bus 525: managerial economics lecture 10 game theory: inside oligopoly

BUS 525: Managerial Economics

Lecture 10

Game Theory: Inside Oligopoly

OverviewOverview

I. Introduction to Game TheoryII. Simultaneous-Move, One-Shot

GamesIII. Infinitely Repeated GamesIV. Finitely Repeated GamesV. Multistage Games

10-2

Game EnvironmentsGame Environments• Players are individuals who make decisions• Players’ planned decisions are called strategies.• Payoffs to players are the profits or losses

resulting from strategies.• Order of play is important:

– Simultaneous-move game: each player makes decisions without knowledge of other players’ decisions.

– Sequential-move game: one player observes its rival’s move prior to selecting a strategy.

• Frequency of rival interaction– One-shot game: game is played once.– Repeated game: game is played more than once; either

a finite or infinite number of interactions.

10-3

Simultaneous-Move, One-Shot Simultaneous-Move, One-Shot Games: Normal Form GameGames: Normal Form Game

• A Normal Form Game consists of:– Set of players i = {1, 2, … n} where n is a

finite number.– Each players strategy set or feasible actions

consist of a finite number of strategies.• Player 1’s strategies are S1 = {a, b, c, …}. • Player 2’s strategies are S2 = {A, B, C, …}.

– Payoffs, to the players are the profits or losses that results from the strategies.

– Due to the interdependence, payoff to a player depends not only the player’s strategy but also on the strategies employed by other players

• Player 1’s payoff: π1(a,B) = ?• Player 2’s payoff: π2(b,C) = ?

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Normal Form GameNormal 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

10-5

Normal form game: A representation of a game indicating the players, their possible strategies, and the payoffs resulting from alternative strategies

In game theory, Strategy, is a decision rule that describes the actions a player will take at each decision point

A Normal Form GameA Normal Form Game

Strategy Left RightUp 10,20 15,8

Down -10,7 10,10

Player B

Pla

yer

A

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Dominant Strategy Dominant Strategy


Down -10,7 10,10

Player B

Pla

yer

A

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A strategy that results in highest payoff to a player regardless of opponents action

Principle: Check to see if you have a dominant strategy. If you have one, play it.

Dominant Strategy in a Simultaneous-Dominant Strategy in a Simultaneous-Move, One-Shot GameMove, One-Shot Game

• A dominant strategy is a strategy resulting in the highest payoff regardless of the opponent’s action.

• If “Up” is a dominant strategy for Player A in the previous game, then:– πA(Up,Left) ≥ πA(Down,Left);

– πA(Up,Right) ≥ πA(Down,Right).

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Secure Strategy Secure Strategy


Down -10,7 10,10

Player B

Pla

yer

A

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Does B have a dominant strategy?

Play secure strategy, i.e. A strategy that guarantees the highest payoff under the worst possible scenario. i.e choose Right; 8>7

Could you do better? Yes, choose left. Because A will choose Up, A’s dominant strategy; 20>8

Principle: Put yourself in rivals shoes. If your rival has a dominant strategy, anticipate that he or she will play it.

Nash Equilibrium Nash Equilibrium


Down -10,7 10,10

Player B

Pla

yer

A

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The Nash equilibrium is a condition in which no player can improve her payoff by unilaterally changing her own strategy, given the other players’ strategy

Suppose A chooses Up, B chooses Left. Either player will have no incentive to change his or her strategy. Given that A’s strategy is Up, the best player B can do is to choose Left. Given that B’s strategy is Left, the best player A can do is to choose Up.

The Nash equilibrium strategy for player A is Up, and for player B it is Left.

Nash Equilibrium

Key InsightsKey Insights

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

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A Pricing GameA Pricing Game• Two managers want to maximize

market share: i = {1,2}.• Strategies are pricing decisions

– SA = {Low Price, High Price}.

– SB = {Low Price, High Price}.

• Simultaneous moves.• One-shot game.

