Chapter 13
Imperfect Competition:
A Game-Theoretic Approach
-A non-passive environment, unlike PC and Monopoly
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Chapter Outline
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An Introduction to the Theory of Games
Some Specific Oligopoly Models
Competition When There are Increasing Returns to Scale
Monopolistic Competition
A Spatial Interpretation of Monopolistic Competition
Historical Note: Hotelling’s Hot Dog Vendors
Consumer Preferences and Advertising
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Prisoner's Dilemma
--difficulty of collusion even with few producers
 Two prisoners are held in separate cells for a serious crime that they did
in fact commit. The prosecutor has only enough hard evidence to convict
them of a minor offense, for which the penalty is a year in jail.
 Each prisoner is told that if one confesses while the other remains silent,
the confessor will go scot-free while the other spends 20 years in prison.
 If both confess, they will get an intermediate sentence 5 years.
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Prisoner’s Dilemma
Prisoner Y
Prisoner X
Strategy
Confess
Don’t Confess
Confess
5 years for X
5 years for Y
0 for X
20 for Y
Don’t Confess 20 for X
0 for Y
1 year for X
1 year for Y
Dominant strategy--- the
strategy in a game that
produces better results
irrespective of the strategy
chosen by one’s opponent.
Yet when each confesses,
each does worse {5 years
each}than if each had not
confessed {1 year for each}.
Profits to Cooperation and Defection in a Prisoner’s Dilemma
Firm 1
Firm 2
Strategy
Cooperate
(P=$10)
Defect
(P=$9)
Cooperate
(P=$10)
Π1=$50
Π1=$50
Π1=$99
Π2=$0
Defect
(P=$9)
Π1=$0
Π2=$99
Π1=$49.50
Π2=$49.50
Dominant strategy- the strategy in a game that produces better
results irrespective of the strategy chosen by one’s opponent.
The dominant strategy is
for each firm to defect, for
doing so, it earns higher
profit no matter which
option its rival chooses.
Yet when both defect, each
earns marginally less
{$49.50 each} than when
each cooperates {$50 each}
Nash equilibrium: the combination of strategies in a game such that neither player has any incentive to
change strategies given the strategy of his opponent.
–A Nash equilibrium does not require both players to have a dominant strategy
A Game in which Firm 2 has no Dominant Strategy – a Maximin Approach
Firm 1
Firm 2
Strategy
Don’t
Advertise
Advertise
Don’t
Advertise
Π1=$500
Π1=$400
Π1=$750
Π2=$100
Advertise
Π1=$200
Π2=$0
Π1=$300
Π2=$200
Firm 1’s dominant strategy is to advertise regardless of what Firm 2 does.
Firm 2 has no dominant strategy. Thus, if Firm 1 advertises, Firm 2 does best by advertising
as well {Π1=$300, Π2=$200}.
BUT if Firm I doesn’t advertise, Firm 2 does best by not advertising as well{Π1=$500
Π1=$400}.
Since Firm 2 doesn’t have a dominant strategy, its response is determined by (a) likelihood it
assigns to Firm 1’s choices and (b) how its own payoffs are affected by (a).
One approach is for Firm 2 to take the maximin approach – choose the option that
maximizes its lowest possible value of its own payoff.
1. If Firm 2 doesn’t advertise, its lowest payoff is $100 if Firm 1 advertises.
2. But if it chooses to advertise, the lowest payoff is $0 if Firm 1 doesn’t advertise.
 Thus, if it follows a maximin strategy, Firm 2 will choose not to advertise.
Maximin strategy--choosing the option that makes the lowest payoff one can receive as large
as possible
Tit-for-Tat
Tit-for-tat strategy- The first time you interact with someone, you
cooperate. In each subsequent interaction you simply do what that
person did in the previous interaction.
 Thus, if your partner defected on your first interaction, you would then
defect on your next interaction with her.
 If she then cooperates, your move next time will be to cooperate as well.
 Requirement: there not be a known, fixed number of future
interactions.
Sequential Games
 Sequential game: one player moves first, and the other is then able to
choose his strategy with full knowledge of the first player’s choice.
– Example - United States and the former Soviet Union (USSR) during
much of the Cold War.
 Strategic entry deterrence – they change potential rivals’ expectations
about how the firm will respond when its market position is threatened.
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Figure 13.1: Nuclear Deterrence
as a Sequential Game
Nash Equilibrium if the USSR does attack initially.
1st move
Points B & C are US response
that depend on Soviet initial
action
“Doomsday” devise
eliminates the
bottom part.
Nash Equilibrium if the USSR does not attack
initially.
