Managerial Economics & Business Strategy

Chapter 10Game Theory: Inside Oligopoly

Introduction: Prisoner’s DilemmaGinger and Rocky rob a 7-eleven. Evidence is good on the robbery but not so good on whether it was an armed robbery.

StrategyDon’t

Confess Confess

Don’t Confess 1, 1 7, Free

Confess Free, 7 5, 5

Rocky

Gin

ger

Introduction: Prisoner’s Dilemma

• Game Theory on Display (tires example)• EC Fines Roche, BASF, Six Other Firms T

otal of $751.7 million for Price Fixing, WSJ 2001

Game Form• A Normal Form Game consists of:

Players = managers or firms Strategies or feasible actions = planned decisions Payoffs = profits or losses, market share

• The order in which players make decisions is important

Simultaneous Sequential

• Type of game is important one-shot repeated games

A Normal Form Game

Strategy A B Cabc

Player 2

Pla

yer

1 12,11 11,12 14,13

11,10 10,11 12,12

10,15 10,13 13,14

Normal Form Game:Scenario Analysis

• Suppose 1 thinks 2 will choose “A”.

Strategy A B Cabc

Player 2

Pla

yer

1 12,11 11,12 14,13

11,10 10,11 12,12

10,15 10,13 13,14

Normal Form Game:Scenario Analysis

• Then 1 should choose “a”. Player 1’s best response to “A” is “a”.

Strategy A B Cabc

Player 2

Pla

yer

1 12,11 11,12 14,13

11,10 10,11 12,12

10,15 10,13 13,14

Normal Form Game:Scenario Analysis

• Suppose 1 thinks 2 will choose “B”.

Strategy A B Cabc

Player 2

Pla

yer

1 12,11 11,12 14,13

11,10 10,11 12,12

10,15 10,13 13,14

Normal Form Game:Scenario Analysis

• Then 1 should choose “a”. Player 1’s best response to “B” is “a”.

Strategy A B Cabc

Player 2

Pla

yer

1 12,11 11,12 14,13

11,10 10,11 12,12

10,15 10,13 13,14

Normal Form GameScenario Analysis

• Similarly, if 1 thinks 2 will choose C… Player 1’s best response to “C” is “a”.

Strategy A B Cabc

Player 2

Pla

yer

1 12,11 11,12 14,13

11,10 10,11 12,12

10,15 10,13 13,14

Dominant Strategy• Regardless of whether Player 2 chooses A, B, or

C, Player 1 is better off choosing “a”!• “a” is Player 1’s Dominant Strategy!

Strategy A B Cabc

Player 2

Pla

yer

1 12,11 11,12 14,13

11,10 10,11 12,12

10,15 10,13 13,14

Strategy A B Cabc

C

Putting Yourself in your Rival’s Shoes

• What should player 2 do? 2 has no dominant strategy! But 2 should reason that 1 will play “a”. Therefore 2 should choose “C”.

Player 2

Pla

yer

1 12,11 11,12 14,13

11,10 10,11 12,12

10,15 10,13 13,14

C

A

The Outcome

• This outcome is called a Nash equilibrium: “a” is player 1’s best response to “C”. “C” is player 2’s best response to “a”.

Strategy A B Cabc

Player 2

Pla

yer

1 12,11 11,12 14,13

11,10 10,11 12,12

10,15 10,13 13,14

Key Insights• Look for dominant strategies• Put yourself in your rival’s shoes

• Nash Equilibrium: doing the best you can given what other players are doing. No player can unilaterally improve his/her payoff by changing his/her strategy.

Not all games will have a Nash Equilibrium (monitoring workers example)

Some games may have more than 1 Nash Equilibrium

A Market Share Game

• Two managers want to maximize market share (the payoff is market share)

• Strategies are pricing decisions• Simultaneous moves• One-shot game

The Market-Share Game in Normal Form

Strategy P=$10 P=$5 P = $1P=$10 .5, .5 .2, .8 .1, .9P=$5 .8, .2 .5, .5 .2, .8P=$1 .9, .1 .8, .2 .5, .5

Manager 2

Man

ager

1

First number in each cell represents the market share for Firm 1, and the second for Firm 2.

Market-Share Game Equilibrium

Strategy P=$10 P=$5 P = $1P=$10 .5, .5 .2, .8 .1, .9P=$5 .8, .2 .5, .5 .2, .8P=$1 .9, .1 .8, .2 .5, .5

Manager 2

Man

ager

1

Nash Equilibrium

A Coordination Game

Strategy Work Together Don’t Work Together

Work Together $100, $80 $40, $40

Don’t Work Together $25, $25 $50, $80

Microsoft Word

Cor

el W

ord

Per

fect

A Coordination Problem: Two Nash Equilibria!

