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Part 3: Game Theory IBasic Concepts, Dominance Solvability
Simultaneous Move Games, Payoff Matrix, Dominant Strategy,Iterated Elimination of Dominated Strategies
May 2016
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Basic Concepts
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Why do we study games?
A game is a way to model strategic behavior – people recognize that their ownbehavior affect the choices of others and that the outcome depends on any oneperson’s choice, not just one’s own
Strategic behavior is important in situations where
A small number of individuals interact (negotiations, auctions)
There is imperfect competition
Externalities and public goods are present
Game theory not limited to economics
e.g. biology, sociology, political science, dating, sports, ...
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An Example: The Prisoners’ Dilemma
The simplest game is a game with two players where players choose actionssimultaneously
Example of a 2x2 matrix (normal form or strategic form) game:
Row
ColumnConfess Don’t
Confess −10,−10 0,−20Don’t −20, 0 −1,−1
The Prisoners’ Dilemma
Matrix entries = payoffs
First Entry = Row’s payoff, second entry = Column’s payoff
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What is a game?
A simultaneous-move game involves:
A list of players
For each player, a set of actions = strategies
For each player, preferences over each possible strategy combination = payoffs
When solving a game we assume player are rational:
They choose their action to maximize their payoff
They form beliefs about what others will do
Those beliefs are correct in equilibrium
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Solving the Prisoners’ DilemmaThe PD has an obvious solution (equilibrium)
Row
ColumnConfess Don’t
Confess −10,−10 0,−20Don’t −20, 0 −1,−1
The Prisoners’ Dilemma
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Solving the Prisoners’ DilemmaThe PD has an obvious solution (equilibrium)
Row
ColumnConfess Don’t
Confess -10 ,−10 0 ,−20Don’t -20 , 0 -1 ,−1
The Prisoners’ Dilemma
No matter what Column chooses, Row does better by confessing
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Solving the Prisoners’ DilemmaThe PD has an obvious solution (equilibrium)
Row
ColumnConfess Don’t
Confess −10, -10 0, -20
Don’t −20, 0 −1, -1
The Prisoners’ Dilemma
The same is true for Column
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Solving the Prisoners’ DilemmaThe PD has an obvious solution (equilibrium)
Row
ColumnConfess Don’t
Confess -10,-100,−20
Don’t −20, 0 −1,−1
The Prisoners’ Dilemma
Confessing is a dominant strategy for each player: it is the best choiceregardless of what the other player does
Confessing is a dominated strategy: it is worse than some other strategyregardless of the other player
Both players confessing is an equilibrium of the game (even though it is worsefor both than if neither confesses)
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Applications of the PD
Many other situations have structures similar to the Prisoners’ Dilemma
ex. 1: working on a joint project (private provision of a public good)
Row
ColumnGoof off Work Hard
Goof off 1, 1 3, 0
Work Hard 0, 3 2, 2
Private Provision of a Public Good
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Applications of the PD
Many other situations have structures similar to the Prisoners’ Dilemma
ex. 1: working on a joint project (private provision of a public good)
Row
ColumnGoof off Work Hard
Goof off 1,13, 0
Work Hard 0, 3 2, 2
Private Provision of a Public Good
Eliminating strictly dominated actions gives (Goof off, Goof off) asequilibrium
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Applications of the PD Cont’d
ex. 2: Two firms produce the same good and can choose prices (duopoly)
Row
ColumnLow Price High Price
Low Price 600, 600 1200,−200High Price −200, 1200 1000, 1000
A Price Setting Duopoly
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Applications of the PD Cont’d
ex. 2: two firms produce the same good and can choose prices (duopoly)
Row
ColumnLow Price High Price
Low Price 600,6001200,−200
High Price −200, 1200 1000, 1000
A Price Setting Duopoly
Eliminating strictly dominated actions gives (Low Price, Low Price) asequilibrium
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Applications of the PD Cont’d
In general, the PD encompasses all situations in which players can‘Cooperate’ (C) or ‘Not cooperate/Defect’ (D):
Row
ColumnC D
C a, b c, d
D e, f g, h
The Prisoners’ Dilemma
Where payoffs are:
e > a > g > c
d > b > h > f
Defecting is a dominant strategy for each player
The equilibrium outcome is (D,D)
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Dominance Solvability
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A Dominance Solvable Game
