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CHAPTER 6: OLIGOPOLY

1. Cover first single-period and multi-period games

2. Analyze standard oligopoly models: Cournot, Bertrand, and Stackelberg

3. The evidence

GAME THEORY: Formal models to analyze conflict (or competition) and cooperation (or collusion). It

is the study of how interdependent decision makers make choices.

Strategic Games: Describes interactions between mutually aware (and usually rational) players.

Why study game theory?

A deeper understanding of Oligopoly may conclude that the S-C-P model may not be a good guide for

public policy. For example, in the Bertrand price competition model, two firms competing is sufficient to

yield a perfectly competitive outcome of P = MC.

Although, game theory also suggests that cooperative behavior is easier to facilitate with more

concentration.

There are two broad types of games we will be looking at:

1. Static: Simultaneous Move Games

2. Dynamic: Sequential Move Games

SIMULTANEOUS MOVE GAMES

Use Game table or Payoff table or Matrix table. May be called the normal form, strategic form, ormatrix form of the game. We will primarily examine the following:

Prisoners Dilemma

Simultaneous, One-Move

Repeated

i) Finite Repetition

ii) Infinite Repetition

SEQUENTIAL MOVE GAMES

Use Game Trees. Tree is called the extensive form or game tree form

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SIMULTANEOUS MOVE GAMES

The starting point for solving simultaneous-move games for the equilibrium is to look for dominant

strategies.

Dominant Strategy: Strategy that gives a player higher payoffs against every opposing strategy.

In a dominant strategy equilibrium, each player chooses an action that is the best response against any

action the other might take.

A more general situation for an equilibrium is a Nash Equilibrium. In a Nash Equilibrium each player is

doing the best it can, given the strategies of the other players. (To change is to be worse off)

Games with dominant strategies yield a Nash Equilibrium. However, a Nash Equilibrium can emerge

without dominant strategies.

(Note: Nash is a nobel prize winner in economics and is the main character in the movie Its a Beautiful

Mind)

PRISONERS DILEMMA

This is the classic game in game theory. It is a simultaneous-move, one-shot game with imperfect

information (do not know what the other player will do)

The characteristics and outcomes of the PD are:

1. Each player has two strategies

Cooperate or collude

Defect, cheat, or compete

2. Each player has a dominant strategy (defect from cooperation)

3. Equilibrium for both players is worse than cooperative outcome

4. Conflict between individual incentives & joint incentives

Importance:

I. Wide ranging applicability

II. Equilibrium outcome is a bad outcome for individual players

Issues:

In Adam Smith, perfect competition world, the dominant strategy is pursuing self-interest

(greed), which yields the best outcome for society

Players can achieve a better outcome for themselves by committing conspiracies againstsociety

It takes only two players to achieve the (desirable) competitive outcome

Note:

Repeated PD games may yield a different outcome than one-shot PD games. A reputation or signals may

be established to encourage or facilitate the cooperative outcome. For example the reputation or signal

might be to:

Punish defection

Reward cooperation

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Consider the examples below

The first case is the classic PD game, where both players choose their dominant strategy Confess

Mumbles Action

Confess Dont Confess

Big Boys Confess 6 years, 6 years 1 year, 10 yearsAction Dont Confess 10 years, 1 year 3 years, 3 years

If someone can signal an alternative outcome for confessing, the outcome of the game can be altered.

Mumbles Action

Confess Dont Confess

Big Boys Confess death, death death, 10 years

Action Dont Confess 10 years, death 3 years, 3 years

For the signal to be credible, a reputation would need to be established which requires Repeated PD

Games.

PRISONERS DILEMMA REPEATED GAMES

Series of simple one-period simultaneous-move games

Because players can react to other players past actions, repeated games allow for equilibrium outcomes

that would not emerge in one-shot games. (Reputation Effects) There are two cases:

Finite:

In a finite game, it is not rational for any player to deviate from the dominant strategy in any period.

