Chapter 13

Chapter 13

Game Theory and Competitive Strategy

Chapter 13 2©2005 Pearson Education, Inc.

Topics to be Discussed

Gaming and Strategic Decisions

Dominant Strategies

The Nash Equilibrium Revisited

Repeated Games


Topics to be Discussed

Sequential Games

Threats, Commitments, and Credibility

Entry Deterrence

Bargaining Strategy

Auctions



Game is any situation in which players (the participants) make strategic decisions Ex: firms competing with each other by

setting prices, group of consumers bidding against each other in an auction

Strategic decisions result in payoffs to the players: outcomes that generate rewards or benefits



Game theory tries to determine optimal strategy for each player

Strategy is a rule or plan of action for playing the game

Optimal strategy for a player is one that maximizes the expected payoff

We consider players who are rational – they think through their actions



“If I believe that my competitors are rational and act to maximize their own profits, how should I take their behavior into account when making my own profit-maximizing decisions?”(Text, p. 474)


Noncooperative vs. Cooperative Games

Cooperative Game Players negotiate binding contracts that allow

them to plan joint strategiesExample: Buyer and seller negotiating the price

of a good or service or a joint venture by two firms (i.e., Microsoft and Apple)

Binding contracts are possible



Noncooperative Game Negotiation and enforcement of binding

contracts between players is not possibleExample: Two competing firms, assuming the

other’s behavior, independently determine pricing and advertising strategy to gain market share

Binding contracts are not possible



“The strategy design is based on understanding your opponent’s point of view, and (assuming your opponent is rational) deducing how he or she is likely to respond to your actions.” (Text, p. 475)



An Example: How to buy a dollar bill1. Auction a dollar bill

2. Highest bidder receives the dollar in return for the amount bid

3. Second highest bidder must pay the amount he or she bid but gets nothing in return

4. How much would you bid for a dollar?

Typically bid more for the dollar when faced with loss as second highest bidder


Acquiring a Company

Scenario Company A: The Acquirer Company T: The Target A will offer cash for all of T’s shares

The value and viability of T depends on the outcome of a current oil exploration project


Acquiring a Company

Project failure: T’s value = $0Project success: T’s value = $100/shareAll outcomes in between equally likelyT’s value will be 50% greater with A’s

management


Acquiring a Company

Scenario A must submit the proposal before the

exploration outcome is known T will not choose to accept or reject until after

the outcome is known only to T Company T will accept any offer that is

greater than the per share value of the company under current management

How much should A offer?


Dominant Strategies

Dominant Strategy is one that is optimal no matter what an opponent does An Example

A and B sell competing productsThey are deciding whether to undertake

advertising campaigns


Payoff Matrix for Advertising Game

Fir

m A

Advertise

Don’tAdvertise

Advertise

Don’tAdvertise

Firm B

10, 5 15, 0

10, 26, 8



Observations A: regardless of B,

advertising is the best B: regardless of A,

advertising is best

Firm A

AdvertiseDon’t

Advertise

Advertise

Don’tAdvertise

Firm B

10, 5 15, 0

10, 26, 8



Observations Dominant strategy for

A and B is to advertise

Do not worry about the other player

Equilibrium in dominant strategy

Firm A

AdvertiseDon’t

Advertise

Advertise

Don’tAdvertise

Firm B

10, 5 15, 0

10, 26, 8


Dominant Strategies

Equilibrium in dominant strategies Outcome of a game in which each firm is

doing the best it can regardless of what its competitors are doing

Optimal strategy is determined without worrying about the actions of other players

However, not every game has a dominant strategy for each player


Dominant Strategies

Game Without Dominant Strategy The optimal decision of a player without a

dominant strategy will depend on what the other player does

Revising the payoff matrix, we can see a situation where no dominant strategy exists


10, 5 15, 0

20, 26, 8

Fir

m A

Advertise

Don’tAdvertise

Advertise

Don’tAdvertise

Firm B

Modified Advertising Game


10, 5 15, 0

20, 26, 8

Firm A

AdvertiseDon’t

Advertise

Advertise

Don’tAdvertise

Firm B

Modified Advertising Game

Observations A: No dominant

strategy; depends on B’s actions

B: Dominant strategy is to advertise

Firm A determines B’s dominant strategy and makes its decision accordingly



A dominant strategy is stable, but in many games one or more party does not have a dominant strategy

A more general equilibrium concept is the Nash Equilibrium introduced in Chapter 12 A set of strategies (or actions) such that each

player is doing the best it can given the actions of its opponents



None of the players have incentive to deviate from its Nash strategy, therefore it is stable In the Cournot model, each firm sets its own

price assuming the other firm’s outputs are fixed. Cournot equilibrium is a Nash Equilibrium.



