Game Theory

“Loretta’s driving because I’m drinking and I’m drinking because she’s driving.”

- The Lockhorns

Topic 2Simultaneous Games

Review

Understanding the game Noting if the rules are flexible Anticipating our opponents’ reactions Thinking one step ahead

Where does this lead us? We’ve defined the “game” but not the outcome

Mike Shor2

Equilibrium

The likely outcome of a game when rational, strategic agents interact Each player is playing his or her best strategy

given the strategy choices of all other players No player has incentive to change his or her action

unilaterally

Outline: Model interactions as games Identify the equilibria Decide when they are likely to occur

Mike Shor3

Cigarette Advertising on TV All US tobacco companies

advertised heavily on TV

Surgeon General issues official warningCigarette smoking may be hazardous

Cigarette companies fear lawsuitsGovernment may recover healthcare costs

Companies strike agreementCarry the warning label and cease

TV advertising in exchange for immunity from federal lawsuits.

Mike Shor4

1964

1970

Strategic Interaction

Players: Reynolds and Philip Morris Strategies: Advertise or Not Advertise Payoffs: Companies’ Profits

Environment: Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor

How to represent this game?

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Strategic Form of a Game

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PLAYERS

STRATEGIESPAYOFFS

Philip Morris

No Ad Ad

Reynolds No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

What to Do?

If you are advising Reynolds, what strategy do you recommend?

Mike Shor7

Philip Morris

No Ad Ad

ReynoldsNo Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

Best Replies

A strategy is a best reply to some opponents’ strategy if it does at least as well as any other strategy

si is a best reply to s-i if

for every si’

Mike Shor8

),(),( iiiiii ssussu

Philip Morris

No Ad Ad

ReynoldsNo Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

Solving the Game

Best reply for Reynolds: If Philip Morris advertises: If Philip Morris does not advertise:

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Dominance

A strategy is dominant if it outperforms all other strategies no matter what opposing players do

Games with dominant strategies are easy to solve No need for “what if …” thinking

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Dominance

si strictly dominates si’ if

for every s-i

(the payoff is strictly higher for every strategy of the other players)

si weakly dominates si’ if

for every s-i, and

for some s-i

Mike Shor11

),(),( iiiiii ssussu

),(),( iiiiii ssussu ),(),( iiiiii ssussu

Dominance

A strategy si is strictly dominant if it strictly dominates all other strategies for that player

A strategy si is weakly dominant if it weakly dominates all other strategies for that player

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Dominance

Example 1

A strictly dominates B

& A strictly dominates C

Therefore A is strictly dominantMike Shor

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X Y Z

A 10 20 30

B 8 18 25

C 5 5 5

Dominance

Example 2

A strictly dominates B

& A weakly dominates C’

Therefore A is weakly dominantMike Shor

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X Y Z

A 10 20 30

B 8 18 25

C’ 10 10 10

Dominance

Example 3

A strictly dominates B

& A does not dominate C’’

Therefore A is not dominantMike Shor

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X Y Z

A 10 20 30

B 8 18 25

C’’ 20 20 20

Dominance

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If you have a dominant strategy (and no ability to agree on an alternate course of action)

use it.

If your opponent has a dominant strategy(and no ability to agree on an alternate course of action)

then expect her to play it.

Prisoner’s Dilemma

Both players have a dominant strategy The equilibrium results in lower

payoffs for each player

Mike Shor17

No Ad Ad

No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

Equilibrium

Optimal

Prisoner’s Dilemma

Both players have a dominant strategy (s1,s1)

u11 > u21 u12 > u22 The equilibrium results in lower payoffs for each player

u22 > u11 The above two statements imply:

u12 > u22 > u11 > u21

Mike Shor18

s1 s2

s1 u11 , u11 u12 , π21

s2 u21 , u12 u22 , π22

Cigarette Advertising After the 1970 agreement:

Cigarette advertising decreased by $63 million Industry Profits rose by $91 million

19Mike Shor

Prisoner’s Dilemma

The dominant strategy will be played

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Social Behavior in Pigs

Two small pigs: First pig gets 8 units of food, second gets 2 If simultaneous, each gets 5 Pushing the lever costs 1

One small, one big: If big pig is first, eats all of the food If small pig is first, it gets 6 units of food If simultaneous, big pig gets 7

Mike Shor21

Baldwin and Meese (1979), “Social Behavior in Pigs Studied by Means of Operant Conditioning,” Animal Behavior

Prisoner’s Dilemma

The dominant strategy will be played

Prisoner’s Dilemma An equilibrium is NOT necessarily efficient Players can be forced to accept

mutually bad outcomes Bad to be playing a prisoner’s dilemma,

but good to make others play

Mike Shor22

How to Win a Bidding War by Bidding Less?

