Cooperative Game
Theory
Jennifer Wilson
Cooperative Game Theory
Outline
Introduction
Jennifer Wilson
Department of Natural Sciences and Mathematics
Eugene Lang College The New School for Liberal Arts
August 6, 2008
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Introduction
Relationship between Non-cooperative and Cooperative
Games
Cooperative GameTheory
A Survey of Different Solution Concepts
A Small Market
Imputations and the Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power Index
UN Security Council
Multichoice Games
Extensions of Cooperative Game Theory
Definitions
Examples
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
Example
Player 3: A
Player 3: B
Player 2
Player 2
A
A
Player 1
B
B
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(1, −2, 1)
(−4, 1, 3)
(3, 1, −4)
(−5, 10, −5)
A
A
Player 1
B
B
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(−2, −1, 3)
(2, −4, 2)
(−6, 12, −6)
(3, −1, −2)
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
Example
A Survey of
Different Solution
Concepts
Player 3: A
A
A
Player 1
B
Player 2
Player 3: B
B
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(1, −2, 1) −→ (−4, 1, 3)

x
y
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(3, 1, −4) −→(−5, 10, −5)
A
A
Player 1
B
Player 2
B
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(−2, −1, 3)←− (2, −4, 2)

x
y
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(−6, 12, −6)−→(3, −1, −2)
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
Example
Player 3: A
Player 3: B
Player 2
Player 2
A
A
Player 1
B
B
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(1, −2, 1)
(−4, 1, 3)
(3, 1, −4)
(−5, 10, −5)
A
A
Player 1
B
B
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(−2, −1, 3)
(2, −4, 2)
(−6, 12, −6)
(3, −1, −2)
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
AA
A
Player 1
B
Player 2 & 3
AB
BA
BB
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(1, −1)
(−2, 2) (−4, 4) (2, −2)
(3, −3)
(−6, 6) (−5, 5) (3, −3)
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Jennifer Wilson
Outline
Introduction
Example
AA
Player 2 & 3
AB
BA
BB
AA
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(1, −1)
(3, −3)
AA
A
Player 1
B
(−2, 2) (−4, 4) (2, −2)
A
Player 1
(−6, 6) (−5, 5) (3, −3)
Player 2 & 3
AB
BA
(1, −1)
(−2, 2) (−4, 4) (2, −2)
(3, −3)
(−6, 6) (−5, 5) (3, −3)
BB
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(1, −1)
(−2, 2) (−4, 4) (2, −2)
(3, −3)
(−6, 6) (−5, 5) (3, −3)
AA
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(1, −1)
(−2, 2) (−4, 4) (2, −2)
(3, −3)
(−6, 6) (−5, 5) (3, −3)
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Jennifer Wilson
Outline
Introduction
Example
AA
Player 2 & 3
AB
BA
BB
AA
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(1, −1)
(3, −3)
AA
A
Player 1
B
(−2, 2) (−4, 4) (2, −2)
A
Player 1
(−6, 6) (−5, 5) (3, −3)
Player 2 & 3
AB
BA
(1, −1)
(−2, 2) (−4, 4) (2, −2)
(3, −3)
(−6, 6) (−5, 5) (3, −3)
So ν(1) = −4 and ν(2, 3) = 4.
BB
B
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........................................................................................................................................................................................................................
(1, −1)
(−2, 2) (−4, 4) (2, −2)
(3, −3)
(−6, 6) (−5, 5) (3, −3)
AA
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Player 2 & 3
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.......................................................................................................................................................................................................................
(1, −1)
(−2, 2) (−4, 4) (2, −2)
(3, −3)
(−6, 6) (−5, 5) (3, −3)
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
Example
Player 1 & 3
AA
A
Player 2
B
AB
BA
Player 1 & 3
(2, −2)
(1, −1)
AA
q
BB
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(−1, 1) (−1, 1) (12, −12)
(−4, 4) (10, −10) (−1, 1)
Ap
Player 2
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1−q
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........................................................................................................................................................................................................................
(2, −2)
B 1 − p (1, −1)
(−1, 1) (−1, 1) (12, −12)
(−4, 4) (10, −10) (−1, 1)
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
Example
Player 1 & 3
AA
A
Player 2
B
AB
BA
Player 1 & 3
(2, −2)
(1, −1)
AA
q
BB
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........................................................................................................................................................................................................................
(−1, 1) (−1, 1) (12, −12)
(−4, 4) (10, −10) (−1, 1)
Solving, gives p = 56 and q = 12 .
So ν(2) = −1.5 and ν(1, 3) = 1.5.
Ap
Player 2
AB
1−q
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BB
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(2, −2)
B 1 − p (1, −1)
(−1, 1) (−1, 1) (12, −12)
(−4, 4) (10, −10) (−1, 1)
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
Example
Player 1 & 3
AA
A
Player 2
B
AB
BA
Player 1 & 3
(2, −2)
(1, −1)
AA
q
BB
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........................................................................................................................................................................................................................