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A Pricing GameA Pricing Game

Strategy Low Price High PriceLow Price 0,0 50,-10High Price -10,50 10,10

Firm BF

irm

A

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•Is there Nash equilibrium?

•Is Nash equilibrium the best?

•Cheating

An Advertising gameAn Advertising game

Strategy Advertise Don't AdvertiseAdvertise $4,$4 $20,$1

Don't Advertise $1, $20 $10, $10

Firm B

Fir

m A

10-14

Dominant strategy for each firm is to advertise. Nash equilibrium for the game is to advertise

Key Insight:Key Insight:

• Game theory can also be used to analyze situations where “payoffs” are non monetary!

• We will, without loss of generality, focus on environments where businesses want to maximize profits.– Hence, payoffs are measured in

monetary units.

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Coordination GamesCoordination Games• In many games, players have

competing objectives: One firm gains at the expense of its rivals.

• However, some games result in higher profits by each firm when they “coordinate” decisions.

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A Coordination GameA Coordination Game

Strategy 220-Volt Outlets 110-Volt Outlets

220-Volt Outlets $100,$100 $0,$0

110-Volt Outlets $0,$0 $100,$100

Firm B

Fir

m A

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The game has two Nash equilibria. Better outcome by coordination. If firms produce different outlets, lower demand as consumers needs to spend more money in wiring their houses

Key Insights:Key Insights:

• Not all games are games of conflict.

• Communication can help solve coordination problems.

• Sequential moves can help solve coordination problems.

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Games With No Pure Strategy Games With No Pure Strategy Nash EquilibriumNash Equilibrium

Strategy Work Shirk

Monitor -1,1 1,-1

Don’t,Monitor 1,-1 -1,1

WorkerM

anag

er

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Suppose Managers strategy is to monitor the worker. Then the best choice for the worker is to work. But if the worker works, the best strategy for the manager not to monitor. Therefore, monitoring is not a Nash equilibrium strategy. Not monitoring is also not a Nash equilibrium strategy. Because, if manager’s strategy is “don’t monitor;” The worker would choose shirking. Manager should monitor.

Strategies for Games With No Strategies for Games With No Pure Strategy Nash EquilibriumPure Strategy Nash Equilibrium

• In games where no pure strategy Nash equilibrium exists, players find it in there interest to engage in mixed (randomized) strategies.– This means players will “randomly”

select strategies from all available strategies.

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A Nash Bargaining GameA Nash Bargaining Game

Strategy 0 50 1000 0,0 0,50 0,10050 50,0 50,50 -1,-1100 100,0 -1,-1 -1,-1

Union

Man

agem

ent

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How much of a $100 surplus is to be given to the union?. The parties write the amount (either 0,50 or100). If the sum exceeds 100, the bargaining ends in a stalemate. Cost of delay -$1 both for the union and the management. Three Nash equilibrium to this bargaining game(100,0), (50,50), and (0,100).

Six of the nine potential payoffs are inefficient as they result in total payoff lesser than amount to be divided. Three negative payoffs. Settle for 50-50 split which is fair, look at history

Lessons in Lessons in Simultaneous BargainingSimultaneous Bargaining

• Simultaneous-move bargaining results in a coordination problem.

• 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.

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Infinitely Repeated GamesInfinitely Repeated Games

• Infinitely Repeated Games• A game that is played over and over again forever and in

which players receive payoffs during each play of the game

• Recall key aspects of present value analysis– Value of the firm for T period with different payoffs at

different periods– Value of the firm for ∞ period with same payoffs at

different periods

• Trigger strategy– A strategy that is contingent on the past play of a game and

in which some particular past action “triggers” a different action by a player

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A Pricing Game that is RepeatedA Pricing Game that is Repeated

Strategy Low Price High PriceLow Price 0,0 50,-40High Price -40,50 10,10

Firm BF

irm

A

10-24

Nash equilibrium in one shot play of this game is for each firm to charge low prices and earn zero profits

Impact of repeated play on the equilibrium outcome of the game

Trigger strategy: ”Cooperate provided no player has ever cheated in the past. If any players cheats, punish the player by choosing one shot Nash equilibrium strategy forever after.”