The battle for Federated (1988)Parent of Bloomingdales

Current share price ≈ $60 Expected post-takeover share price ≈ $60

Macy’s offers $70/share contingent on receiving 50% of the shares

Do you tender your shares to Macy’s?

Mike Shor23

How to Win a Bidding War (continued)

Robert Campeau bids $74 per share not contingent on amount acquired

Campeau’s Mixed Scheme: If less than 50% tender their shares,

each receives: $74 per share

If X>50% tender, each receives:

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60$%

%50% 74$

%

%50

X

X

X

The Federated Game

To whom do you tender your shares?

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Majority of Others

Macy’s Campeau

YouMacy’s $70 $60

Campeau $74 $67+

How to Win a Bidding War

Each player has a dominant strategy: Tender shares to Campeau

Resulting Price:

(½ x 74) + (½ x 60) = $67

BUT: Macy’s offered $70 !

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Dominant Strategies

“ The biggest, looniest deal ever. ” – Fortune Magazine, July 1988 on Campeau’s acquisition of Federated Stores

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Prisoner’s Dilemma Examples

Pricing by Firms High or low prices? Value menus and loyalty programs

Divorce Hire attorneys or proceed amicably?

Nuclear Weapons Build or don’t build weapons?

State governments Inducements to attract business to a state

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Dominated Strategies

Two restaurants compete Can charge price of $30, $50, or $60

Customer base consists of tourists and natives 600 tourists pick randomly 400 natives select the lowest price

Marginal costs are $10

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Tourists & Natives

Example scenario: Restaurant 1: $50, Restaurant 2: $60

Restaurant 1 gets:300 tourists + 400 natives = 700 customers x ($50-$10) = $28K

Restaurant 2 gets:300 tourists + 0 natives= 300 customers x ($60-$10) = $15K

Mike Shor30

Tourists & Natives

in thousands of dollars

Mike Shor31

$30 $50 $60

R. 1

$30 10 , 10 14 , 12 14 , 15

$50 12 , 14 20 , 20 28 , 15

$60 15 , 14 15 , 28 25 , 25

R. 2

Dominance

A strategy si is strictly dominated if some strategy si’ strictly dominates it

A strategy si is weakly dominated if some strategy si’ weakly dominates it

Mike Shor32

Iterated Deletion of Strictly Dominated Strategies

Does any player have a (strictly) dominated strategy?

Eliminate the strictly dominated strategy Reduce the size of the game Repeat: Iterate the above procedure

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$30 $50 $60

R. 1

$30 10 , 10 14 , 12 14 , 15

$50 12 , 14 20 , 20 28 , 15

$60 15 , 14 15 , 28 25 , 25

Iterated Deletion of Dominated Strategies

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R. 2

No Dominated Strategies

Often there are no dominated strategies Some games may have multiple equilibria Equilibrium selection becomes an issue

Method:For each player, find the best response to every strategy of the other player

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Equilibrium

An outcome in which every player is playing a best response to the strategies of all other players.

An equilibrium is a strategy profile s such that si is a best reply to s-i for all i.

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Equilibrium IllustrationThe Lockhorns

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Games of Coordination

Complements & technology adoption Two complementing firms Must use same technology,

but each firm has a preferred technology

Equilibrium does not offer a unique prediction Commit (or go first) to win!

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Firm 2A B

Firm 1A 100 , 50 0 , 0

B 0 , 0 50 , 100

Games of Assurance

Joint research ventures Each firm may invest $50,000 into an R&D project Project succeeds only if both invest If successful, each nets $75,000

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Firm 2$50K $0

Firm 1$50K 75 , 75 -50 , 0

$0 0 , -50 0 , 0

Games of Chicken

Entry into small markets

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Firm 2Stay Swerve

Firm 1Stay -50 , -50 100 , 0

Swerve 0 , 100 50 , 50

The Right Game to Play

Why do we “solve” games?

To know which one to play! How do internal corporate changes impact

the outcome of strategic interaction?

Some games are better than others

Mike Shor41

Capacity Constraints

Can decreasing others’ added value increase our profits?

Can decreasing total industry value increase our profits?

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Multiple Equilibria

What is the predictive power of game theory when there are multiple equilibria?

Sometimes nothing ?

Refinements Focal points Efficiency Evolutionary stability Fairness Risk dominance

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Summary

Games have predictable outcomes Notice dominant & dominated strategies

Select the right game to play

Looking ahead: Sequential Games:

How do games unfold over time?

Mike Shor44