criteria for stability.
Consider, for example, the decision of U.S. auto makers
to refrain from building small cars in the 1950s and
early 1960s. Such restraint appeared to many observers
to be unstable-a company that built small cars could
have made a significant profit by doing so. But the threat
that its competitors would quickly enter the market also
(thereby seriously eroding profits on both small and
large cars) kept anyone company from proceeding. Of
course, this cooperative solution ultimately failed be­
cause the rapid growth of imported small cars forced the
U.S. companies to respond. The paralysis induced by
the strategic situation of the 19505 placed U.S. makers
at a considerable disadvantage in later years because
they had not developed the production expertise that
they needed.
A related situation concerns the pricing of gasoline
on major highways. Although there are relatively many
sellers, all seem to charge about the same price, and they
change prices together, usually on the same day. Presum­
ably, anyone seller faces a fairly elastic demand for its
gasoline and could gain significant sales by dropping its
price a few cents a gallon. But its competitors would
probably retaliate, and since total demand for gasoline
along the highway is relatively inelastic, all firms would
be worse off. Gas "wars" arise because of the periodic
breakdown of the cooperative solution, and prices drop
dramatically (your author once purchased gasoline for
$.12 per gallon in the 19605). But these price breaks
are usually quite short-lived (the $.12 price lasted only
about four hours), and discipline quickly returns to the
market.
Chicken, Nuclear Strategy, and the "Star Wars" Program
Some authors claim that the game of Chicken provides
the best representation of the strategic nuclear deter­
rence practiced by the United States and the Soviet
UnionY In this game-theoretic interpretation, both
superpowers refrain from the use of nuclear weapons
because of the threat that the other side will use them
too (similar arguments were also made about chemical
and biological weapon use in World War II). Although
this cooperative solution is unstable in that one party
could gain by being the first to use such weapons, the
threat that the other will follow suit maintains this un­
stable solution. Some authors have criticized this inter­
pretation because, in their view, the threat of "Mutually
Assured Destruction" is fundamentally "irrational"­
only a Dr. Strangelove would threaten to destroy the
world if attacked. This has led to considerable analysis
USee, for example, S. J. Beams, Superpower Games: Applying
Game Theory to the Superpower Conflict (New Haven, Conn.:
Yale University Press, 1985).
644
PART VI • Models of Imperfect Competition
of the nature of threats in the Chicken game including
the issues of which threats are credible and whether
threats can be probabilistic in nature. No simple, ac­
ceptable model of the nuclear stalemate has yet been de­
veloped, however.
Because we do not have a complete model of nuclear
deterrence, it is difficult to predict how technical inno­
vations will affect the strategic balance. This is especially
important in appraising the Strategic Defense Initiative
(SOl or "Star Wars") proposed in the early 1980s. Anti­
missile defenses have both desirable and undesirable
effects on the stability of strategic deterrence. The prin­
cipal positive effect is that such defenses reduce the
probability that accidents or third parties could initiate
a major superpower confrontation. They offer the chance
to destroy a few missiles that are launched in error. But
such defensive systems also introduce some uncertainty
into the deterrence calculation because they make retali­
ation more difficult. From the perspective of the Chicken
game, they reduce the credibility of threats and therebv
may make the cooperative solution less stable. Henc~
even discounting the cost and technical difficulties tha;
SDI poses, the overall desirability of the system cannot
be appraised fully until better theoretical representa­
tions of the strategic balance become available.
PROBLEMS
--------~--------------
21.1 Players A and B are engaged in a coin-matching game. Each shows a coin as either heads or tails. If the coins match, B pays A $1. If they differ, A pays B $1. a. Write down the payoff matrix for this game, and
show that it does not contain a Nash equilibrium.
b. How might the players choose their strategies in
this case?
21.2 Smith and Jones are playing a number-matching game. Each chooses either 1, 2, or 3. If the numbers match, Jones pays Smith $3. If they differ, Smith pays Jones $1. a. Describe the payoff matrix for this game, and
show that it does not possess a Nash equilibrium
strategy pair.
b. Show that with mixed strategies this game does
have a Nash equilibrium if each player plays
each number with probability!. What is the
value of this game?
21.3
Suppose firms A and B each operate under conditions of
10,
constant average and marginal COSt, bur that MeA
MC B = 8. The demand for the firms' output is given by
Qn
500 - 20P.
a. If the firms practice Bertrand competition, what
will be the market price under a Nash equI­
librium?
b. What will the profits be for each firm?
c. Will this equilibrium be Pareto efficient?
