Game theory inside oligopoly
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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

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Game Theory: Inside Oligopoly

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Game theory inside oligopoly

Game Theory:

Inside Oligopoly


Introduction

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

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

11,10

10,11

12,12

10,15

10,13

13,14

A Normal Form Game

  • Elements of the game: Players

    • Strategies or feasible actionsPayoff matrix

Player 2

Planed decision or actions

12,11

11,12

14,13

Player 1

Results from strategy dependent on the strategies of all the players


Dominant strategy

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”!(Indiana Jones and the Holy Grail)

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

Player 2

2’s beststrategy c

c a

12,11

11,12

14,13

Player 1

1’s best a a astrategy


The outcome

11,10

10,11

12,12

10,15

10,13

13,14

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

12,11

11,12

14,13

14,13*

Player 1

  • 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”.


Best response strategy

Best Response Strategy

Try to predict the likely action of competitor to identify your best response:

  • Conjecture choice of rival

  • Select your own best response

  • Was conjecture reasonableor

  • Look for dominant strategies

  • Put yourself in your rival’s shoes


Market share game equilibrium

Market-Share Game Equilibrium

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

Manager 2

Manager 1

Nash Equilibrium


Dominated strategy

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

BestDefenseRun

Pass

2

6

14

Offense

7*

8

10

Best Offense Pass Pass Run


Maximin or secure strategy

Firm 2

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

Firm 1

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

Maximin or Secure Strategy

In 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).

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

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

A Coordination Problem: Three Nash Equilibria!

Player 2

Player 1


Key insights

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.

Suspect 2

Best Maxfor 2 for 1Confess ConfessDo Not Confess

Suspect 1

Best for 1 ConfessDo NotMax for 2ConfessConfess


One shot advertising game equilibrium

One-Shot Advertising Game Equilibrium

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

General Mills

Kellogg’s

Nash Equilibrium


Repeating the game 2 times will not improve outcome

Repeating the game 2 times will not improve outcome

  • 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.

General Mills

*

Kellogg’s

Nash Equilibrium


Can collusion work if firms play the game each year forever

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

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

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

General Mills

Kellogg’s


Kellogg s gain to cheating

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!

General Mills

Kellogg’s


Benefits costs of cheating

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”.

General Mills

Kellogg’s


Key insight

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

Real World Examples of Collusion: Garbage Collection Industry

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

Firm 2

One-Shot Bertrand (Nash) Equilibrium

Firm 1

Firm 2

Repeated Game Equilibrium

Firm 1


Real world examples of collusion opec

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

Venezuela

One-Shot Cournot (Nash) Equilibrium

Saudi Arabia

Venezuela

Repeated Game EquilibriumAssuming a Low Interest Rate

Saudi Arabia


Opec s demise

OPEC’s Demise

Low Interest

Rates

High Interest

Rates


Simultaneous move bargaining

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

Union

Three Nash Equilibriums in Normal Form

Management


Single offer bargaining

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 move step 2 add the union s move step 3 add the payoffs

Accept

100, 500

Union

-100, -3

Reject

10

Accept

300, 300

5

Firm

Union

-100, -3

Reject

1

Accept

500, 100

Union

-100, -3

Reject

To get The Game in Extensive Form

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


Game theory inside oligopoly

Step 4: Identify Nash Equilibriums

Accept

100, 500

Union

-100, -3

Reject

10

Accept

300, 300

5

Firm

Union

-100, -3

Reject

1

Accept

500, 100

Union

-100, -3

Reject

Outcomes such that neither player has an incentive to change its strategy, given the strategy of the other


Step 5 find the subgame perfect nash equilibriums

Accept

100, 500

Union

-100, -3

Reject

10

Accept

300, 300

5

Firm

Union

-100, -3

Reject

1

500, 100

Accept

Union

-100, -3

Reject

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

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.


Game theory inside oligopoly

-1, 1

Hard

Incumbent

Enter

Soft

5, 5

Entrant

Out

0, 10

Pricing to Prevent Entry: An Application of Game Theory

Two 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

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

$

(DM – QL)

Entrant's residual

demand curve

M

P

L

P

AC

P = AC

M

D

Quantity

L

Q

M

Q

Q

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.


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