EC 3322 Semester I – 2008/2009

Yohanes E. RiyantoEC 3322 (Industrial Organization

I) 1

EC 3322 Semester I – 2008/2009

Topic 5:Static Games

Cournot Competition


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Introduction A monopoly does not have to worry about how rivals will

react to its action simply because there are no rivals. A competitive firm potentiall faces many rivals, but the

firm and its rivals are price takers also no need to worry about rivals’ actions.

An oligopolist operating in a market with few competitors needs to anticipate rivals’ actions/ strategies (e.g. prices, outputs, advertising, etc), as these actions are going to affect its profit.

The oligopolist needs to choose an appropriate response to those actions similarly, rivals also need to anticipate the firm’s response and act accordingly interactive setting.

Game theory is an appropriate tool to analyze strategic actions in such an interactive setting important assumption: firms (or firms’ managers) are rational decision makers.


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Introduction … Consider the following story (taken from Dixit and Skeath

(1999), Games of Strategy)

…”There were two friends taking Chemistry at Duke. Both had done pretty well on all of the quizzes, the labs, and the midterm, so that going to the final they had a solid A. They were so confident that the weekend before the final exam they decided to go to a party at the University of Virginia. The party was so good that they overslept all day Sunday, and got back too late to study for the Chemistry final that was scheduled for Monday morning. Rather than take the final unprepared, they went to the professor with a sob story. They said they had gone to the University of Virginia and had planned to come back in good time to study for the final but had had a flat tire on the way back. Because they did not have a spare, they had spent most of the night looking for help. Now they were too tired, so could they please have a make-up final the next day?

The two studied all of Monday evening and came well prepared on Tuesday morning. The professor placed them in separate rooms and handed the test to each. Each of them wrote a good answer, and greatly relieved, but …

when they turned to the last page. It had just one question, worth 90 points. It was: “Which tire?”….


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Introduction … Why are Professors So Mean (taken from Dixit and

Skeath (1999), Games of Strategy)

Many professors have an inflexible rule not to accept late submission of problem sets of term papers. Students think the professors must be really hard heartened to behave this way.

However, the true strategic reason is often exactly the opposite. Most professors are kindhearted , and would like to give their students every reasonable break and accept any reasonable excuse. The trouble lies in judging what is reasonable. It is hard to distinguish between similar excuses and almost impossible to verify the truth. The professor knows that on each occasion he will end up by giving the student the benefit of the doubt. But the professor also knows that this is a slippery slope, As the students come to know that the professor is a soft touch, they will procrastinate more and produce even flimsier excuses. Deadline will cease to mean anything.


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Introduction … Non-cooperative game theory vs. cooperative game

theory. The former refers to a setting in which each individual firm (a player) behave non-cooperatively towards others (rivals players). The latter refers to a setting in which a group of firms cooperate by forming a coalition.

We focus on non-cooperative game theory. Different in timing of actions: simultaneous vs. sequential

move games. Different in the nature of information: complete vs.

incomplete information. Oligopoly theory no single unified theory, unlike

theory of monopoly and theory of perfect competition theoretical predictions depend on the game theoretical tools chosen.

Need a concept of equilibrium to characterize the chosen optimal strategies.


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Introduction … A ‘game’ consists of:

A set of players (e.g. 2 firms (duopoly)) A set of feasible strategies (e.g. prices, quantities, etc)

for all players A set of payoffs (e.g. profits) for each player from all

combinations of strategies chosen by players. Equilibrium concept first formalized by John Nash

no player (firm) wants to unilaterally change its chosen strategy given that no other player (firm) change its strategy.

Equilibrium may not be ‘nice’ players (firms) can do better if they can cooperate, but cooperation may be difficult to enforced (not credible) or illegal.

Finding an equilibrium: one way is by elimination of all (strictly) dominated strategies, i.e. strategies that will never be chosen by players the elimination process should lead us to the dominant strategy.


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Introduction … Ways of representing a Game:

Extensive Form Representation (Game Tree) Normal Form Representation

Extensive FormExtensive Form

1H

L

2

H

L

2H

L

In $ millions

1,1

0,2

2,0

½, ½

1H

L

2

H

L

2H’

L’

In $ millions

1,1

0,2

2,0

½, ½sequential move simultaneous move


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Introduction …

Normal FormNormal Form

Definition: A strategy is a complete contingent plan (a full specification of a player’s behavior at each of his/ her decision points) for a player in the game.