Strategy Work Together Don’t Work Together

Work Together $100, $80 $40, $40

Don’t Work Together $25, $25 $50, $80

Microsoft Word

Cor

el W

ord

Per

fect

Key Insights:

• Not all games are games of conflict (e.g., industry standards existed for floppy disc sizes, electrical currents, new “industries,” HDTV)

• Communication can help solve coordination problems.

• Sequential moves can help solve coordination problems.

• Government could set a standard to solve coordination problems: electrical current.

An Advertising Game

• Two firms (Kellogg’s & General Mills) managers want to maximize profits

• Strategies consist of advertising campaigns• Simultaneous moves• One-shot interaction• Repeated interaction

A One-Shot Advertising Game

Strategy None Moderate HighNone 12,12 1, 20 -1, 15

Moderate 20, 1 6, 6 0, 9High 15, -1 9, 0 2, 2

General Mills

Kel

logg

’s

Equilibrium to the One-Shot Advertising Game

Strategy None Moderate HighNone 12,12 1, 20 -1, 15

Moderate 20, 1 6, 6 0, 9High 15, -1 9, 0 2, 2

General Mills

Kel

logg

’s

Nash Equilibrium

Can collusion work if the game is repeated 1 time?

Strategy None Moderate HighNone 12,12 1, 20 -1, 15

Moderate 20, 1 6, 6 0, 9High 15, -1 9, 0 2, 2

General Mills

Kel

logg

’s

No (by backwards induction).• In period 2, the game is a one-shot game, so

equilibrium entails High Advertising in the last period.

• The same holds true if we repeat the game any known, finite number of times.

Can collusion work if firms play the game each year, forever?

• Yes, unless a rival “cheats.” Collusion can be supported with trigger strategies.

• Consider the following “trigger strategy” by each firm:

“Don’t advertise, provided the rival has not advertised in the past. If the rival ever advertises, “punish” it by engaging in a high level of advertising forever after.”

• In effect, each firm agrees to “cooperate” so long as the rival hasn’t “cheated” in the past. “Cheating” triggers punishment in all future periods.

Suppose General Mills adopts this trigger strategy. Kellogg’s profits?

Cooperate = 12 +12/(1+i) + 12/(1+i)2 + 12/(1+i)3 + …

= 12 + 12/i

Strategy None Moderate HighNone 12,12 1, 20 -1, 15

Moderate 20, 1 6, 6 0, 9High 15, -1 9, 0 2, 2

General Mills

Kel

logg

’s

Value of a perpetuity of $12 paid at the end of every year

Cheat = 20 +2/(1+i) + 2/(1+i)2 + 2/(1+i)3 + = 20 + 2/i

Kellogg’s Gain to Cheating:• Cheat - Cooperate = 20 + 2/i - (12 + 12/i) = 8 - 10/i

Suppose i = .05

• Cheat - Cooperate = 8 - 10/.05 = 8 - 200 = -192

• It doesn’t pay to deviate. Collusion is a Nash equilibrium in the infinitely repeated

game!

Strategy None Moderate HighNone 12,12 1, 20 -1, 15

Moderate 20, 1 6, 6 0, 9High 15, -1 9,0 2, 2

General Mills

Kel

logg

’s

UGH!

Key Insight• Collusion can be sustained as a Nash

equilibrium when there is no certain “end” to a game.

• Doing so requires: Know your rival, your rival’s customers, and be able to

monitor Ability (and reputation for) punishing cheaters Low interest rate (in this example, i ≤ 5/4) High probability of future interaction

• In infinitely repeated games there is always a tomorrow to consider. Must weigh the benefits and costs of cheating.

Real World Example of Collusion: OPEC

• Cartel founded in 1960 by Iran, Iraq, Kuwait, Saudi Arabia, and Venezuela.

• OPEC's mission is to coordinate and unify the petroleum policies of Member Countries and ensure the stabilization of oil markets in order to secure an efficient, economic and regular supply of petroleum to consumers, a steady income to producers and a fair return on capital to those investing in the petroleum industry. (www.opec.org)

Cournot Game in Normal Form

Strategy High Q Med Q Low QHigh Q 5, 3 9,4 3, 6Med Q 6, 7 12,10 20, 8Low Q 8, 1 10, 18 18, 15

Venezuela

Sau

di A

rab

ia

High Q is never a good strategy for either country; this essentially, means the game is between Med and Low.

One-Shot Cournot (Nash) Equilibrium

Strategy High Q Med Q Low QHigh Q 5, 3 9,4 3, 6Med Q 6, 7 12,10 20, 8Low Q 8, 1 10, 18 18, 15

Venezuela

Sau

di A

rab

ia

Repeated Game Equilibrium*

Venezuela

Strategy High Q Med Q Low QHigh Q 5, 3 9,4 3, 6Med Q 6, 7 12,10 20, 8Low Q 8, 1 10, 18 18, 15

* (Assuming a sufficiently low interest rate)

Sau

di A

rab

ia

Caveat

• Collusion is a felony under Section 2 of the Sherman Antitrust Act.

