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Game Theory
“Loretta’s driving because I’m drinking and I’m drinking because she’s driving.”
- The Lockhorns
Topic 2Simultaneous Games
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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
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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
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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
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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?
Mike Shor5
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Strategic Form of a Game
Mike Shor6
PLAYERS
STRATEGIESPAYOFFS
Philip Morris
No Ad Ad
Reynolds No Ad 50 , 50 20 , 60
Ad 60 , 20 30 , 30
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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
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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
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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:
Mike Shor9
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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
Mike Shor10
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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
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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
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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
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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
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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.
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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
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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
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Cigarette Advertising After the 1970 agreement:
Cigarette advertising decreased by $63 million Industry Profits rose by $91 million
19Mike Shor
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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
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Baldwin and Meese (1979), “Social Behavior in Pigs Studied by Means of Operant Conditioning,” Animal Behavior
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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
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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
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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: $74 per share
If X>50% tender, each receives:
Mike Shor24
60$%
%50% 74$
%
%50
X
X
X
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The Federated Game
To whom do you tender your shares?
Mike Shor25
Majority of Others
Macy’s Campeau
YouMacy’s $70 $60
Campeau $74 $67+
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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 !
Mike Shor26
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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
Mike Shor28
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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
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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
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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
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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
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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!
Mike Shor38
Firm 2A B
Firm 1A 100 , 50 0 , 0
B 0 , 0 50 , 100
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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
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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
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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
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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?
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