Game Theory
Topic 2
Simultaneous Games
“Loretta’s driving because I’m
drinking and I’m drinking
because she’s driving.”
- The Lockhorns
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
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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
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Cigarette Advertising on TV

All US tobacco companies
advertised heavily on TV
1964  Surgeon General issues official warning
 Cigarette smoking may be hazardous

Cigarette companies fear lawsuits
 Government
may recover healthcare costs
1970  Companies strike agreement
 Carry the warning label and cease
TV advertising in exchange for
immunity from federal lawsuits.
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Strategic Interaction

Players:
Strategies:
Payoffs:

Environment:






Reynolds and Philip Morris
Advertise or Not Advertise
Companies’ Profits
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
PLAYERS
Philip Morris
No Ad
Ad
No Ad
50 , 50
20 , 60
Ad
60 , 20
30 , 30
Reynolds
STRATEGIES
PAYOFFS
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What to Do?
Reynolds
No Ad
Ad
Philip Morris
No Ad
Ad
50 , 50
20 , 60
60 , 20
30 , 30
If you are advising Reynolds,
what strategy do you recommend?
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Mike Shor
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’
ui (si , si )  ui (si, si )
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Mike Shor
Solving the Game
Reynolds

No Ad
Ad
Philip Morris
No Ad
Ad
50 , 50
20 , 60
60 , 20
30 , 30
Best reply for Reynolds:
If Philip Morris advertises:
 If Philip Morris does not advertise:

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Mike Shor
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
ui (si , si )  ui (si, si ) for every s-i
(the payoff is strictly higher for every
strategy of the other players)

si weakly dominates si’ if
ui (si , si )  ui (si, si ) for every s-i, and
ui (si , si )  ui (si, si ) for some s-i
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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
B
C
X
10
8
5
Y
20
18
5
Z
30
25
5
A strictly dominates B
&
A strictly dominates C
Therefore A is strictly dominant
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Dominance

Example 2
A
B
C’
X
10
8
10
Y
20
18
10
Z
30
25
10
A strictly dominates B
&
A weakly dominates C’
Therefore A is weakly dominant
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Mike Shor
Dominance

Example 3
A
B
C’’
X
10
8
20
Y
20
18
20
Z
30
25
20
A strictly dominates B
&
A does not dominate C’’
Therefore A is not dominant
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Dominance
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
Optimal
No Ad
Ad
No Ad
50 , 50
60 , 20
Ad
20 , 60
30 , 30
Equilibrium
Both players have a dominant strategy
 The equilibrium results in lower
payoffs for each player

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Prisoner’s Dilemma
s1
s2
Mike Shor
u11
s1
, u11
u21 , u12
u12
s2
, π21
u22 , π22

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
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Cigarette Advertising

After the 1970 agreement:


Cigarette advertising decreased by $63 million
Industry Profits rose by $91 million
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Prisoner’s Dilemma

The dominant strategy will be played
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Social Behavior in Pigs
Baldwin and Meese (1979), “Social Behavior in Pigs Studied by
Means of Operant Conditioning,” Animal Behavior

Two small pigs:




One small, one big:



Mike Shor
First pig gets 8 units of food, second gets 2
If simultaneous, each gets 5
Pushing the lever costs 1
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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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?
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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:
 50% 
 X %  50% 

  $74  
  $60
X%
 X% 


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The Federated Game
Majority of Others
Macy’s
Campeau
You

Macy’s
$70
$60
Campeau
$74
$67+
To whom do you tender your shares?
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Mike Shor
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



Divorce


Hire attorneys or proceed amicably?
Nuclear Weapons


High or low prices?
Value menus and loyalty programs
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
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Tourists & Natives
$30
R. 1 $50
$60
R. 2
$30
$50
$60
10 , 10 14 , 12 14 , 15
12 , 14 20 , 20 28 , 15
15 , 14 15 , 28 25 , 25
in thousands of dollars
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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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Mike Shor
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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Iterated Deletion of
Dominated Strategies
R. 1
R. 2
$30
$50
$60
$30 10 , 10 14 , 12 14 , 15
$50 12 , 14 20 , 20 28 , 15
$60 15 , 14 15 , 28 25 , 25
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Mike Shor
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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Mike Shor
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 Illustration
The 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
Firm 2
Firm 1


A
A
100 , 50
B
0 ,
0
B
0 ,
0
50 , 100
Equilibrium does not offer a unique prediction
Commit (or go first) to win!
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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
Firm 2
Firm 1
$50K
$0
$50K
75 , 75
$0
-50 ,
0
0 , -50
0 ,
0
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Games of Chicken

Entry into small markets
Firm 1
Stay
Swerve
Firm 2
Stay
Swerve
-50 , -50 100 , 0
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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