Managerial Economics Ko University Graduate School of Business - - PDF document

1

Managerial Economics

Koç University Graduate School of Business MGEC 501 Levent Koçkesen

Game Theory

• Game theory studies strategic interactions

within a group of individuals

• actions of each individual have an effect on the
• utcome that is of interest to all
• individuals are aware of that fact
• Individuals are rational
• have well-defined objectives over the set of possible
• utcomes
• implement the best available strategy to pursue

them

• Rules of the game and rationality are common

knowledge

2

Price Competition

Toys“R”Us and Wal-Mart have to decide whether to sell a particular toy at a high or low price They act independently and without knowing the choice of the other store

Toys“R”Us Wal-Mart 5,5 15,2 Low 2,15 10,10 High Low High

What should Toys“R”Us play? Does that depend on what it thinks Wal-Mart will do? Low is an example of a dominant strategy it is optimal independent of what other players do How about Wal-Mart? (Low, Low) is a dominant strategy equilibrium A lesson we learned from oligopoly models Individual rationality does not imply collective rationality

Toys“R”Us Wal-Mart 5,5 15,2 Low 2,15 10,10 High Low High

Strategic Form Games

It is used to model situations in which players choose strategies without knowing the strategy choices of the other players Three components:

• 1. Players: N = {Toys“R”Us, Wal-Mart}
• 2. Strategies: ST = {High, Low}, SW = {High, Low}

Outcomes S = {(High, High), (High, Low), (Low, High), (Low, Low)}

• 3. Payoffs: For each player assigns a number to each outcome

uT (High ,High) = 10 Reflects players’ rankings of outcomes

3

Low Price Guarantee

Toys“R”Us web page has the following advertisement How does this change the game? Sounds like a very good deal for consumers

5,5 5,5 2,15 Low 5,5 15,2 Low Toys“R”Us Wal-Mart 10,10 10,10 Match 10,10 10,10 High Match High

• What happens if we add price matching as a strategy for both stores?
• Match: post a high price and match the other store’s price

Low Price Guarantee

• Is High ever an optimal strategy?
• High is weakly dominated by Match
• Is Match a dominant strategy?
• A rational player should not use a dominated strategy
• What happens to the game once you eliminate the dominated strategies?
• Is there a dominated or dominant strategy in the new game?
• Low becomes weakly dominated ↔ Match becomes weakly dominant
• Unique solution is (Match, Match)
• The above procedure is known as Iterated Elimination of Dominated Strategies (IEDS)

4

5,5 5,5 2,15 Low 5,5 15,2 Low Toys“R”Us Wal-Mart 10,10 10,10 Match 10,10 10,10 High Match High Iterated Elimination of Dominated Strategies (IEDS)

High is weakly dominated and Toys“R”Us is rational → it should not use High High is weakly dominated and Wal-Mart is rational → it should not use High If Toys“R”Us knows that Wal-Mart is rational, it knows that Wal-Mart will not use High This is where we use common knowledge of rationality To be a good strategist try to see the world from the perspective of your rivals and understand that they will most likely do the same This makes Low a weakly dominated strategy for both

Dominant Strategy Equilibrium

• A strategy X strictly dominates another strategy Y, if X always gives a strictly higher payoff than Y no

matter what other players do

• Low strictly dominates High
• A strategy X weakly dominates another strategy Y, if X never gives less payoff than Y and sometimes

gives a strictly higher payoff

• Right weakly dominates Left
• dominant strategy: it dominates every other strategy
• it is optimal independent of what other players do
• strictly dominant: strictly dominates every other strategy
• Low is strictly dominant
• weakly dominant: weakly dominates every other strategy
• Right is weakly dominant
• If every player has a dominant strategy, then the corresponding outcome is a dominant strategy

equilibrium

• (Low, Low) is a strictly dominant strategy equilibrium
• (Right, Right) is a weakly dominant strategy equilibrium

5,5 15,2 Low 2,15 10,10 High Low High 5,5 15,5 Down 5,15 10,10 Up Right Left

5

Iterated Elimination of Dominated Strategies

dominated strategy: never optimal no matter what other players do strictly dominated: there is a strategy that strictly dominates it