This can be seen by working backward. In the last period, with no future to worry about, it is rational to

play the dominant strategy. If it is known that everyone will play the dominant strategy in the lastperiod, then it is rational to play it in the next-to-the-last period. Repeating this logic, rational players

should play the dominant strategy in each period.

Chain-store paradox: Paradox is that despite the conclusion that predatory pricing is irrational to deter

entry in finite PD games, many firms are commonly perceived as engaging in predatory pricing. The

paradox is resolved by introducing uncertainty. If potential entrants lack certainty, then price cutting by

incumbents may alter entrants expectations about the incumbents future pricing strategy.

Infinite:

In infinite repeated games, dynamic pricing rivalry emerges and reputations can be established.

Cooperative (monopoly) pricing may be sustainable as an equilibrium, even though firms are makingdecisions non-cooperatively (w/o explicit collusion).

High market concentration facilitates cooperative pricing (cheaters are more easily detected)

A tit-for-tat strategy, in which a firm is prepared to match whatever strategy a competitor makes, is one

of the more compelling strategies that may lead to cooperative pricing.

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SEQUENTIAL GAMES

Sequence of decisions made over time.

These games utilize a game tree and are solved using the fold-back method or backward induction.

Consider the game in the handout.

Beta

Do not Expand Small Expand

Do not Expand $18, $18 15, 20 9, 18Alpha Small 20, 15 16, 16 8, 12

Large 18, 9 12, 8 0, 0

In the simultaneous-move game the Nash Equilibrium is

SMALL, SMALL.

Now suppose that the game is sequential and Alpha chooses first. Seek the subgame perfect Nash

Equilibrium. Subgames are a game within a game.

Game tree for capacity game.

,

Do Not Expand 18, 18Do Not Expand Beta Small 15, 20

Large 9, 18

Do Not Expand 20, 15

Alpha Small Beta Small 16, 16

Large 8, 12

Do Not Expand 18, 9

Large Beta Small 12, 8

Large 0, 0

Alpha chooses first, but decides on the basis of Betas expected choices.

If Alpha goes Beta maximizes payoff going Alpha payoff

Do Not Expand Small (Beta payoff $20) $15

Small Small (Beta payoff $16) $16

Large Do Not Expand (Beta payoff $9) $18

Given Betas choices, Alpha will maximize payoff by moving first and going LARGE. Beta does not

expand.

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Standard Models of Oligopoly Theory

(Single-period models of non-cooperative Oligopoly)

Single Period Static Models:

C: Cournot Duopoly Model (1838)

B: Bertrand Model (1883)

S: Stackelberg Model (1934)

Common Assumptions:

1. Number of firms is given & does not change

2. Produce a homogenous product (relaxed in B)

3. Firms have the same (constant) MC

Differences:

C, S: choice variable is quantity

B: choice variable is price

C, B: firms act simultaneously

S: interaction is sequential

Cournot Duopoly (2-firm) Model

Each firm chooses quantity given the other firm maintains their current output. [Output maintenance

assumption]

Market Demand: P = 30 ( q1 + q2 )

where q1 and q2 are perfect substitutes

Cost: AC = MC = $6

Monopoly Solution: Produce where MR = MC

MR = 30 2 Q = $6 = MC

Q = 12 P = $18 Profit = TR TC = 216 72 = 144

Competitive Solution: P = AC = MC = $6

Q = 24 P = $6 Profit = 0

Cournot Duopoly Solution:

Firm 1 faces the (inverse) demand curve (with choice variable q1)

P = ( 30 q2 ) q1

[Note typo in text on page 161, eq. (6.2)]

The demand curve is a residual demand (the market demand - q2 ). This firm will want to maximize

profit where MR1 = MC

[MR1 is the residual marginal revenue curve]

MR1 = (d R1 / d q1 ) = ( 30 q2 ) 2 q1 and MC = 6

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Solve MR1 = MC for q1 taking q2 as given. This will yield Firm 1s reaction function or best-response

function.