Player 2

normal form - sequential move

H L

HH’

HL’

LH’

LL’

Player 1

1 , 1 2 , 0

1 , 1 ½ , ½

0 , 2 2 , 0

0 , 2 ½ , ½

extensive form - sequential move

1H

L

2

H

L

2H’

L’

In millions

1,1

0,2

2,0

½, ½


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Introduction …

1H

L

2H

L

2H

L

In millions1,1

0,2

2,0

½, ½

L H

L

H

Player 2

Player 1

1/2 , 1/2

0 , 2

2 , 0

1 , 1

extensive form - simultaneous move normal form - simultaneous move


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Introduction … When players can choose infinite number of actions,

instead of only 2 actions e.g. quantities, advertising expenditures, prices, etc.

12

1-a,0

a-a2, ¼ - a/2stayin

exit1

2

1-a,0

a-a2, ¼ - a/2

exit

stayin

aa aa

sequential move simultaneous move


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Example … Two airline companies, e.g. SIA and Qantas offering a

daily flight from Singapore to Sydney. Assume that they already have set a price for the flight,

but the departure time is still undecided the departure time is the strategy choice in this game.

70% of consumers prefer evening departure while 30% prefer morning departure.

If both airlines choose the same departure time, they share the market share equally.

Payoffs to the airlines are determined by the market share obtained.

Both airlines choose the departure time simultaneously we can represent the payoffs in a matrix firm.


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Example …The Pay-Off Matrix

Qantas

SIA

Morning

Morning

Evening

Evening

(15, 15)

The left-handnumber is the

pay-off toSIA

(30, 70)

(70, 30) (35, 35)

What is theequilibrium for this

game?

The right-handnumber is the

pay-off toQantas


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Qantas

SIA

Morning

Morning

Evening

Evening

(15, 15)

If Qantaschooses a morning

departure, SIAwill choose

evening

(30, 70)

(70, 30) (35, 35)

If Qantaschooses an evening

departure, SIAwill also choose

evening

The morning departureis a dominated

strategy for SIABoth airlines

choose an evening

departure

(35, 35)

The morning departureis also a dominated strategy for Qantas


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Example … Suppose now that SIA has a frequent flyer

program.

Thus, when both airlines choose the same departure times, SIA will obtain 60% of market share.

This will change the payoff matrix.


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Qantas

SIA

Morning

Morning

Evening

Evening

(18, 12) (30, 70)

(70, 30) (42, 28)

However, amorning departureis still a dominated

strategy for SIA

If SIAchooses a morningdeparture, Qantas

will chooseevening

But if SIAchooses an eveningdeparture, Qantas

will choosemorning

Qantas has no dominated strategy

Qantas knowsthis and so

chooses a morningdeparture

(70, 30)


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Example … What if there are no dominated strategies? We need to

use the Nash Equilibrium concept. To show this consider a modified version of our

airlines game instead of choosing departure times, firms choose prices For simplicity, consider only two possible price levels.

Settings: There are 60 consumers with a reservation price of $500 for

the flight, and another 120 consumers with the lower reservation price of $220.

Price discrimination is not possible (perhaps for regulatory reasons or because the airlines don’t know the passenger types).

Costs are $200 per passenger no matter when the plane leaves.

airlines must choose between a price of $500 and a price of $220

If equal prices are charged the passengers are evenly shared. The low price airline gets all passengers.


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ExampleThe Pay-Off Matrix

Qantas

SIA

PH = $500

($9000,$9000) ($0, $3600)

($3600, $0) ($1800, $1800)

PH = $500

PL = $220

PL = $220

If both price highthen both get 30

passengers. Profitper passenger is

$300

If SIA prices highand Qantas lowthen Qantas gets

all 180 passengers.Profit per passenger

is $20

If SIA prices lowand Qantas high

then SIA getsall 180 passengers.

Profit per passengeris $20

If both price lowthey each get 90

passengers.Profit per passenger

is $20


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Nash EquilibriumThe Pay-Off Matrix

Qantas

SIA

PH = $500

($9000,$9000) ($0, $3600)

($3600, $0) ($1800, $1800)

PH = $500

PL = $220

PL = $220

(PH, PL) cannot bea Nash equilibrium.