If the USSR attacked, the best response of the US is not to retaliate {Point E}
If the USSR doesn’t attack, the best US response is not to attack {Point G}
Since the USSR gets a higher payoff from attacking {Point E} than not attacking {Point G},
the US assumed (like the USSR) that the USSR would attack – reason to the Cold War built-up.
However, if the US maximizes its payoff, its threat to retaliate {Point D} is not credible since 50 > -100.
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Nash Equilibrium if X
knows Sears payoff
Figure 13.2: The Decision to Build
the Tallest Building
Figure 13.3: Strategic Entry Deterrence
Assume that at construction, Sears had the option to build a platform that allows to create it
to build a higher building if it so chose later. Cost of this is 10 units but the presence of a
platform reduces building higher floors by 20 units.
Given this provision, X (a rival to Sears) knows that Sears can add floors if X enters. If X
enters and Sears builds, the outcome is D {Sears =40; X =-50}. But if X enters and Sears does
not build, the outcome is E {Sears = 30; X =60}. Problem – X is not sure of outcome E.
The Nash Equilibrium is C {Sears=90; X =0}, i.e. the existence of a platform has acted a
strategic deterrence to X’s entry! Note that X doesn’t enter: 90 at C {Fig 13.3}= 100 at
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C{Fig.13.2} +(-10)
Figure 13.4: The Profit-Maximizing Cournot Duopolist
The portion to the right of the vertical at
Q1 is the demand curve for Firm , i.e.
Residual demand
The Cournot Model--oligopoly model in which each firm assumes that rivals will continue
producing their current output levels (assumes a naïve rival – not a convincing assumption)
Main assumption - each duopolist treats the other’s quantity as a fixed number, one that will
not respond to its own production decisions.
Reaction function- a curve that tells the profit-maximizing level of output for one oligopolist for
each amount supplied by another.
Suppose Market demand is: P = a – b(Q1 + Q2) with MC = 0;
Firm 1’s demand: P1=(a – bQ2) – bQ1 implies TR1 = P1Q1 = Q1(a – bQ2) – bQ12
MR1 =dTR1/dQ1 = a – bQ2 – 2bQ1 and set MR1 = MC and solve for Q1
a – bQ2 – 2bQ1 = 0 or Q1 = (a - bQ2)/2b = RN1 function for Firm 1. Similarly, Q2 = (a – bQ1)/2b
=RN2 since these are symmetric.
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Figure 13.5: Reaction Functions
for the Cournot Duopolists
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Figure 13.6: Deriving the Reaction Functions for Specific Duopolists
P=56 –2Q and MC =20
P1 = 56 – 2Q1- 2Q2; TR1 = 56Q1 – 2Q12 –
2Q1Q2; MR1 = 56 – 4Q1 – 2Q2
Set MR1 = MC and solve for Q1 = 9 – ½ Q2.
Similarly for Q2= 9 – ½ Q1
Q1 = 9 – ½ Q2 = 9 -½(9 – ½ Q1) 
3Q1 = 18 or Q1 = 6 = Q2.
Residual Demand for Firm 1
The Bertrand Model
Bertrand model - oligopoly model in which each firm assumes that rivals will
continue charging their current prices (again – a naïve assumption about pricing
behavior of a rival)
Example: Duopolist demand function: P =56 -2Q, MC =20.
Set P =MC but note that industry output and price: S =MC and D
So 20 = 56 – 2Q so that Q = 18 and since they share the market equally, each firm
produces 9 units. Naturally, P = 56 – 2*18 = 56 -36 = 20 which is the MC.
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Stackelberg Model
Figure 13.7: The Stackelberg
Leader’s Demand and Marginal
Revenue Curves
Figure 13.8: The Stackelberg Equilibrium
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Comparison Of Outcomes
Figure 13.9: Comparing Equilibrium Price and Quantity
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Competition When There Are Increasing Returns To Scale
 In markets for privately sold goods, buyers are often too numerous to organize
themselves to act collectively
 Where it is impractical for buyers to organize direct collective action, it
may nonetheless be possible for private agents to accomplish much the
same objective on their behalf.
Figure 13.10: Sharing a Market with Increasing Returns to Scale
With 2 firms in the market, costs are higher than with 1 firm (AC’ versus AC0).
Despite lower costs for the natural monopolist (AC0), it doesn’t follow that the incumbent
will successfully prevent entry or drive-off potential entrants into the market.
Reason - problem of collective action – many consumers are too difficult to organize to
boycott the natural monopolist that charges higher prices – Mancur Olson: The Logic of
Collective Action (1965).
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