A Pricing Game that is RepeatedA Pricing Game that is Repeated• The benefit of cheating today on the collusive agreement is earning $50

instead of $10 today. The cost of cheating today is earning $0 instead of $10 in each future period. If the present value of cost of cheating exceeds one time benefit of cheating, it does not pay for a firm to cheat, and high prices can be sustained.

• In the example, if the interest rate is less than 25 percent both firms A and B will loose more (in present value) from cheating. That’s why we see collusion in oligopoly.

• Condition for sustaining cooperative outcome with trigger strategies (ΠCheat- πCoop)/(πCoop- πN)≤ 1/i or ΠCheat- πCoop ≤ 1/i(πCoop- πN)• The LHS represents the one time gain from cheating. The RHS represents the present value of what is given up in the future by cheating

today. In one-shot games there is no tomorrow, and threats have no bite.When interest rate is low firms may find it in their interest to collude and charge high prices

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Factor Affecting Collusion in Pricing Factor Affecting Collusion in Pricing GamesGames

10-26

To sustain collusive arrangements via punishment strategies firms must know: (i) who the rivals are; (ii) who the rivals customers are; (iii) When the rivals deviate from collusive arrangement; and (iv) must be able to successfully punish rivals for cheating.

These factors are related to:

(1) Number of firms; (2) Firm size; (3) History of the market; (4) Punishment mechanism

Finitely Repeated GamesFinitely Repeated Games

• Finitely Repeated Games– Games in which players do not know when the game will

end– Games in which players know when the game will end

• Games with an uncertain final period– When the game is repeated a finite but uncertain times, the

benefits of cooperating look exactly the same as benefits from cooperating in an infinitely repeated game except that 1-θ plays the role of 1/(1+i); PLAYERS DISCOUNT THEIR FUTURE NOT BECAUSE OF THE INTEREST RATE BUT BECAUSE THEY ARE NOT CERTAIN FUTURE PLAYS WILL OCCUR

• In our example Firm A has no incentive to cheat if

ΠFirm ACheat=50 ≤10/θ = ΠFirm A

Coperate, which is true if θ ≤1/5, probability the

game will end is less than 20 percent, If θ=1, one shot game, dominant strategy for both firm is to charge the low price

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Repeated Games with a Known Repeated Games with a Known Final periodFinal period

• Provided the players know when the game will end, the game has only one Nash equilibrium.– Same outcome as one shot game

• Trigger strategies would not work– Rival cannot be punished in the last period– Trigger strategies wont work

• Backward unraveling.• End of period problem

– Workers work hard due to threat of firing– Announces plans to quit, the cost of shirking is reduced– Manager’s response, fire immediately after announcements– Worker responds by telling you at the very end– What to do? Incentives, gratuity Same outcome as one shot game

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Multistage GamesMultistage Games

• Multistage framework permits players to make sequential rather than simultaneous decisions

• Differs from one-shot and infinitely repeated games • Extensive-form game A representation of a game that summarizes the players, the information available to them at each stage, the strategies available to them, the sequence of moves, and the payoff resulting from alternative strategies

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Pricing to Prevent Entry: An Pricing to Prevent Entry: An Application of Game TheoryApplication of Game Theory

• Two firms: an incumbent and potential entrant.

• Potential entrant’s strategies: – Enter.– Stay Out.

• Incumbent’s strategies: – {if enter, play hard}.– {if enter, play soft}.– {if stay out, play hard}.– {if stay out, play soft}.

• Move Sequence: – Entrant moves first. Incumbent observes entrant’s

action and selects an action.