If Qantas priceslow then SIA should

also price low

($0, $3600)

(PL, PH) cannot bea Nash equilibrium.

If Qantas priceshigh then SIA should

also price high

($3600, $0)

(PH, PH) is a Nashequilibrium.

If both are pricinghigh then neither wants

to change

($9000, $9000)

(PL, PL) is a Nashequilibrium.

If both are pricinglow then neither wants

to change

($1800, $1800)

There are two Nashequilibria to this version

of the game

There is no simpleway to choose between

these equilibria Custom and familiaritymight lead both to

price high “Regret” might

cause both toprice low


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Nash Equilibrium Another very common game prisoner’s dilemma game

illustrates that the resulting NE outcome may be ‘inefficient’.

So the only Nash equilibrium for this game is (C,C), even though (D,D) gives both 1 and 2 better jail terms. The only Nash equilibrium is inefficient.

criminal 1(6,6) (1,10)

(10,1) (3,3)

Confess

Don’t confess

Confess Don’t Confesscriminal 2


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Nash Equilibrium

Firm A(100, 100) (25, 140)

(140, 25) (80, 80)

H

L

H LFirm B

Consider the following price game between Firm A and Firm B

Had the firms been able to cooperate, they would have been able to obtain higher payoffs.


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Mathematical Presentation of Nash Eq. Suppose that there are 2 firms, 1 and 2 it can be

generalized to n firms.

The profit of each firm is denoted by with

is the set of all feasible strategies from which i can choose. Thus, are the pair of strategies chosen by players i and j from the set of feasible strategies.

Then, a pair of strategies is a Nash equilibrium if, for each firm i:

Thus, for a strategy combination to be a Nash eq., the strategy si* must be firm i’s best response to firm j’s strategy, sj* , and conversely sj* must be firm j’s best response to strategy si*.

, i i i js s 1,2, . i j i

, 1,2,iS i and i js s

* *,i js s

* * *, , for all , 1,2, . i i ii j i j is s s s s S i j i


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Example:

is firm i’s profit and are firms i and j’s quantities (outputs). If the profit function is continuous, concave and differentiable, we can solve for the optimal strategy si* by solving the first-order condition for the max. problem:

Similarly firm j will also choose its strategy optimally:

Finally the pair of Nash eq. outputs can be obtained by solving the system of equation (1) and (2) simultaneously. To guarantee that are the maximands we have to check for the second order condition for profit maximization.

Mathematical Presentation of Nash Eq.

i ,i js s

*,0 then solve f or (1)

i i j

ii jss s

s s

*,0 then solve f or (2)

j i j

jj iss s

s s

* *,i js s * *,i js s


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Oligopoly Models There are three dominant oligopoly models

Cournot Bertrand

Stackelberg

They are distinguished by

the decision variable that firms choose

the timing of the underlying game

We will start first with Cournot Model.


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The Cournot Model Consider the case of duopoly (2 competing firms) and

there are no entry..

Firms produce homogenous (identical) product with the market demand for the product:

Marginal cost for each firm is constant at c per unit of output. Assume that A>c.

To get the demand curve for one of the firms we treat the output of the other firm as constant. So for firm 2, demand is

It can be depicted graphically as follows.

1 2

1

2

quantity of firm 1 quantity of firm 2

P A BQ A B q qqq

1 2 P A Bq Bq


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The Cournot Model

P = (A - Bq1) - Bq2

$

Quantity

A - Bq1

If the output offirm 1 is increasedthe demand curvefor firm 2 moves

to the left

A - Bq’1

The profit-maximizing choice of output by firm 2 depends upon the output of firm 1

DemandMarginal revenue for firm 2 isMR2 = = (A - Bq1) - 2Bq2

MR2MR2 =

MCA - Bq1 - 2Bq2 = c

Solve thisfor output

q2

q*2 = (A - c)/2B - q1/2

c MC

q*2

2

2

TRq


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The Cournot Model We have

this is the best response function for firm 2 (reaction function for firm 2).

It gives firm 2’s profit-maximizing choice of output for any choice of output by firm 1.

In a similar fashion, we can also derive the reaction function for firm 1.

Cournot-Nash equilibrium requires that both firms be on their reaction functions.