• High is strictly dominated

weakly dominated: there is a strategy that weakly dominates it

• Up is weakly dominated

Iterated elimination of strictly dominated strategies: every strategy eliminated is a strictly dominated strategy (U, M) is the unique outcome that survives IESDS 0,1 1,2 M 2,0 0,3 D 0,1 1,0 U R L Iterated elimination of weakly dominated strategies: at least one strategy eliminated is a weakly dominated strategy (Match, Match) is the unique outcome that survives IEWDS

1 2 3

5,5 5,5 2,15 Low 15,2 Low Toys“R”Us Wal-Mart 10,10 Match 10,10 High High

What if only Toys“R”US is aware of this smart strategy?

Low Price Guarantee: It Takes Two to Tango

Low becomes weakly dominant for Toys“R”US If Wal-Mart believes that Toys“R”US is rational, it will play Low as well

6

Two-Tiered Tender Offer

• Robert Campeau made a tender offer for Federated Department Stores in 1998
• The following is a simplified version of the actual offer
• Pre-takeover price of a Federated share is $100
• Campeau offers to pay $105 for the first 50% of the shares tendered and $90 for the remainder
• All shares, however, are bought at the weighted average price. If s is the percentage share

tendered, then the price of each share tendered is given by

     ≥ − × + × < = 50 , 50 90 50 105 50 , 105 s s s s s p

• If the takeover succeeds (s ≥ 50), the shares that were not tendered is worth $90 each; if it does not

succeed they are worth $100

• How much does each share cost Campeau if everybody tenders?
• There are 100 Federated shareholders (including you) each of whom owns one share
• What would you do: Tender or Not?
• Let s* be the number of shares tendered – not including you

100 105

s* = 49

90 100 Not 90 + 15(50/(s*+1)) 105 Tender

s* ≥ 50 s* < 49

Game of Chicken

• There are two providers of satellite radio: XM and Sirius
• XM is the industry leader with 5 million subscribers; Sirius has 2.2 million
• In the long-run the market can sustain only one provider

XM Sirius 0,0 0,300 Exit 300,0

– 200, – 200

Stay Exit Stay

• Is there a dominated strategy?
• What are the likely outcomes?
• Could (Stay, Stay) be an outcome?
• If XM expects Sirius to exit, what is its best strategy (best response)?
• If Sirius expects XM to stay what is its best response?
• (Stay, Exit) is an outcome such that

1. Each player best responds, given what they believe the other will do 2. Their beliefs are correct

• It is a Nash Equilibrium

7

XM Sirius 0,0 0,300 Exit 300,0

– 200, – 200

Stay Exit Stay

Nash Equilibrium

• Nash equilibrium is a strategy profile (a collection of strategies, one for each player) such that each

strategy is a best response (maximizes payoff) to all the other strategies

• Nash equilibrium is self-enforcing: no player has an incentive to deviate unilaterally
• One way to find Nash equilibrium is to first find the best response correspondence for each player
• Best response correspondence gives the set of payoff maximizing strategies for each strategy

profile of the other players

• … and then find where they “intersect”
• XM’s best response to Stay is Exit
• Its best response to Exit is Stay
• Sirius’ best response to Stay is Exit and to Exit is Stay
• Best response correspondences intersect at (Stay, Exit) and (Exit, Stay)
• These two strategy profiles are the two Nash equilibria of this game
• We would expect one of the companies to exit in the long-run

Sequential Move Games

Strategic form games are used to model situations in which players choose strategies without knowing the strategy choices of the other players In some situations players observe other players’ moves before they move Consider the following entry game: Kodak is contemplating entering the instant photography market and Polaroid can either fight the entry or accommodate

K P Enter Stay out Fight Accommodate 0, 20 – 5, 0 10, 10

Sequential moves games can be represented using a game tree What should Polaroid do if Kodak enters? Given what it knows about Polaroid’s response to enter, what should Kodak do? This is an example of a backward induction equilibrium

8

Backward Induction Equilibrium

At a backward induction equilibrium each player plays optimally at every decision node in the game tree (i.e., plays a sequentially rational strategy) (Enter, Accommodate) is the unique backward induction equilibrium of the entry game (Stay out, Fight) is a Nash equilibrium of this game