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The Pricing to Prevent EntryThe Pricing to Prevent Entry Game in Extensive FormGame in Extensive Form

A(Entrant)

Out

Enter

B (Incumbent)

Price war

Share

-1, 1

5, 5

0, 10

10-31

Identify Nash and Identify Nash and Subgame Perfect Subgame Perfect

EquilibriaEquilibria

A (Entrant)

Out

Enter

B (Incumbent)

-1, 1

5, 5

0, 10

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Price war

Share

Two Nash EquilibriaTwo Nash Equilibria

A (Entrant)

Out

Enter

B (Incumbent)

-1, 1

5, 5

0, 10

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Share

Price war

One Subgame Perfect One Subgame Perfect EquilibriumEquilibrium

A (Entrant)

Out

Enter

B (Incumbent)

-1, 1

5, 5

0, 10

Subgame Perfect Equilibrium Strategy:{enter; If enter, Share market}

10-34

Share

Price war

InsightsInsights• 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.

10-35

An Innovation Game: An An Innovation Game: An Application of Game TheoryApplication of Game Theory

• Class Exercise• Two firms: Innovator and Rival• Innovator’s strategies:

– Introduce.– Don’t introduce.

• Rival’s strategies: – {if introduced, clone}.– {if introduced, don’t clone}.

• Move Sequence: – Innovator moves first. Rival observes innovator’s

action and selects an action.

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Class ExerciseClass Exercise• If you do not introduce the new product, you and your rival

will earn $1 million each• If you introduce the new product

– Rival clones : you will loose $ 5 million and your rival will earn $20million

– Rival does not clone: you will make $100 million and your rival will make $0 million

• Set up the extensive form of the game• Should you introduce the new product?• How would your answer change if your rival has promised

not to clone the product?• What would you do if patent law prevented your rival from

cloning the product?

Don’tIntroduce

Introduce

Clone

Don’t clone

-5,20

100, 0

1, 1

10-38

A (Innovator)

B (Rival)

An Innovation Game: An An Innovation Game: An Application of Game TheoryApplication of Game Theory

Simultaneous-Move Simultaneous-Move BargainingBargaining

• 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.

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Fairness: The “Natural” Fairness: The “Natural” Focal PointFocal Point

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

10-40

Single Offer BargainingSingle 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. – Sequence of moves.– Each player’s payoff.

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Firm

10

5

1

Step 1: Management’s Step 1: Management’s MoveMove

10-42

Firm

10

5

1

Union

Union

Union

Accept

Reject

Accept

Accept

Reject

Reject

Step 2: Add the Union’s MoveStep 2: Add the Union’s Move10-43

Firm

10

5

1

Union

Union

Union

Accept

Reject

Accept

Accept

Reject

Reject

Step 3: Add the PayoffsStep 3: Add the Payoffs

100, 500

-100, -3

300, 300

-100, -3

500, 100

-100, -3

10-44

Firm

10

5

1

Union

Union

Union

Accept

Reject

Accept

Accept

Reject

Reject

The Game in Extensive FormThe Game in Extensive Form

100, 500

-100, -3

300, 300

-100, -3

500, 100

-100, -3

10-45

Step 4: Identify the Firm’s Step 4: Identify the Firm’s Feasible StrategiesFeasible Strategies

• Management has one information set and thus three feasible strategies:– Offer $10.– Offer $5.– Offer $1.

10-46

Step 5: Identify the Step 5: Identify the Union’s Feasible Union’s Feasible

StrategiesStrategies• The Union has three information set and

thus eight feasible strategies:– Accept $10, Accept $5, Accept $1– Accept $10, Accept $5, Reject $1– Accept $10, Reject $5, Accept $1– Accept $10, Reject $5, Reject $1 – Reject $10, Accept $5, Accept $1– Reject $10, Accept $5, Reject $1– Reject $10, Reject $5, Accept $1– Reject $10, Reject $5, Reject $1

10-47

Step 6: Identify Nash Step 6: Identify Nash Equilibrium OutcomesEquilibrium Outcomes

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

10-48

Finding Nash Finding Nash Equilibrium OutcomesEquilibrium Outcomes

Accept $10, Accept $5, Accept $1Accept $10, Accept $5, Reject $1Accept $10, Reject $5, Accept $1Reject $10, Accept $5, Accept $1Accept $10, Reject $5, Reject $1Reject $10, Accept $5, Reject $1Reject $10, Reject $5, Accept $1Reject $10, Reject $5, Reject $1

Union's Strategy Firm's Best Response

Mutual Best Response?