Another example: Microsoft vs Start-up in a market for new online service
Mircosoft
Start-upEnter Don’t
Enter 2,−2 5, 0
Don’t 0, 5 0, 0
An Entry Game
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A Dominance Solvable Game
Another example: Microsoft vs Start-up in a market for new online service
Microsoft
Start-upEnter Don’t
Enter 2 ,−2 5 , 0
Don’t 0 , 5 0 , 0
An Entry Game
Microsoft has a dominant strategy: Enter
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A Dominance Solvable Game
Another example: Microsoft vs Start-up in a market for new online service
Microsoft
Start-upEnter Don’t
Enter 2, - 2 5, 0
Don’t 0, 5 0, 0
An Entry Game
Microsoft has a dominant strategy: Enter
Start-up has no dominant strategy: if MS enters, Start-up should stay out; ifMS stays out, Start-up should enter
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Dominance Solvability (cont’d)
we can solve the game by iterated elimination of dominated strategies
Mircosoft
Start-upEnter Don’t
Enter 2,−2 5, 0
Don’t 0, 5 0, 0
An Entry Game
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Dominance Solvability (cont’d)
we can solve the game by iterated elimination of dominated strategies
Microsoft
Start-upEnter Don’t
Enter 2,−2 5, 0
Don’t 0, 5 0, 0
An Entry Game
eliminate first MS’s dominated strategy (Don’t enter)(seems reasonable if Start-up knows MS’s payoffs)
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Dominance Solvability (cont’d)
we can solve the game by iterated elimination of dominated strategies
Microsoft
Start-upEnter Don’t
Enter 2,−2 5, 0
Don’t 0, 5 0, 0
An Entry Game
eliminate first MS’s dominated strategy (Don’t enter)
with remaining game eliminate Start-up’s dominated strategy (Enter)
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Dominance Solvability (cont’d)
we can solve the game by iterated elimination of dominated strategies
Microsoft
Start-upEnter Don’t
Enter 2,−2 5, 0
Don’t 0, 5 0, 0
An Entry Game
eliminate first MS’s dominated strategy (Don’t enter)
with remaining game eliminate Start-up’s dominated strategy (Enter)
only (Enter, Don’t enter) is left → equilibrium outcome
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Dominance Solvability (cont’d)
Another example
Row
ColumnLeft Center Right
Top −5,−1 2, 2 3, 3
Middle 1,−3 1, 2 1, 1
Bottom 0, 10 0, 0 0,−10
A Dominance Solvable Game
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Dominance Solvability (cont’d)
Another example
Row
ColumnLeft Center Right
Top −5,−1 2, 2 3, 3
Middle 1,−3 1, 2 1, 1
Bottom 0, 10 0, 0 0,−10
A Dominance Solvable Game
‘Middle’ dominates ‘Bottom’ for Row → drop ‘Bottom’
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Dominance Solvability (cont’d)
Another example
Row
ColumnLeft Center Right
Top −5,−1 2, 2 3, 3
Middle 1,−3 1, 2 1, 1
Bottom 0, 10 0, 0 0,−10
A Dominance Solvable Game
‘Middle’ dominates ‘Bottom’ for Row → drop ‘Bottom’
‘Left’ is dominated for Column in remaining game → drop ‘Left’
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Dominance Solvability (cont’d)
Another example
Row
ColumnLeft Center Right
Top −5,−1 2, 2 3, 3
Middle 1,−3 1, 2 1, 1
Bottom 0, 10 0, 0 0,−10
A Dominance Solvable Game
‘Middle’ dominates ‘Bottom’ for Row → drop ‘Bottom’
‘Left’ is dominated for Column in remaining game → drop ‘Left’
‘Top’ dominates ‘Middle’ for Row in remaining game → drop ‘Middle’
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Dominance Solvability (cont’d)
Another example
Row
ColumnLeft Center Right
Top −5,−1 2, 2 3, 3
Middle 1,−3 1, 2 1, 1
Bottom 0, 10 0, 0 0,−10
A Dominance Solvable Game
‘Middle’ dominates ‘Bottom’ for Row → drop ‘Bottom’
‘Left’ is now dominated for Column → drop ‘Left’
‘Top’ dominates ‘Middle’ for Row in remaining game → drop ‘Middle’
‘Right’ is better than ‘Center’ for Column → drop ‘Center’
Equilibrium outcome is (Top, Right)
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Dominance Solvability (cont’d)
Another example
Row
ColumnLeft Center Right
Top −5,−1 2, 2 3, 3
Middle 1,−3 1, 2 1, 1
Bottom 0, 10 0, 0 0,−10
A Dominance Solvable Game
Outcome (Top, Right) reasonable if players know each others’ payoffs; plus:no player would want to change their behavior given the behavior of others
But iterated elimination of dominated strategies does not always give anoutcome ...
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A Coordination Game
A coordination game
Harry
SallyRenaissance Starbucks
Renaissance 2, 1 0, 0
Starbucks 0, 0 1, 2
Battle of the Sexes
Both players prefer to cooperate (meet at same location) but disagree onabout best outcome (location)
→ No dominated strategies
But: if Sally chooses Starbucks, Harry should also choose Starbucks (and viceversa) → reasonable to presume that they will meet
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