(−1, 1) (−1, 1) (12, −12)
(−4, 4) (10, −10) (−1, 1)
Ap
Player 2
(2, −2)
B 1 − p (1, −1)
Solving, gives p = 56 and q = 12 .
So ν(2) = −1.5 and ν(1, 3) = 1.5.
Similarly ν(3) = −4.48 and ν(1, 2) = 4.48.
AB
1−q
BA
BB
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........................................................................................................................................................................................................................
(−1, 1) (−1, 1) (12, −12)
(−4, 4) (10, −10) (−1, 1)
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Example
Summary:
ν(1) = −4, ν(2) = −1.5, ν(3) = −4.48,
ν(2, 3) = 4, ν(1, 2) = 4.48 and ν(1, 3) = 1.5.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Example
Summary:
ν(1) = −4, ν(2) = −1.5, ν(3) = −4.48,
ν(2, 3) = 4, ν(1, 2) = 4.48 and ν(1, 3) = 1.5.
It’s advantageous to form a coalition.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Example
Summary:
ν(1) = −4, ν(2) = −1.5, ν(3) = −4.48,
ν(2, 3) = 4, ν(1, 2) = 4.48 and ν(1, 3) = 1.5.
It’s advantageous to form a coalition.
Which coalitions should form?
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Example
Summary:
ν(1) = −4, ν(2) = −1.5, ν(3) = −4.48,
ν(2, 3) = 4, ν(1, 2) = 4.48 and ν(1, 3) = 1.5.
It’s advantageous to form a coalition.
Which coalitions should form?
Player 1 versus Players 2 & 3: (−4, 1, 3)
Player 2 versus Players 1 & 3: (−0.58, 1.5, 2.08)
Player 3 versus Players 1 & 2: (−0.56, 5.04, −4.48)
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
A Three-Person Zero-Sum Game
[Rapoport, 1970]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Example
Summary:
ν(1) = −4, ν(2) = −1.5, ν(3) = −4.48,
ν(2, 3) = 4, ν(1, 2) = 4.48 and ν(1, 3) = 1.5.
It’s advantageous to form a coalition.
Which coalitions should form?
Player 1 versus Players 2 & 3: (−4, 1, 3)
Player 2 versus Players 1 & 3: (−0.58, 1.5, 2.08)
Player 3 versus Players 1 & 2: (−0.56, 5.04, −4.48)
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
What should each player receive from the game?
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Assumptions
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
I
I
Players may communicate and form coalitions with one
another.
Players may make sidepayments to each other.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Assumptions
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
I
I
Players may communicate and form coalitions with one
another.
Players may make sidepayments to each other.
I
Utility is transferable between players.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Assumptions
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
I
I
Players may communicate and form coalitions with one
another.
Players may make sidepayments to each other.
I
I
Utility is transferable between players.
Players’ utility values are comparable.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Assumptions
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
I
I
Players may communicate and form coalitions with one
another.
Players may make sidepayments to each other.
I
I
Utility is transferable between players.
Players’ utility values are comparable.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
We will look at transferable utitility(TU) games.
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Characteristic Function Form
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
The characteristic function ν of a game is a function that
assigns a value for each coalition (subset) of players.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Characteristic Function Form
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
The characteristic function ν of a game is a function that
assigns a value for each coalition (subset) of players.
If the game is constant-sum then
I
ν(∅) = 0
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Characteristic Function Form
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
The characteristic function ν of a game is a function that
assigns a value for each coalition (subset) of players.
If the game is constant-sum then
I
ν(∅) = 0
I
ν(N) = K
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Characteristic Function Form
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
The characteristic function ν of a game is a function that
assigns a value for each coalition (subset) of players.
If the game is constant-sum then
I
ν(∅) = 0
I
ν(N) = K
I
ν(S) + ν(N \ S) = K
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Games
Cooperative Game
Theory
Jennifer Wilson
Definition
A cooperative game is a set N of players, with a function
ν : 2N → < such that ν(∅) = 0.
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Cooperative Games
Jennifer Wilson
Outline
Definition
Introduction
A cooperative game is a set N of players, with a function
ν : 2N → < such that ν(∅) = 0.
A Survey of
Different Solution
Concepts
Definition
A cooperative game is called superadditive if
ν(S ∪ T ) ≥ ν(S) + ν(T )
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
for all S ∩ T = ∅.
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Cooperative Games
Jennifer Wilson
Outline
Definition
Introduction
A cooperative game is a set N of players, with a function
ν : 2N → < such that ν(∅) = 0.
A Survey of
Different Solution
Concepts
Definition
A cooperative game is called superadditive if
ν(S ∪ T ) ≥ ν(S) + ν(T )
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
for all S ∩ T = ∅.
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Every normal form game can be put in characteristic
(cooperative) form.