Dominant Strategy “I’m doing the best I can no matter what you

do. You’re doing the best you can no matter what I do.”

Nash Equilibrium “I’m doing the best I can given what you are

doing. You’re doing the best you can given what I am doing.”

Dominant strategy is a special case of Nash equilibrium



Two cereal companies face a market in which two new types of cereal can be successfully introduced, provided each type is introduced by only one firm

Product Choice Problem Market for one producer of crispy cereal Market for one producer of sweet cereal Each firm only has the resources to introduce one

cereal Noncooperative


Product Choice ProblemF

irm

1

Crispy Sweet

Crispy

Sweet

Firm 2

-5, -5 10, 10

-5, -510, 10


Product Choice Problem

Firm 1

Crispy Sweet

Crispy

Sweet

Firm 2

-5, -5 10, 10

-5, -510, 10

If Firm 1 hears Firm 2 is introducing a new sweet cereal, its best action is to make crispy

Bottom left corner is Nash equilibrium

What is other Nash Equilibrium?


Beach Location Game

Scenario Two competitors, Y and C, selling soft drinks Beach is 200 yards long Sunbathers are spread evenly along the

beach Price Y = Price C Customer will buy from the closest vendor


Beach Location Game

Where will the competitors locate (i.e., where is the Nash equilibrium)?

Will want to all locate in center of beach Similar to groups of gas stations, car

dealerships, etc.

Ocean

0 B Beach A 200 yards

C



Maximin Strategies - Scenario Two firms compete selling file encryption

software They both use the same encryption standard

(files encrypted by one software can be read by the other - advantage to consumers)

Firm 1 has a much larger market share than Firm 2

Both are considering investing in a new encryption standard


Maximin StrategyF

irm

1

Don’t invest InvestFirm 2

0, 0 -10, 10

20, 10-100, 0

Don’t invest

Invest


Maximin Strategy

Firm 1


0, 0 -10, 10

20, 10-100, 0

Don’t invest

Invest

Observations Dominant strategy

Firm 2: Invest Firm 1 should

expect Firm 2 to invest

Nash equilibriumFirm 1: investFirm 2: Invest

This assumes Firm 2 understands the game and is rational


Maximin Strategy

Firm 1


0, 0 -10, 10

20, 10-100, 0

Don’t invest

Invest

Observations If Firm 2 does not

invest, Firm 1 incurs significant losses

Firm 1 might play don’t invest

Minimize losses to 10 – maximin strategy


Maximin Strategy

If both are rational and informed Both firms invest Nash equilibrium

If Player 2 is not rational or completely informed Firm 1’s maximin strategy is to not invest Firm 2’s maximin strategy is to invest If 1 knows 2 is using a maximin strategy, 1

would invest


Maximin Strategy

If Firm 1 is unsure about what Firm 2 will do, it can assign probabilities to each possible action Could use a strategy that maximizes its

expected payoff Firm 1’s strategy depends critically on its

assessment of probabilities for Firm 2


Prisoners’ DilemmaP

riso

ner

A

Confess Don’t Confess

Confess

Don’tConfess

Prisoner B

-5, -5 -1, -10

-2, -2-10, -1


Prisoners’ Dilemma

Prisoner A

Confess Don’t Confess

Confess

Don’tConfess

Prisoner B

-5, -5 -1, -10

-2, -2-10, -1

What is the: Dominant strategy Nash equilibrium Maximin solution

Dominant strategies are also maximin strategies

Both confess is both Nash equilibrium and maximin solution


Mixed Strategy

Pure Strategy Player makes a specific choice or takes a

specific action

Mixed Strategy Player makes a random choice among two or

more possible actions, based on a set of chosen probabilities


Matching PenniesP

lay

er

A

Heads Tails

Heads

Tails

Player B

1, -1 -1, 1

1, -1-1, 1


Matching Pennies

Pure strategy: No Nash equilibrium No combination of

head and tails leaves both players better off

Mixed strategy: Random choice is a Nash equilibrium

Player A

Heads Tails

Heads

Tails

Player B

1, -1 -1, 1

1, -1-1, 1


Matching Pennies

Player A might flip coin playing heads with ½ probability and tails with ½ probability