2

* 1

2 2

A c qqB

1

* 2

2 2

A c qqB


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q2

q1

The reaction functionfor firm 1 is

q*1 = (A-c)/2B - q2/2(A-c)/B

(A-c)/2B

Firm 1’s reaction function

The reaction functionfor firm 2 is

q*2 = (A-c)/2B - q1/2(A-c)/2B

(A-c)/B

If firm 2 producesnothing then firm1 will produce themonopoly output

(A-c)/2B

If firm 2 produces(A-c)/B then firm1 will choose to

produce no output


The Cournot-Nashequilibrium is atthe intersectionof the reaction

functions

C

qC1

qC2

The Cournot Model


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q2

q1

(A-c)/B

(A-c)/2B


(A-c)/2B

(A-c)/B


C

q*1 = (A - c)/2B - q*2/2

q*2 = (A - c)/2B - q*1/2

q*2 = (A - c)/2B - (A - c)/4B + q*2/4

3q*2/4 = (A - c)/4B q*2 = (A - c)/3B

(A-c)/3B q*1 = (A - c)/3B

(A-c)/3B

The Cournot Model


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The Cournot Model In equilibrium each firm produces

Total output is therefore

Demand is P=A-BQ, thus price equals to

Profits of firms 1 and 2 are respectively

A monopoly will produce

1

* *2 3

c c A cq q

B

* 23A c

QB

* 2 23 3A c A cP A

* * * * * *1 2 1 2

2* *1 2 9

c cP c q P c q

A cB

1

1 1 1 1max M

qP c q A Bq c q

1 2M A cq

B

2

1 4

M A cB


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The Cournot Model Competition between firms leads them to overproduce.

The total output produced is higher than in the monopoly case. The duopoly price is lower than the monopoly price.

It can be verified that, the duopoly output is still lower than the competitive output where P=MC.

The overproduction is essentially due to the inability of firms to credibly commit to cooperate they are in a prisoner’s dilemma kind of situation more on this when we discuss collusion.

1

* 23 2

MA c A cQ q

B B

*1

2 because 3 2

mA c A cP P A Bq A c

2 2

MR

A cP MC c c A BQ QB


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The Cournot Model (Many Firms)

Suppose there are N identical firms producing identical products.

Total output:

Demand is:

Consider firm 1, its demand can be expressed as:

Use a simplifying notation:

So demand for firm 1 is:

1 2 3 ... NQ q q q q

1 2 3 ... NP A BQ A B q q q q

2 3 1... NP A BQ A B q q q Bq

1 2 3 ... NQ q q q

This denotes outputof every firm other

than firm 1

1 1P A BQ Bq


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P = (A - BQ-1) - Bq1

$

Quantity

A - BQ-1

If the output ofthe other firms

is increasedthe demand curvefor firm 1 moves

to the leftA - BQ’-

1

The profit-maximizing choice of output by firm 1 depends upon the output of the other firms Deman

d

Marginal revenue for firm 1 is

MR1 = (A - BQ-1) - 2Bq1

MR1MR1 =

MCA - BQ-1 - 2Bq1 = c


q1

q*1 = (A - c)/2B - Q-

1/2

c MC

q*1



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q*1 = (A - c)/2B - Q-1/2How do we solve this

for q*1?The firms are identical.So in equilibrium they

will have identicaloutputs

Q*-1 = (N - 1)q*1

q*1 = (A - c)/2B - (N - 1)q*1/2

(1 + (N - 1)/2)q*1 = (A - c)/2B q*1(N + 1)/2 = (A - c)/2B q*1 = (A - c)/(N + 1)B

Q* = N(A - c)/(N + 1)B P* = A - BQ* = (A + Nc)/(N + 1)

As the number offirms increases output

of each firm falls As the number of

firms increasesaggregate output

increases As the number offirms increases price

tends to marginal cost

Profit of firm 1 is Π*1 = (P* - c)q*1 = (A - c)2/(N + 1)2B

As the number offirms increases profit

of each firm falls


lim1N

A Ncc

N

*

2

11

A cQN B N


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Cournot-Nash Equilibrium: Different Costs

Marginal costs of firm 1 are c1 and of firm 2 are c2.

Demand is P = A - BQ = A - B(q1 + q2)

We have marginal revenue for firm 1 as before.