.... -~-~-....-
-...- -
.. -~-----
21.4 Suppose the two firms in a duopoly pursue Cournot competition as described in Equation 20.10. Suppose each firm operates under conditions of increasing mar­
ginal cost but that firm A has a larger scale of operations than does firm B in the sense that MC A < MC B for any given output level. In a Nash equilibrium, will marginal cost necessarily be equalized across the two firms? Will total output be produced as cheaply as possible? 21.5 In the ice cream cone stand example of Chapter 20, as­
sume each stand has five possible locational strategies­ locating 0, 25, 50, 75, or 100 yards from the left end of the beach. Describe the payoff matrix for this game, and explain whether it has an equilibrium strategy pair. 21.6 The world's entire supply of kryptonite is controlled by 20 people with each having 10,000 grams of this potent mineral. The world demand for kryptonite is given by Q
=
10,000 - 1,000P,
where P is the price per gram.
a. If all owners could conspire to rig the price of
kryptonite, what price would they set, and how
much of their supply would they sell?
b. Why is the price computed in part (a) an un­
stable equilibrium?
c. Does a price for kryptonite exist that would be
a stable equilibrium in the sense that no firm
CHAPTER 21
could gain by altering its output from that re­
quired to maintain this market price?
21.7
Smith and Jones are stranded on a desert island with
fixed initial endowments of clams and bread. Instead of
seeking mutually beneficial trades directly, they opt for
a bidding strategy in which they use their initial clam
endowments to bid for bread. That is, the bread is de­
posited in a safe place, and each person states how many
clams he or she will offer for the bread. When the bids
are revealed, the available bread is split in proportion to
each bid. If, for example, Smith bids one clam and Jones
bids two clams, Smith gets ! of the bread and Jones
gets !.
a. Sketch an Edgeworth box diagram that shows
the initial endowments of Smith and Jones.
b. On your Edgeworth box, show how a final al-
• Strategy and Game Theory
645
location is determined given a particular set of
bids by Smith and Jones.
c. Use your construction from part (b) to show
Smith's optimal reply to a particular bid by
Jones. Develop a similar construction for Jones.
d. Is there an eqUllibrium pair of strategies for the
situation described in part (c)? That is, is there a
bid for Smith that is optimal given Jones's bid
and vice versa?
e. Is the equilibrium described in parr (d) neces­
sarily on the contract curve for this exchange
economy?
(Note: This is an example of a "strategic market game."
For a further discussion, see M. Shubik, Game Theory
in the Social Sciences [Cambridge, Mass.: MIT Press,
1982], pp. 316-322.)
EXTENSIONS StrategiC Substitutes and Complements
(ii)
One way to conceptualize the relationships between the
choices of firms in an imperfectly competitive market is
to introduce the ideas of strategic substitutes and com­
plements. By drawing analogies to similar definitions
from consumer and producer theory, game theorists de­
fine firms' activities to be strategic substitutes if an in­
crease in the level of an activity (say, output, price, or
spending on product differentiation) by one firm is met
by a decrease in that activity by its rival. On the other
hand, activities are strategic complements if an increase
in an activity by one firm is met by an increase in that
activity by its rival.
To make these ideas precise, suppose that profits for
firm A ('lT A ) depend on the level of an activity it uses
itself (SA) and on use of a similar activity by its rival.
The firm's goal, therefore, is to maximize 'IT,4(SA' S8)'
Obviously, the optimal choice of SA specified by Equa­
tion (i) will differ for different values of S8' We can re­
cord this relationship by A's reaction function (R A )
(iii)
The strategic relationship between SA and SB is implied
by this reaction function. If R~ > 0, .)4 and S8 are stra­
tegic complements. If R~ < 0, SA and SB are strategic
substitutes.
E21.2 Inferences from the Profit Function.
It is usually more convenient to use the profit function
directly to examine strategic relationships. Substituting
Equation (iii) into the first-order condition (i) gives
o.
(iv)
Partial differentiation with respect to SB yields
E21.1 Optimality Conditions and Reaction Functions.
The first-order condition for A's choice of its own stra­
tegic activity is
0,
(v)
Therefore
(i)
where the subscripts for 'IT represent partial derivatives
with respect to its various arguments. For a maximum
we also require that
so, in view of the second-order condition (ii), 'lT~ > 0
implies R~ > 0 and 'lT~ < 0 implies R~ < O. Strategic
-
646
PART VI • Models of Imperfect Competition
relationships can therefore be inferred directly from the
derivatives of the profit function.