If both players follow this strategy, there is a Nash equilibrium – both players will be doing the best they can given what their opponent is doing

Although the outcome is random, the expected payoff is 0 for each player


Mixed Strategy

One reason to consider mixed strategies is when there is a game that does not have any Nash equilibriums in pure strategy

When allowing for mixed strategies, every game has a Nash equilibrium

Mixed strategies are popular for games like poker

A firm might not find it reasonable


The Battle of the Sexes

Jim

Wrestling Opera

Wrestling

Opera

Joan

2,1 0,0

1,20,0


The Battle of the Sexes

Pure Strategy Both watch wrestling Both watch opera

Mixed Strategy Jim chooses

wrestling Joan chooses

wrestling

Jim

Wrestling Opera

Wrestling

Opera

Joan

2,1 0,0

1,20,0


Repeated Games

Game in which actions are taken and payoffs are received over and over again

Oligopolistic firms play a repeated gameWith each repetition of the Prisoners’

Dilemma, firms can develop reputations about their behavior and study the behavior of their competitors


Pricing Problem

Firm 1

Low Price High Price

Low Price

High Price

Firm 2

10, 10 100, -50

50, 50-50, 100


Pricing Problem

How does a firm find a strategy that would work best on average against all or almost all other strategies?

Tit-for-tat strategy Repeated game strategy in which a player

responds in kind to an opponent’s previous play, cooperating with cooperative opponents and retaliating against uncooperative ones


Tit-for-Tat Strategy

What if the game is infinitely repeated? Competitors repeatedly set price every

month, forever Tit-for-tat strategy is rational

If competitor charges low price and undercuts firm

Will get high profits that month but know I will lower price next month

Both of us will get lower profits if keep undercutting, so not rational to undercut



What if repeated a finite number of times? If both firms are rational, they will charge high prices

until the last month After the last month, there is no retaliation possible But in the month before last month, knowing that will

charge low price in last month, will charge low price in month before

Keep going and see that only rational outcome is for both firms to charge low price every month



If firms don’t believe their competitors are rational or think perhaps they aren’t, cooperative behavior is a good strategy

Most managers don’t know how long they will be competing with their rivals

In a repeated game, prisoner’s dilemma can have cooperative outcome


Repeated Games

Conclusion Cooperation is difficult at best since these

factors may change in the long run Need a small number of firms Need stable demand and cost conditions

This could lead to price wars if don’t have them


Oligopolistic Cooperationin the Water Meter Industry

Characteristics of the Market Four producers of water meters

Rockwell InternationalBadger MeterNeptune Water Meter CompanyHersey ProductsRockwell has about 35% of market shareBadger, Neptune, and Hersey combined have

about a 50 to 55% share



Most buyers are municipal water utilitiesVery inelastic demand

Not a significant part of the budget for providing water

Demand is stable Demand grows steadily with population

Utilities have long-standing relationships with suppliers Reluctant to switch



Significant economies of scaleBoth long term relationship and

economies of scale represent barriers to entry Hard for new firms to enter market

If firms were to cooperate, could earn significant monopoly profits

If compete aggressively to gain market share, profits will fall to competitive levels



This is a Prisoners’ Dilemma – what should the firms do? Lower price to a competitive level Cooperate

Companies have been playing repeated game for decades

Cooperation has prevailed given market characteristics


Sequential Games

Players move in turn, responding to each other’s actions and reactions Ex: Stackelberg model (ch. 12) Responding to a competitor’s ad campaign Entry decisions Responding to regulatory policy


Sequential Games

Going back to the product choice problem Two new (sweet, crispy) cereals Successful only if each firm produces one

cereal Sweet will sell better Both still profitable with only one producer


Modified Product Choice Problem

If firms both announce their decisions independently and simultaneously, they will both pick sweet cereal and both will lose money