( 30 q2 ) 2 q1 = 6

q1 = 12 ( q2 / 2 )

By symmetry, firm 2s reaction function is

q2 = 12 ( q1 / 2 )

In equilibrium, each firm makes -max decision, anticipating a -max decision of all competitors.

Called the Cournot-Nash Equilibrium. It is a Nash equilibrium since each is doing the best given the

others choice.

The equilibrium must satisfy all reactions functions simultaneously.

q2 = 12 ( q1 / 2 ) = 12 [6 (q2 /4)]

Solving for q2 yields

q2 = q1 = 8 and P = $14

andProfit = $64 for each firm

Monopoly Duopoly

Perfect Competition

(Social Optimum)

Price $18 $14 $6

Industry Output 12 16 24

Industry Profit $144 $128 0

Generalization: As the number of firms increases, quantity competition equilibrium approaches the

perfectly competitive outcome. This supports the S-C-P model (see Table 6.2)

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Interpretation

The Cournot adjustment process shown above leads to the equilibrium where the two reaction functions

intersect. This suggests a sequential or dynamic process w/o learning.

A better interpretation is to consider this a Nash equilibrium game as a simultaneous output choice,

single-period model where all decision makers are rational (optimize) and strategic (consider rivalsactions).

But How do firms form their beliefs?

Problems with Cournot Model:

1. Single-period, static model is not realistic

2. The Cournot-Nash equilibrium is not a dominant strategy equilibrium. The output maintenance

assumption leads to this outcome.

3. The output maintenance assumption is nave. With learning and experience the players could

choose:

Monopoly outcome: (q1 = q2 = 6) and industry profit is maximized, or

Stackelberg outcome: Firm 1 commits to q1 = 12 and maximizes individual profit at $72 >

$64. Firm 2 passively reacts with q2 = 6 and profit = $36 < $64.

4. No basis for the underlying beliefs of the players

5. These criticisms have led to multiperiod models

q2

q1

Firm 1 Reaction Function


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BERTRAND MODEL OF PRICE COMPETITION

(firms set prices rather than output)

Features:

1. Duopolists sell homogeneous good at identical and constant MC

2. Consumers are astute and always purchase at the lower price

3. Low price firm gains the entire market. High price firm sells nothing

4. Each firm believes its rivals price is fixed

Consider the Prisoners Dilemma as a Price War

Firm 2

High Price (HP) Low Price (LP)

Firm High Price (HP) 10, 10 5, 12

1 Low Price (LP) 12, 5 7, 7

Now add the Bertrand assumption that low price always takes the entire market

Firm 2High Price (HP) Lower Price

Firm (HP) 10, 10 - FC, 15

1 Lower Price 15, - FC 0, 0

Each firm will want to undercut the rival to avoid losing all sales. In the limit each firm will price at MC

with profit of zero.

The Bertrand model has the extreme result that price competition by as few as two firms, yields the

perfectly competitive outcome. Lower price strategy always dominates, resulting in MC pricing.

Caveat:

1. Firms need unlimited capacity (there is no equilibrium with capacity constraints no one can

capture the entire market)

2. Must have homogeneous product

3. A differentiated product with soften the price competition

STACKELBERG MODEL

(sequential leader-follower model)

Go from a one-period simultaneous-move model to a two-stage sequential game (still considered static).Firms set output (still quantity competition like Cournot) but one firm acts as the leader and chooses

before the others (first-mover). The others choose according to their Cournot reaction function.

If Firm 1 chooses first, they would select the best point for them on Firm 2s reaction function.

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Compared to the Cournot solution, the Stackelberg solution is for the leader to increase output and the

follower to produce less. Total market output increases and market price falls. Firm 2 passively follows

with a smaller output and smaller profit while the leader gains at its expense. However, the output is

below the competitive output and price is above competitive price.

Strategic and credible commitment by Firm 1 gives it an advantage when other firms follow passively.

q2

q1



12

Firm 1 selects q1

= 12, Firm 2 selects q2

= 6

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