• Conviction can result in both fines and jail-time (at the discretion of the court).

• OPEC isn’t illegal; thus, US laws don’t apply

Simultaneous-Move Bargaining Game (SMBG)

• Management and a union are negotiating a wage increase.

• Strategies are wage offers & wage demands• Successful negotiations lead to $600 million in

surplus, which must be split among the parties• Failure to reach an agreement results in a loss to

the firm of $100 million and a union loss of $3 million

• Simultaneous moves, and time permits only one-shot at making a deal.

The SMBG in Normal Form

Strategy W = $10 W = $5 W = $1W = $10 100, 500 -100, -3 -100, -3W=$5 -100, -3 300, 300 -100, -3W=$1 -100, -3 -100, -3 500, 100

Union

Man

agem

ent

Three Nash Equilibria!

Strategy W = $10 W = $5 W = $1W = $10 100, 500 -100, -3 -100, -3W=$5 -100, -3 300, 300 -100, -3W=$1 -100, -3 -100, -3 500, 100

Union

Man

agem

ent

Fairness

Strategy W = $10 W = $5 W = $1W = $10 100, 500 -100, -3 -100, -3W=$5 -100, -3 300, 300 -100, -3W=$1 -100, -3 -100, -3 500, 100

Union

Man

agem

ent

SMBG are difficult to predict because of multiple Nash Equilibria. Experimental evidence suggests that bargainers often perceive a 50-50 split to be fair.

Single Offer Bargaining

• Now suppose the game is sequential in nature, and management gets to make the union a “take-it-or-leave-it” offer.

• Do the following: write the game in extensive form

Summarize the players Their potential actions Their information at each decision point The sequence of moves and Each player’s payoff

Firm

10

5

1

Step 1: Management’s Move

Firm

10

5

1

Union

Union

Union

Accept

Reject

Accept

Accept

Reject

Reject

Step 2: Add the Union’s Move

Firm

10

5

1

Union

Union

Union

Accept

Reject

100, 500

-100, -3

Accept

Accept

300, 300

-100, -3Reject

Reject

500, 100

-100, -3

Step 3: Add the Payoffs

Step 6: Identify Nash Equilibrium Outcomes

• Outcomes such that neither the firm nor the union has an incentive to change its strategy, given the strategy of the other

Firm

10

5

1

Union

Union

Union

Accept

Reject

100, 500

-100, -3

Accept

Accept

300, 300

-100, -3Reject

Reject

500, 100

-100, -3

There are 3 Nash Equilibrium Outcomes!

Firm

10

5

1

Union

Union

Union

Accept

Reject

100, 500

-100, -3

Accept

Accept

300, 300

-100, -3Reject

Reject

500, 100

-100, -3

Only 1 Subgame-Perfect Nash Equilibrium Outcome!

Re-Cap• In take-it-or-leave-it bargaining, there is a

first-mover advantage (what if union made offer, and firm had to take or leave it?). But...

• Management should be careful, however; real world evidence suggests that people sometimes reject offers on the the basis of “principle” instead of cash considerations.

• While the outcome was obvious, the framework can be used to analyze more complicated bargaining issues.

Pricing to Prevent Entry: An Application of Game Theory

• Two firms: an incumbent and potential entrant

• Incumbent threatens to start a price war. Is this credible?

• The game in extensive form:

The Entry Game in Extensive Form

Entrant

Out

Enter

Incumbent

Price War

No Price War

-1, 1

5, 5

0, 10

One Subgame Perfect Equilibrium

Entrant

Out

Enter

Incumbent

Price War

No Price War

-1, 1

5, 5

0, 10

Incumbant VS. Entrant:Incumbant and Credible Threat

Incumbent

Don’t Expand Q

Expand Q

Entrant

Enter

Don’t Enter

($2m), ($2m)

$4m, $0

Entrant

Enter

Don’t Enter$6m, $0

$2m, $2m

Incumbant VS. Entrant:Incumbant and Credible Threat

Incumbent

Don’t Expand Q

Expand Q

Entrant

Enter

Don’t Enter

($2m), ($2m)

$4m, $0

Entrant

Enter

Don’t Enter$6m, $0

$2m, $2m

Nash and Subgame Perfect Nash: Expand and Don’t Enter

Note: would see different outcome if entrant goes first (and has started its entry), then incumbant threat not credible.

Insight

• It does not pay to heed threats made by rivals because not all threats are credible.