What if Firm 1 sped up production and introduced new cereal first? Now there is a sequential game Firm 1 will think about what Firm 2 will do


Modified Product Choice Problem

Fir

m 1

Crispy Sweet

Crispy

Sweet

Firm 2

-5, -5 10, 20

-5, -520, 10


Extensive Form of a Game

Extensive Form of a Game Representation of possible moves in a

game in the form of a decision tree Allows one to work backward from the best

outcome for Firm 1


Product Choice Game in Extensive Form

Crispy

Sweet

Crispy

Sweet

-5, -5

10, 20

20, 10

-5, -5

Firm 1

Crispy

Sweet

Firm 2

Firm 2


Sequential Games

The Advantage of Moving First In this product-choice game, there is a clear

advantage to moving first The first firm can choose a large level of

output, thereby forcing second firm to choose a small level

Can show the firm’s mover advantage by revising the Stackelberg model and comparing to Cournot


First Mover Advantage

Assume: Duopoly

Firm and

Production Total

/1001010

:

0

30

21

21

PQQ

Cournot

MC

QQ Q

QP


First Mover Advantage

Duopoly

25.56 50.112

50.7 and 5.7 15

rg)(StackelbeFirst Moves Firm

Firm/50.112 15 and 5.7

CollusionWith

21

21

21

PQQ

PQQ


Choosing OutputF

irm

1

7.5

Firm 2

112.50, 112.50 56.25, 112.50

0, 0112.50, 56.25

125, 93.75 50, 75

93.75, 125

75, 50

100, 100

10 15

7.5

10

15


Choosing Output

This payoff matrix illustrates various outcomes Move together, both

produce 10 If Firm 1 moves first

(Q=15), best Firm 2 can do is 7.5

Firm 1

7.5

Firm 2

112.50, 112.50 56.25, 112.50

0, 0112.50, 56.25

125, 93.75 50, 75

93.75, 125

75, 50

100, 100

10 15

7.5

10

15



Strategic Moves What actions can a firm take to gain

advantage in the marketplace?Deter entryInduce competitors to reduce output, leave,

raise priceImplicit agreements that benefit one firm



Strategic Move Action that gives a player an advantage by

constraining his behavior Firm 1 must constrain his behavior to the

extent Firm 2 is convinced that he is committed



How to Make the First Move Demonstrate Commitment Firm 1 must do more than announce they will

produce sweet cerealInvest in expensive advertising campaignBuy large order of sugar and send invoice to

Firm 2 Commitment must be enough to induce Firm

2 to make the decision Firm 1 wants it to make



Empty Threats If a firm will be worse off if it charges a low

price, the threat of a low price is not credible in the eyes of the competitors

When firms know the payoffs of each other’s actions, firms cannot make threats the other firm knows they will not follow

In our example, Firm 1 will always charge high price and Firm 2 knows it


Pricing of Computers and Word Processors

Firm 1

High Price Low Price

High Price

Low Price

Firm 2

100, 80 80, 100

10, 2020, 0



Sometimes firms can make credible threats

Scenario Race Car Motors, Inc. (RCM) produces cars Far Out Engines (FOE) produces specialty

car engines and sells most of them to RCM Sequential game with RCM as the leader FOE has no power to threaten to build big

engines since RCM controls output


Production Choice Problem

Far Out Engines

Small cars Big cars

Small engines

Big engines

Race Car Motors

3, 6 3, 0

8, 31, 1



RCM does best by producing small carsKnows that Far Out will then produce

small enginesFar Out prefers to make big enginesCan Far Out induce Race Car to produce

big cars instead?