E2 t.3 The Cournot Model.
In the Coumot model (Equation 20.10) profits are given
as a function of the two firms' quantities as
TC(qA).
Using this notation,
TTA
P"q,
TC(qA)
(x)
Hence
TC'DA
(vi)
(xi)
and
In this case
'r. A
TC'
"
and
°
(viii)
0, the sign of
will depend on the con­
Since P'
cavity of the demand curve (pn). With a linear demand curve, P"
so TT~2 is dearly negative. Quantities are strategic substitutes in the COUrllot model with linear demand. This will generally be true unless the demand curve is relatively convex (P" > 0). °
E21.4 Strategic Relationship between Prices. If we view the duopoly problem as one of setting prices, both q, and qR will be functions of prices charged by the two firms: q.,
DA (P,\> PB)
TC' D~\
(vii)
(ix)
DR(P4 , PRJ.
-
TC'D~lD ;'.
Obviously, interpreting this mass of symbols is no easy
task. In the special case of constant marginal COSt ('TC:
= 0) and linear demand (Dj\
0), the sign of
IS
given by the sign of D;-that is how increases in PH
affect qA. In the usual case when the two goods are
themselves substitutes, D 2' > 0, so TT'~2 > 0. That is,
prices are strategic complements. Firms in such a duop­
oly would either raise or lower prices together.
REFERENCES
Bulow, J., Geanakoplous, G., and Klemperer, P. "Multi­
market Oligopoly: Strategic Substitutes and Comple·
ments." Journal of Political Economy (June 1985):
488-511.
Tirole, J The Theory of Industrial Organization. Cam­
bridge, Mass.: MIT Press, 1988. Pp. 326-336.
SUGGESTED READINGS Brams, J Superpower Games. New Haven, Conn.: Yale
University Press, 1985.
equilibrium concepts although examples are sometimes
difficult to follow. Application of game theory to the study of nuclear
strategy ami arms control. Brams has several other books
011 game theory (including one on game theory and the
Bible) that also make interesting reading.
Luce, R. D., and RaIffa, H. Games and Decisions. New York: John Wiley & Sons, 1957. Classic text 011 game theory. Very readable and com· plete-but only through mid·1950s. Friedman, J. W. Oligopoly and the Theory of Games.
Amsterdam: North-Holland Publishing Co., 1977.
Raiffa, H. Decision Analysis: Introductory Lectures 011 Choices under Uncertainty. Reading, Mass.: Addison· Wesley, 1968. ExtC1lstve theoretical analysis of game-theoretic inter­
pretatio/ls of oligopoly models. Excellent treatment of
Extended, easy-co-follow analysis of a sillgle problem ill repeated games.
decision theory. For more formal treatment. see Raiffa Kreps, D. M. A Course in Microeconomic Theory.
Princeton, N.J: Princeton University Press, 1990.
Part III contains a detailed summary of much recent
work in game theory. Useful discusswn of a number of
and Schlaifler, Applied Statistical Decision Theory. Schelling, T. C. Micromotives and Macrohehavior. New York: Norton, 1978. 1I
!
CHAPTER 21 • Strategy and Game Theory
647
Informal analysis of many situations with game­
theoreth' interpretations. Very readable.
Tirole, J. The Theory of Industrial Organization. Cam­
bridge, Mass.: MIT Press, 1988.
Schotter, A., and Schwodiauer, G. "Economics and
Game Theory; A Survey." Journal of EconomIc Litera­
ture (June 1980): 479-527.
Chapter 11 provides a helpful "users' manual" for game
theory. Game-theoretic techniques are used throughout
the text, especially in the discussions of tacit collusion
(Chapter 6) and entry (Chapter 8).
Useful suwey of the basic corlcepts of n-person game
theory. Good reference for some of the uses of game
theory in explaining institutional relationships.
Shubik, M. Game Theory in the Social Sciences. Cam­
bridge, Mass.: MIT Press, 1982.
Complete sun'ey of many aspects ofgame theory. Many
instructive examples make the book fairly informal and
readable.
von Neumann, J., and Morgenstern, O. Theory of
Games and Economic Behallior. Princeton, N.J.: Prince­
ton University Press, 1944.
Classic work with rather heary mathematics. Although
many of the proofs are now available in simpler form,
much of the conceptual background material here IS still
useful reading.