Kodak Polaroid 10,10 – 5,0 Enter 0,0 0,0 Stay Out Accommodate Fight

(Stay out, Fight) benefits Polaroid It can only be sustained by issuing a threat If Polaroid could commit itself to fight it would benefit Commitments (tying your own hands) could be beneficial in strategic interactions Commitments to what seem to be ex post irrational strategies could be beneficial We have already seen such a beneficial commitment in Stackelberg game Hernan Cortes ordered his men to burn all but one ship in his conquest of Mexico Credibility of the commitment is key Kodak entered instant photography in late 1970s, but in a gentle fashion Designed cameras and film that were incompatible with Polaroid’s cameras reassuring Polaroid that it would be able to continue to make money, selling to its customer base Yet Polaroid chose war: launched patent infringement lawsuits, lowered prices, introduced new products Edwin Land (founder of Polaroid) “This is our very soul … our whole life. For them, it’s just another field.” (NYT: April 28, 1976) Is the threat credible? Polaroid limited itself to only one industry. Loosing that market would mean the end of the company, whereas Kodak was diversified. Polaroid won the war

Commitments

9

Price Protection

General Electric and Westinghouse were the only manufacturers of large electric turbine generators during 60s and 70s 1963: GE offered its customers “price protection” for 6 months This is known as most-favored-customer-clause (MFCC) Westinghouse responded by offering the same deal This looks like a policy that benefits the customers Antitrust Division of the U.S. Department of Justice demanded that GE and Westinghouse refrain from this policy claiming that it harmed the customers What is going on here?

• 50

15 150

• 80

100 10 10 45 15 135

• 60

180 20 9 120 15 120

• 40

240 30 8 175 15 105

• 20

280 40 7 210 15 90 300 50 6 225 15 75 20 300 60 5 220 15 60 40 280 70 4 195 15 45 60 240 80 3 150 15 30 80 180 90 2 85 15 15 100 100 100 1 Profit MC TC MR TR Reservation Price Potential Buyer

• Suppose it costs GE $15,000 per unit to build a turbine
• There are 10 potential customers with reservation prices as in the table below
• What is the profit maximizing price for GE?
• GE sets its price at $60,000 and sells a turbine to the first 5 customers
• There are still 5 more customers who are willing to do business with GE

Price Protection

10

• After GE sells to the first 5 customers the

remaining market is as follows

• What is the profit maximizing price for GE

now?

• GE sets its price at $40,000 and sells a turbine

to customers 6 and 7

• 25

15 75

• 30

50 10 10 20 15 60

• 10

80 20 9 45 15 45 10 90 30 8 50 15 30 30 80 40 7 35 15 15 50 50 50 6 Profit MC TC MR TR Reservation Price Potential Buyer

• 15

15 45

• 10

30 10 10 10 15 30 10 40 20 9 15 15 15 30 30 30 8 Profit MC TC MR TR Reservation Price Potential Buyer

• GE sets its price at $20,000 and sells a turbine to

customer 9

• No more profitable trade for GE
• 10

15 30 20 10 10 5 15 15 20 20 20 9 Profit MC TC MR TR Reservation Price Potential Buyer

• The market becomes
• GE sets its price at $30,000 and sells a turbine

to customer 8

Price Protection

9 20 4 8 30 3 6-7 40 2 1-5 60 1 Customers Price Period

Price Protection

• So, if the customers are not willing to wait, i.e., they are impatient, GE can implement price

discrimination over time

• What happens if the customers are patient?
• Anticipating a price cut by GE, customer 8 will wait until the next period
• In general, as long as the current price is greater than the MC the monopolist will have an incentive

to reduce the price next period. Customers anticipate this and do not buy unless P = MC

• The unique backward induction equilibrium of the game is P = MC
• Coase conjecture: a monopolist selling a durable good to very patient customers has no monopoly

power

• GE would like to commit itself to not lower the price in the future. If it can do that, then it would set

price at the monopoly price of $60,000

• Most-favored-customer-clause provides the commitment mechanism: GE cannot price discriminate
• ver time. Lowering the price next period means lowering the price for all the units that had been

sold before.

• MFCC reduces incentives to cut prices
• It also makes it easier to collude on monopoly price since price cutting is costly