$1 Yes$5$1$1$10

YesYesYesYes

$5 Yes$1

NoYes

$10, $5, $1

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Step 7: Find the Subgame Step 7: Find the Subgame Perfect Nash Equilibrium Perfect Nash Equilibrium

OutcomesOutcomes• Outcomes where no player has an

incentive to change its strategy, given the strategy of the rival, and

• The outcomes are based on “credible actions;” that is, they are not the result of “empty threats” by the rival.

10-50

Checking for Credible Checking for Credible ActionsActions


Union's StrategyAre all Actions

Credible?

YesNoNoNoNoNoNoNo

10-51

The “Credible” Union The “Credible” Union StrategyStrategy


Union's StrategyAre all Actions

Credible?

YesNoNoNoNoNoNoNo

10-52

Finding Subgame Perfect Finding Subgame Perfect Nash Equilibrium StrategiesNash Equilibrium Strategies


Union's Strategy Firm's Best Response

Mutual Best Response?

$1 Yes$5$1$1$10

YesYesYesYes

$5 Yes$1

NoYes

$10, $5, $1

Nash and Credible Nash Only Neither Nash Nor Credible

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To Summarize:To Summarize:

• We have identified many combinations of Nash equilibrium strategies.

• In all but one the union does something that isn’t in its self interest (and thus entail threats that are not credible).

• Graphically:

10-54

Firm

10

5

1

Union

Union

Union

Accept

Reject

Accept

Accept

Reject

Reject

There are 3 Nash There are 3 Nash Equilibrium Outcomes!Equilibrium Outcomes!

100, 500

-100, -3

300, 300

-100, -3

500, 100

-100, -3

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Firm

10

5

1

Union

Union

Union

Accept

Reject

Accept

Accept

Reject

Reject

Only 1 Subgame-Perfect Nash Only 1 Subgame-Perfect Nash Equilibrium Outcome!Equilibrium Outcome!

100, 500

-100, -3

300, 300

-100, -3

500, 100

-100, -3

10-56

Bargaining Re-CapBargaining 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; real

world evidence suggests that people sometimes reject offers on the the basis of “principle” instead of cash considerations.

10-57

An Advertising GameAn 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.

10-58

A One-Shot Advertising A One-Shot Advertising GameGame

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

’s10-59

Equilibrium to the One-Equilibrium to the One-Shot Advertising GameShot Advertising Game


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

General Mills

Kel

logg

’s

Nash Equilibrium

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Can collusion work if the Can collusion work if the game is repeated 2 times?game is repeated 2 times?


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

General Mills

Kel

logg

’s

10-61

No (by backwards No (by backwards induction).induction).

• In period 2, the game is a one-shot game, so equilibrium entails High Advertising in the last period.

• This means 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.

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

• 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.

10-63

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

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

= 12 + 12/i


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

10-64

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!


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

General Mills

Kel

logg

’s

10-65

Benefits & Costs of Benefits & Costs of CheatingCheating

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 > 0– Pays to “cheat”.

• If Immediate Benefit - PV of Future Cost 0– Doesn’t pay to “cheat”.


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

General Mills

Kel

logg

’s

10-66

Key InsightKey Insight

• Collusion can be sustained as a Nash equilibrium when there is no certain “end” to a game.

• Doing so requires:– Ability to monitor actions of rivals.– Ability (and reputation for) punishing

defectors.– Low interest rate.– High probability of future interaction.

10-67

bus 525: managerial economics lecture 10 game theory: inside oligopoly

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