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Chapter 13

- Imperfect CompetitionA Game-Theoretic Approach
- -A non-passive environment, unlike PC and Monopoly

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Chapter Outline

- 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 Hotellings 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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Prisoners Dilemma

- Dominant strategy--- the strategy in a game that

produces better results irrespective of the

strategy chosen by ones opponent. - Yet when each confesses, each does worse 5 years

eachthan if each had not confessed 1 year for

each.

Prisoner Y Prisoner Y

Strategy Confess Dont Confess

Prisoner X Confess 5 years for X 5 years for Y 0 for X 20 for Y

Prisoner X Dont Confess 20 for X 0 for Y 1 year for X 1 year for Y

Profits to Cooperation and Defection in a

Prisoners Dilemma

Firm 1 Firm 1

Strategy Cooperate (P10) Defect (P9)

Firm 2 Cooperate (P10) ?150 ?150 ?199 ?20

Firm 2 Defect (P9) ?10 ?299 ?149.50 ?249.50

- 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

- Dominant strategy- the strategy in a game that

produces better - results irrespective of the strategy chosen by

ones opponent.

- 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 1

Strategy Dont Advertise Advertise

Firm 2 Dont Advertise ?1500 ?1400 ?1750 ?2100

Firm 2 Advertise ?1200 ?20 ?1300 ?2200

- Firm 1s 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 ?1300, ?2200. - BUT if Firm I doesnt advertise, Firm 2 does best

by not advertising as well?1500 - ?1400.
- Since Firm 2 doesnt have a dominant strategy,

its response is determined by (a) likelihood it

assigns to Firm 1s 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. - If Firm 2 doesnt advertise, its lowest payoff is

100 if Firm 1 advertises. - But if it chooses to advertise, the lowest payoff

is 0 if Firm 1 doesnt 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 players 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 Deterrenceas 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 doesnt 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 gt -100.

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Nash Equilibrium if X knows Sears payoff

Figure 13.2 The Decision to Buildthe 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 Sears90 X 0, i.e.

the existence of a platform has acted a strategic

deterrence to Xs entry! Note that X doesnt

enter 90 at C Fig 13.3 100 at CFig.13.2

(-10)

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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

others 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 1s 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 Functionsfor the Cournot

Duopolists

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Figure 13.6 Deriving the Reaction Functions for

Specific Duopolists

P56 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

56 218 56 -36 20 which is the MC.

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Stackelberg Model

Figure 13.7 The Stackelberg Leaders 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 doesnt 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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