MR1 = (A - Bq2) - 2Bq1

Equate to marginal cost: (A - Bq2) - 2Bq1 = c1


q1

q*1 = (A - c1)/2B - q2/2

A symmetric resultholds for output of

firm 2

q*2 = (A - c2)/2B - q1/2


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q2

q1

(A-c1)/B

(A-c1)/2B

R1

(A-c2)/2B

(A-c2)/B

R2C

q*1 = (A - c1)/2B - q*2/2

q*2 = (A - c2)/2B - q*1/2

q*2 = (A - c2)/2B - (A - c1)/4B + q*2/4

3q*2/4 = (A - 2c2 + c1)/4B q*2 = (A - 2c2 + c1)/3B

q*1 = (A - 2c1 + c2)/3B

What happens to thisequilibrium when

costs change?

If the marginalcost of firm 2

falls its reactioncurve shifts to

the right

The equilibriumoutput of firm 2increases and of

firm 1 falls


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In equilibrium the firms produce:

The demand is P=A-BQ, thus the eq. price is:

Profits are:

Equilibrium output is less than the competitive level.

Output is produced inefficiently the low cost firm should produce all the output.

1 2 2 11 2

* 1 21 2

2 2 and

3 32

3

C C

C C

A c c A c cq q

B BA c c

Q q qB

* 1 2 1 223 3

A c c A c cP A

2 2* *1 2 2 11 2

2 2 and

9 9

A c c A c c

B B


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Concentration and Profitability Consider the case of N firms with different marginal

costs.

We can use the N-firms analysis with modification.

Recall that the demand for firm 1 is

So then the demand for firm 1 is : , so the MR can be derived as

Equate MR=MC and denote the equilibrium solution by *.

1 1P A BQ Bq

i iP A BQ Bq2i iMR A BQ Bq

* * * * *2 i i ii i i iA BQ Bq c A BQ Bq Bq c

* * * 0i ii i

P

A B Q q Bq c

* * 0 i iP Bq c

But Q*-i + q*i = Q*and A - BQ* = P*

* * i iP Bq c


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Concentration and ProfitabilityP* - ci = Bq*i

Divide by P* and multiply the right-hand side by Q*/Q*

P* - ci

P*=

BQ*P*

q*i

Q*

But BQ*/P* = 1/ and q*i/Q* = si

so: P* - ci

P*=

si

The price-cost marginfor each firm is

determined by itsmarket share anddemand elasticity

Extending this we haveP* - c

P*= H

Average price-costmargin is

determined by industryconcentration

**

*

* * ** *

* 1 1* * *

1

2*

1

ii

N N

i i iNi ii

ii

N

ii

sP cP

s P s cP c P csP P P

s H


I) 40

Final Remarks So far we consider only “pure” strategy equilibria a

player picks the strategy with certainty (prob.=1), e.g. choosing ‘kick the ball to the middle’ in a soccer penalty shootout..

“Mixed” strategies the player uses a probabilistic mixture of the available strategies, e.g. left, middle, right thus randomize the strategies sometimes aims the left, middle or right.Burger King

Low Price

HeavyAdvertising

Low Price

HeavyAdvertising

McDonalds(60, 35) (56, 45)

(58, 50) (60, 40)

No PureStrategy Eq.


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Suppose Burger King believes that McDonald will play strategy L with prob and H with prob. . When BK plays L, its expected payoff is:

If BK plays H, its expected payoff is:

BK will be indifferent between L and H iff:

Thus, when McDonald plays the optimal mixed strategy eq. with the above prob. distribution then BK will be indifferent between playing L or H.

Final Remarks

35 50 1 45 40 1

50 15 40 5

1 1 and 12 2

L L L L

L L

L L

M M M M

M M

M M

p p p p

p p

p p

LMp 1

H LM Mp p

35 50 1 L LM Mp p

45 40 1 L LM Mp p


I) 42

Final Remarks

60 56 1 58 60 1

56 4 60 2

2 1 and 13 3

L L L L

L L

L L

B B B B

B B

B B

p p p p

p p

p p

Similarly, when BK plays its optimal mixed strategy eq. then McDonald will be indifferent between playing L or H.

Burger King

Low Price

HeavyAdvertising

Low Price

HeavyAdvertising

McDonalds(60, 35) (56, 45)

(58, 50) (60, 40)

2 / 3LB

p 1 1/ 3 LB

p

1/ 2LM

p

1 1/ 2 LM

p


I) 43

Next … (Bertrand Price Competition)

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EC 3322 Semester I – 2008/2009

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