Game Theory
“Loretta’s Driving Because I’m
Drinking and I’m Drinking
Because She’s Driving”
- The Lockhorns Cartoon
Mike Shor
Lecture 3
Review

Understand the game you are in

Note if the rules are flexible

Anticipate your opponents’ reactions

Understand the assumptions
• Recognize that not everyone else understands them
Game Theory - Mike Shor
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Game Theory - Mike Shor
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Equilibrium

Nash Equilibrium:
• A set of strategies, one for each player, such that
each player’s strategy is best for her given that all
other players are playing their equilibrium strategies

Best Response:
• The best strategy I can play given the strategy
choices of all other players

Everybody is playing a best response
• No incentive to unilaterally change my strategy
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Identifying the Equilibrium

Pure strategy equilibrium
• Consider mixed later

Dominance
• Dominance solvable
• Only one dominant strategy

Successive elimination of
dominated strategies

Cell-by-cell inspection
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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’ reaction
• Fear of potential liability lawsuits
1970  Companies strike agreement
• Carry the warning label and cease TV
advertising in exchange for immunity from
federal lawsuits.
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Strategic Interactions



Players:
Strategies:
Payoffs:
Reynolds and Philip Morris
{ Advertise , Do 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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Normal (Strategic) Form
PLAYERS
Reynolds
No Ad
Ad
Philip Morris
No Ad
Ad
50 , 50
20 , 60
60 , 20
STRATEGIES
30 , 30
PAYOFFS
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Normal Form
No Ad
Reynolds
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:

advertise
advertise
Regardless of what you think Philip Morris will do
Advertise!
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Dominant Strategy
A strategy that outperforms all other choices
no matter what opposing players do
 Firm 1’s strategies: { A, B, C }
 Firm 2’s strategies: { X, Y, Z }
 C is strictly dominant for Firm 1 if:




P(C,X)>P(A,X)
P(C,Y)>P(A,Y)
P(C,Z)>P(A,Z)
P(C,X)>P(B,X)
P(C,Y)>P(B,Y)
P(C,Z)>P(B,Z)
C is weakly dominant for Firm 1 if:

Some inequalities are weak (), at least one is strong(>)
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Dominance Solvable
COMMANDMENT
If you have a dominant strategy, use it.
Expect your opponent to use her
dominant strategy if she has one.


If each player has a dominant strategy,
the game is dominance solvable
What is the equilibrium of the cigarette
advertising game?
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Cigarette Advertising





After the 1970 agreement, cigarette
advertising decreased by $63 million
Profits rose by $91 million
Prisoner’s Dilemma
An equilibrium is NOT necessarily efficient
What if the game is not
dominance solvable?
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A Strategic Situation
Two firms competing over sales

Time and The Economist must decide
upon the cover story to run some week.

The big stories of the week are:
• A presidential scandal (labeled S), and
• A proposal to deploy US forces to Grenada (G)

Neither knows which story the other
magazine will choose to run
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One Dominant Strategy
Time
The Economist
G
S
0 , 90
S 100 , 100
G 95 , 100 95 , 90
Who has a dominant strategy?
 Assume it will be played!
 Other player can plan accordingly.

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Dominated Strategies
Time


The Economist
G
S
0 , 90
S 100 , 100
G 95 , 100 95 , 90
For The Economist:
G dominant = S dominated
Dominated Strategy:
• There exists another strategy which always does
better regardless of opponents’ actions
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Successive Elimination of
Dominated Strategies
If a strategy is dominated,
eliminate it
 The size and complexity of the game
is reduced
 Eliminate any dominant strategies
from the reduced game
 Continue doing so successively

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Example: Tourists & Natives
• Two bars (bar 1, bar 2) compete
• Can charge price of $2, $4, or $5
• 6000 tourists pick a bar randomly
• 4000 natives select the lowest price bar
Bar 2
$2
$4
$5
$2 10 , 10 14 , 12 14 , 15
Bar 1 $4 12 , 14 20 , 20 28 , 15
$5 15 , 14 15 , 28 25 , 25
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Successive Elimination of
Dominated Strategies
Does any player have a
dominant strategy?
 Does any player have a
dominated strategy?

• Eliminate the dominated strategies
• Reduce the normal-form game
• Iterate the above procedure

What is the equilibrium?
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Successive Elimination of
Dominated Strategies
Bar 2
$2
$4
$5
$2 10 , 10 14 , 12 14 , 15
Bar 1 $4 12 , 14 20 , 20 28 , 15
$5 15 , 14 15 , 28 25 , 25
Bar 2
$4
$5
$4 20 , 20 28 , 15
Bar 1
$5 15 , 28 25 , 25
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No Dominated Strategies

Often there are no dominated strategies
• Or: reducing the game is not sufficient



There may be multiple equilibria
Method:
Cell-by-cell inspection
Ask:
Is each player playing the best response
to the other player?
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Types of Games
Games of Assurance
 Games of Coordination
 Games of Chicken

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Games of Assurance




Two firms each earning $45,000
Both can invest the $45,000 into R&D
R&D successful only if both invest
If R&D successful, each earns $95,000
Invest
Firm 1
Don’t
Firm 2
Invest
Don’t
50 , 50 0 , 45
45 , 0 45 , 45
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Cell-by-cell Inspection

Consider { Invest , Don’t }
Invest
Firm 1
Don’t

Firm 2
Invest
Don’t
50 , 50 0 , 45
45 , 0 45 , 45
Both players have an incentive to change
their strategy: NOT an equilibrium
Game Theory - Mike Shor
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Assurance Outcomes
Two equilibria exist
 Both firms prefer (I ,I) to (D,D)

• Payoffs of 50 to each firm instead of 45

However, investing is risky
• Must have assurances

How to achieve assurance?
• Strategic moves:
• Sequential moves:
commit to choosing I
leader chooses
the equilibrium
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Games of Coordination



Joint ventures and the choice of supplier
Two firms engaged in joint venture
Must use the same supplier,
but each firm has a preferred supplier
Firm 2
A
B
A 100 , 50 0 , 0
Firm 1
0 , 0 50 , 100
B
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Coordination Outcomes
Two equilibria exist
 Firms prefer different equilibria
 How to achieve the most
desirable outcome for you?

• Strategic moves:
• Sequential moves:
commit to choosing A
leader chooses
the equilibrium
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Summary

You must put yourself
your rival’s shoes

Recognize dominant and
dominated strategies

Anticipate that your opponent
will recognize them as well
Game Theory - Mike Shor
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