Suppose Far Out threatens to produce big engines no matter what RCM does? Not credible since once RCM announces

they are producing small cars, FOE will not have incentive to carry out threat

Can make threat credible by altering pay off matrix by constraining its own choices

Shutting down or destroying some small engine production capacity


Modified Production Choice Problem

0, 6 0, 0

8, 31, 1

Far Out Engines

Small cars Big cars

Small engines

Big engines

Race Car Motors


Modified Production Choice Problem

Strategic commitments can be effective but not without risk Rely heavily on accurate knowledge of payoff

matrix and industry May have competitors out there that they

don’t know about and lose sales


Role of Reputation

If Far Out gets the reputation of being irrational They threaten to produce large engines no

matter what Race Car doesThreat might be credible because

irrational people don’t always make profit maximizing decisions

A party thought to be crazy can lead to a significant advantage


Bargaining Strategy

Bargaining situation can depend on ability to affect relative bargaining position

Consider two firms introducing one of two complementary goods: Firm 1 has cost advantage in Good A Firm 2 has cost advantage in Good B


Bargaining Strategy

Firm 1

Produce A Produce B

Produce A

Produce B

Firm 2

40, 5 50, 50

5, 4560, 40


Bargaining Strategy

With collusion: Firm 1 Produces A

and Firm 2 produces B (50,50)

Without collusion: Firm 1 produces A

and Firm 2 produces B (50,50)

Nash equilibrium

Firm 1

Produce A Produce B

Produce A

Produce B

Firm 2

40, 5 50, 50

5, 4560, 40


Bargaining Strategy

Suppose each firm is also bargaining on the decision to join in a research consortium with a third firm

Dominant strategy is for both firms to enter consortium


Bargaining Strategy

Firm 1

Work alone Enter consortium

Work alone

Enterconsortium

Firm 2

10, 10 10, 20

40, 4020, 10


Bargaining Strategy

Linking the Bargaining Problem Firm 1 announces it will join the consortium

only if Firm 2 agrees to produce A and Firm 1 will produce B

Firm 2’s best interest is to produce A with Firm 1 producing B

Firm 1’s profit increases from 50 to 60


Bargaining Strategy

Strategic moves can be used in bargaining

Combining issues in bargaining can benefit one side at other’s expense


Wal-Mart Stores’ Preemptive Investment Strategy

How did Wal-Mart become the largest retailer in the U.S. when many established retail chains were closing their doors? Gained monopoly power by opening in small

towns with no threat of other discount competition

Preemptive game with Nash equilibrium


The Discount Store Preemption Game

Wal-Mart

Enter Don’t enter

Enter

Don’t enter

Company X

-10, -10 20, 0

0, 00, 20


The Discount Store Preemption Game

Two Nash equilibrium Low left Upper right

Must be preemptive to win

Wal-Mart

Enter Don’t enter

Enter

Don’t enter

Company X

-10, -10 20, 0

0, 00, 20


Entry Deterrence

Barriers to entry important for monopoly power Economies of scale, patents and licenses,

access to critical inputs Firms can also deter entry

To deter entry, the incumbent firm must convince any potential competitor that entry will be unprofitable


Entry Possibilities

Incumbent (I)

Enter Stay out

High price(accommodation)

Low Price(warfare)

Potential Entrant (X) ($80 fixed costs)

100, 20 200, 0

130, 070, -10


Entry Deterrence

Scenario If X does not enter, I makes a profit of $200

million If X enters and charges a high price, I earns

a profit of $100 million and X earns $20 million

If X enters and charges a low price, I earns a profit of $70 million and X earns $-10 million


Entry Deterrence

Could threaten X with warfare if enters market Not credible because once X has entered, it

is in your best interest to accommodate and maintain high price


Entry Deterrence

What if firm I makes an investment before entry to increase capacity? Irrevocable commitment

Gives new payoff matrix since profits will be reduced by investment

Threat is completely credibleRational for Firm X to stay out of market


Entry Deterrence

Incumbent (I)

Enter Stay out

High price(accommodation)

Low price(warfare)

Potential Entrant (X)

50, 20 150, 0

130, 070, -10


Entry Deterrence

If incumbent has reputation of price cutting competitors even at loss, then threat will be credible

Short run losses may be offset by long run gains as monopolist


Entry Deterrence

Production of commercial airlines exhibit significant economies of scale

Airbus and Boeing considering new aircraft

Suppose it is not economical for both firms to produce the new aircraft


Development of a New Aircraft

Boeing

Produce Don’t produce

Airbus

-10, -10 100, 0

0, 00, 120

Produce

Don’t produce



Boeing has head start

Boeing will produceAirbus will not

produce

Boeing


Airbus

-10, -10 100, 0

0, 00, 120

Produce

Don’t produce



Governments can change outcome of game

European government agrees to subsidize Airbus before Boeing decides to produce

With Airbus being subsidized, the payoff matrix for the two firms would differ significantly


