Game Theory:Inside Oligopoly

Introduction• Behavior of competitors, or impact of own actions,

cannot be ignored in oligopoly• Managers maximize profit or market share by

outguessing competitors• Insight into oligopolistic markets by using GAME

THEORY (Von Neumann and Morgenstern in 1950): designed to evaluate situations with conflicting objectives or bargaining processes between at least two parties.

Types of Games• Normal Form vs. Extensive Form• Simultaneous vs. Sequential

(act without knowing (one player moves other player’s strategy) after observing others)

• One Shot vs. Repeated (infinite and finite with uncertain and certain final period)

• Zero Sum vs. Non-zero Sum (market share) (profit maximization)

A Normal Form Game

Strategy A B Cabc

Player 2

Play

er 1 12,11 11,12 14,13

11,10 10,11 12,1210,15 10,13 13,14

Elements of the game: PlayersStrategies or feasible

actionsPayoff matrix

Results from strategy dependent on the strategies of all the players

Planed decision or actions

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

C, Player 1 is better off choosing “a”!(Indiana Jones and the Holy Grail)

• “a” is Player 1’s Dominant Strategy!

Strategy A B Cabc

Player 2

Play

er 1 12,11 11,12 14,13

11,10 10,11 12,1210,15 10,13 13,14

1’s best a a astrategy

2’s beststrategy c

c a

The Outcome• What should player 2 do?

• 2 has no dominant strategy, but should reason that 1 will play “a”.• Therefore 2 should choose “C”.

Player 2

Play

er 1 12,11 11,12 14,1311,10 10,11 12,1210,15 10,13 13,14

• This outcome is called a Nash equilibrium (set of strategies were no player can improve payoffs by unilaterally changing own strategy given other player’s strategy)

•“a” 1’s best response to “C” and “C” is 2’s best response to “a”.

Strategy A B Cabc

14,13*

Best Response StrategyTry to predict the likely action of competitor to identify your best response:

• Conjecture choice of rival• Select your own best response• Was conjecture reasonable

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

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

• Two managers want to maximize market share (zero-sum game)• Strategies are pricing decisions• Simultaneous moves• One-shot game

Dominated Strategy• Dominance exception rather than rule• In absence of dominance it might be possible to simplify the game by eliminating

dominated strategy (never played: lowest payoff regardless of other player’s strategy) • Steelers trial by 2, have the ball & enough time for 2 plays• Payoff matrix in yards gained by offense: no dominant strategy• Pass dominant offense without Blitz (dominated defense)

Defense

Off

ense 2 6 14

8 7* 10

Best Offense Pass Pass Run

BestDefenseRun

Pass

Defend DefendStrategy Run Pass Blitz

RunPass

Maximin or Secure StrategyIn absence of dominant strategy risk averse players may abandon Nash or best response (*) and seek maximin option (^) that maximizes the minimum possible payoff.This is not design to maximize payoff but rather to avoid highly unfavorable outcomes (choose the best of all worst).

Strategy None New ProductNone 4, 4^ 3, 6*

New Product 6, 3* 2, 2 Firm

1

Firm 2

Best for 1 New 6 None 3Min for 2 None 3 New 2

Best Minfor 2 for 1New 6 New 3None 3 None 2

Board of Getty Oil agreed to sell 40% stake to Pennzoil @ $128.5 in Jan 1984. Board of Getty Oil subsequently accepted Texaco’s offer for 100% @ $128. Pennzoil sued Texaco for breach of contract & received $10 bill jury award in 1985. Texaco appealed. Before Supreme Court’s decision, they settled for $3 bill in 1987 .

Examples of Coordination Games

• Industry standards• size of floppy disks• size of CDs• etc.

• National standards• electric current• traffic laws• etc.

A Coordination Problem: Three Nash Equilibria!

Strategy A B C1 0,0 0,0 $10,$10*2 $10,$10* 0,0 0,03 0,0 $10, $10* 0,0

Player 2

Play

er 1

Key Insights:• In some cases one-shot, non-cooperative games result in

undesirable outcome for individuals (prisoner’s dilemma) and some times for society (advertisement).

• Communication can help solve coordination problems.• Sequential moves can help solve coordination problems.• Time in jail, Nash (*) and Maximin (^) equilibrium in

Prisoner’s dilemma.

Strategy Confess Do NotConfess 5, 5*^ 2, 15Do Not 15, 2 0, 0*

Susp

ect 1

Suspect 2

Best for 1 Confess Do NotMax for 2 Confess Confess

Best Maxfor 2 for 1Confess ConfessDo Not Confess

One-Shot Advertising Game Equilibrium

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

• Kellogg’s & General Mills want to maximize profits• Strategies consist of advertising campaigns• Simultaneous moves• One-shot interaction• Repeated interaction

Repeating the game 2 times will not improve outcome

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

• In the last period the game is a one-shot game, so equilibrium entails High Advertising.

• Period 1 is “really” the last period, since everyone knows what will happen in period 2.

• Equilibrium entails High Advertising by each firm in both periods.

• 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?

• Consider the “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.”

• Each firm agrees to “cooperate” so long as the rival hasn’t “cheated”, which triggers punishment in all future periods.

