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Game Theory: Inside Oligopoly

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Game Theory:

Inside Oligopoly

- 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.

- 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)

11,10

10,11

12,12

10,15

10,13

13,14

- 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

11,10

10,11

12,12

10,15

10,13

13,14

- 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

11,10

10,11

12,12

10,15

10,13

13,14

- 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”.

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

• 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

- 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

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

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 .

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

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

Player 2

Player 1

- 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

• 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

- 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

- 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

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

- 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

- 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

- 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

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

- 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

Low Interest

Rates

High Interest

Rates

- 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

- 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

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 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

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

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”).

- 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.

-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.

- 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.

$

(DM – QL)

Entrant's residual

demand curve

M

P

L

P

AC

P = AC

M

D

Quantity

L

Q

M

Q

Q

- 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.