Development of an AircraftAfter European Subsidy

Boeing


Airbus

-10, 10 100, 0

0, 00, 120

Produce

Don’t produce


Boeing


Airbus

-10, 10 100, 0

0, 00, 120

Produce

Don’t produce

Development of an AircraftAfter European Subsidy

Airbus will produceBoeing will not

produce


Diaper Wars

Even though there are only two major firms, competition is intense

The competition occurs mostly in the form of cost-reducing innovation

Small cost savings can lead to capturing of market share

Both firms spend significantly on R&D


Competing Through R & D

P&G

R&D No R&D

R&D

No R&D

Kimberly-Clark

40, 20 80, -20

60, 40-20, 60


Competing Through R & D

Both spend on R&D Dominant strategy

Why not cooperate?Strengthening

Bargaining Power Credibility Reducing flexibility

P&G

R&D No R&D

R&D

No R&D

Kimberly-Clark

40, 20 80, -20

60, 40-20, 60


Auctions

Markets in which products are bought and sold through formal bidding processes Encourages competition that increases

seller’s revenue Low cost of transactions Useful for unique items or those with

fluctuating valueTokyo fish market


Auction Formats

1. Traditional English (oral) Seller actively solicits progressively higher

bids from a group of potential buyers Buyers are always aware of highest bid Stops when no one passes highest bid


Auction Formats

2. Dutch auction Seller begins by offering item at relatively

high price, then reduces it by fixed amounts until item is sold

First buyer accepting offered price can buy item at that price


Auction Formats

3. Sealed-bid All bids are made simultaneously in sealed

envelopes, where winning bid is the one who submitted highest bid

A. First price Sales price equals highest bid

B. Second price Sales price equals second highest bid


Valuation and Information

How to choose an auction format1. Private-value auction – bidder knows

individual valuations of object, but valuations differ from bidder to bidder Signed baseball

2. Common-value auction: bidders uncertain what the value is Offshore oil reserve


Price-Value Auctions

Each bidder must choose bidding strategy

Payoff for winning is reservation price minus price paid

Payoff for losing is zero


Private Value Auction

English oral auction and second–price sealed bid auctions Bidding truthfully is dominant strategy Pay based on value of second highest bidder

so no incentive not to bid reservation price Risk to bidding higher than reservation price


Private Value Auctions

English auction Continue bidding until second person is

unwilling to make bid

Sealed-bid auction Winning bid approximately equal to the

second highest bidder’s reservation price

Both yield the same revenue


Common Value Auctions

Winner’s Curse The winner is worse off because they

overestimated the value of the item and thereby overbid

Must reduce bid by amount equal to the expected error of the winning bidder

If a lot of variation in other bidders, then estimates are fairly imprecise


Maximizing Auction Revenue

1. Private Value Auction Encourages many bidders to increase

expected bid of winner

2. Common Value Auction Uses open rather than sealed bid

Generates greater revenue Reveals information about true value,

reducing concern of winner’s curse


Maximizing Auction Revenue

3. Private value auction Sets min bid equal to or higher than value

to you of keeping good for future sale Protects against loss if bidders are unaware

of value Increases size of bids by letting bidders

think item is valuable No sale could make bidders think item is

low quality


Bidding and Collusion

Buyers can allow benefit from collusion Can be done legally through buying groups Can be done illegally through collusive

agreements that violate antitrust laws Collusion is not easy because of large

incentive to cheat Repeated auctions allow for penalizing

participants that break agreement


Bidding and Collusion

Examples

1. Collusion among baseball owners to limit their bidding for free agent players in the 1980’s

2. Two of the world’s most successful auction houses were found guilty of agreeing to fix prices of commissions

Sotheby’s and Christie’s


Internet Auctions

Popularity of auctions has skyrocketed with growth of internet

Most popular site is eBay Dominates online person-person auction

industry

Subject to large network externalities Choose auction site with largest number of

potential bidders


Internet Auctions

eBay auctions are somewhat different from types discussed

A few caveats: No quality control function Poor seller feedback Bid manipulation may occur

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Chapter 13

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