• “Tit-for-tat strategy” of copying opponents move from the previous period dominates “trigger strategy” for:

• Simple to understand• Never invites nor rewards cheating• Forgiving: allows cheater to restore cooperation by reversing actions

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

Benefits & Costs of Cheating Cheat - Cooperate = 8 - 10/i

• 8 = Immediate Benefit (20 - 12 today) • 10/i = PV of Future Cost (12 - 2 forever after)

• If Immediate Benefit > PV of Future Cost• Pays to “cheat”.

• If Immediate Benefit PV of Future Cost• Doesn’t pay to “cheat”.

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

Key Insight

• Collusion can be sustained as a Nash equilibrium when game lasts infinitely many periods or finitely many periods with uncertain “end”.

• Doing so requires:• Ability to monitor actions of rivals• Ability (and reputation for) punishing defectors • Low interest rate• High probability of future interaction

Real World Examples of Collusion: Garbage Collection Industry

Homogeneous products Known identity of customers Bertrand oligopoly Known identity of competitors

Strategy Low Price High PriceLow Price 0,0* 20,-1High Price -1, 20 15, 15

Firm 1

Firm 2

Strategy Low Price High PriceLow Price 0,0 20,-1High Price -1, 20 15, 15*

Firm 1

Firm 2

One-Shot Bertrand (Nash) Equilibrium

Repeated Game Equilibrium

Real World Examples of Collusion: OPEC

• Cartel founded in 1960 by Iran, Iraq, Kuwait, Saudis and Venezuela: “to co-ordinate and unify petroleum policies among Members in order to secure fair and stable prices”

• Absent collusion: PCompetition < PCournot (OPEC) < PMonopoly

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

VenezuelaSa

udi A

rabi

a

One-Shot Cournot (Nash) Equilibrium

VenezuelaStrategy High Q Med Q Low Q

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

Saud

i Ara

bia

Repeated Game Equilibrium

Assuming a Low Interest Rate

OPEC’s Demise

-5

0

5

10

15

20

25

30

35

40

1970 1972 1974 1976 1978 1980 1982 1984 1986

Real Interest Rate Price of Oil

Low Interest Rates

High Interest Rates

Simultaneous-Move Bargaining• Management and a union are negotiating a wage increase.• Strategies are wage offers & wage demands.• Simultaneous, one-shot move at making a deal.• Successful negotiations lead to $600 million in surplus (to be split among the

parties), failure results in a $100 million loss to the firm and a $3 million loss to the union.

• Experiments suggests that, in the absence of any “history,” real players typically coordinate on the “fair outcome”

• When there is a “bargaining history,” other outcomes may prevail

Three Nash Equilibriums 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

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.

• Analysis Tool: 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

Step 1: Management’s MoveStep 2: Add the Union’s MoveStep 3: Add the Payoffs

Firm

10

5

1

Union

Union

Union

Accept

Reject

100, 500

-100, -3

Accept

Accept

300, 300

-100, -3Reject

Reject

500, 100

-100, -3

To get The Game in Extensive Form

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 4: Identify Nash EquilibriumsOutcomes such that neither player 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

Step 5: Find the Subgame Perfect Nash Equilibriums

Outcomes where no player has an incentive to change its strategy, given the strategy of the rival, that are based on “credible actions”: not the result of “empty threats” (not in its “best self interest”).

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

mover advantage.• Management can gain by making a take-it or

leave-it offer to the union. 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.

Entrant

Out

EnterIncumbent

Hard

Soft

-1, 1

5, 5

0, 10

Pricing to Prevent Entry: An Application of Game TheoryTwo firms: an incumbent and potential entrant.Identify Nash and then Subgame Perfect Equilibria.

*

Establishing a reputation for being unkind to entrants can enhance long-term profits.It is costly to do so in the short-term, so much so that it isn’t optimal to do so in a one-shot game.

The Value of a Bad Reputation: Price Retaliation

• In early 1970s General Foods’ Maxwell House dominated the non-instant coffee market in the Eastern USA, while Proctor & Gamble’s Folgers dominate Western USA.

• In 1971 P&G started advertising & distributing Folgers in Cleveland and Pittsburgh.

• GF’s immediately increased advertisement & lowered prices (sometimes below cost) in these regions & midwestern cities (Kansas City) where both marketed.

• GF’s profit dropped from 30% in 1970 to –30% in 1974. When P&G reduced its promotional activities, GF’s price increased and profits were restored.

Limit Pricing• Strategy where an incumbent prices

below the monopoly price in order to keep potential entrants out of the market.

• Goal is to lessen competition by eliminating potential competitors’ incentives to enter the market.

• Incumbent produces QL instead of monopoly output QM.

• Resulting price, PL, is lower than monopoly price PM.

• Residual demand curve is the market demand DM minus QL.

• Entry is not profitable because entrant’s residual demand lies below AC

• Optimal limit pricing results ina residual demand such that, if the entrant entered and produced Q units, its profits would be zero.

Quantity

$

Q M

P M

AC

Entrant's residualdemand curve

D MP = AC

P L

Q LQ

(DM – QL)

Q

$

Q M

PM

AC

MC

MR

DMACM

M