There are cooperative games that do not correspond to any
normal form games.
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Games
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
In non-cooperative game theory, the goals are to
I
determine the optimal strategies for each player, and to
I
find the equilibria of the game when the players play
their optimal strategies.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Games
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
In non-cooperative game theory, the goals are to
I
determine the optimal strategies for each player, and to
I
find the equilibria of the game when the players play
their optimal strategies.
In cooperative game theory the goals are to
I
determine which coalitions should form, and to
I
decide how each player should be rewarded.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Bankruptcy Game [O’Neill, 1982]
Cooperative Game
Theory
Jennifer Wilson
Example
A small company goes bankrupt owing money to three
creditors. It owns $10, 000 to creditor A, $20, 000 to creditor
B and $30, 000 to creditor C . If the company has a total of
$36, 000 in assets, how should the money be divided?
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Bankruptcy Game [O’Neill, 1982]
Cooperative Game
Theory
Jennifer Wilson
Example
Outline
A small company goes bankrupt owing money to three
creditors. It owns $10, 000 to creditor A, $20, 000 to creditor
B and $30, 000 to creditor C . If the company has a total of
$36, 000 in assets, how should the money be divided?
v (A) = 0 v (B) = 0 v (C ) = 6
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Bankruptcy Game [O’Neill, 1982]
Cooperative Game
Theory
Jennifer Wilson
Example
Outline
A small company goes bankrupt owing money to three
creditors. It owns $10, 000 to creditor A, $20, 000 to creditor
B and $30, 000 to creditor C . If the company has a total of
$36, 000 in assets, how should the money be divided?
v (A) = 0 v (B) = 0 v (C ) = 6
v (A, B) = 6 v (A, C ) = 16 v (B, C ) = 26 and v (A, B, C ) = 36
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Bankruptcy Game [O’Neill, 1982]
Cooperative Game
Theory
Jennifer Wilson
Example
Outline
A small company goes bankrupt owing money to three
creditors. It owns $10, 000 to creditor A, $20, 000 to creditor
B and $30, 000 to creditor C . If the company has a total of
$36, 000 in assets, how should the money be divided?
v (A) = 0 v (B) = 0 v (C ) = 6
v (A, B) = 6 v (A, C ) = 16 v (B, C ) = 26 and v (A, B, C ) = 36
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
What are the minimum requirements for a settlement?
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Bankruptcy Game [O’Neill, 1982]
Cooperative Game
Theory
Jennifer Wilson
Example
Outline
A small company goes bankrupt owing money to three
creditors. It owns $10, 000 to creditor A, $20, 000 to creditor
B and $30, 000 to creditor C . If the company has a total of
$36, 000 in assets, how should the money be divided?
v (A) = 0 v (B) = 0 v (C ) = 6
v (A, B) = 6 v (A, C ) = 16 v (B, C ) = 26 and v (A, B, C ) = 36
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
What are the minimum requirements for a settlement?
Let x1 , x2 and x3 be the payments offered to each creditor.
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Bankruptcy Game [O’Neill, 1982]
Cooperative Game
Theory
Jennifer Wilson
Example
Outline
A small company goes bankrupt owing money to three
creditors. It owns $10, 000 to creditor A, $20, 000 to creditor
B and $30, 000 to creditor C . If the company has a total of
$36, 000 in assets, how should the money be divided?
v (A) = 0 v (B) = 0 v (C ) = 6
v (A, B) = 6 v (A, C ) = 16 v (B, C ) = 26 and v (A, B, C ) = 36
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
What are the minimum requirements for a settlement?
Let x1 , x2 and x3 be the payments offered to each creditor.
x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36
.
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Bankruptcy Game [O’Neill, 1982]
A small company goes bankrupt owing money to three
creditors. It owns $10, 000 to creditor A, $20, 000 to creditor
B and $30, 000 to creditor C . If the company has a total of
$36, 000 in assets, how should the money be divided?
v (A) = 0 v (B) = 0 v (C ) = 6
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Bankruptcy Game [O’Neill, 1982]
A small company goes bankrupt owing money to three
creditors. It owns $10, 000 to creditor A, $20, 000 to creditor
B and $30, 000 to creditor C . If the company has a total of
$36, 000 in assets, how should the money be divided?
v (A) = 0 v (B) = 0 v (C ) = 6
v (A, B) = 6 v (A, C ) = 16 v (B, C ) = 26 and v (A, B, C ) = 36
x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36
.
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Bankruptcy Game [O’Neill, 1982]
A small company goes bankrupt owing money to three
creditors. It owns $10, 000 to creditor A, $20, 000 to creditor
B and $30, 000 to creditor C . If the company has a total of
$36, 000 in assets, how should the money be divided?
v (A) = 0 v (B) = 0 v (C ) = 6
v (A, B) = 6 v (A, C ) = 16 v (B, C ) = 26 and v (A, B, C ) = 36
x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36
.
What additional requirements should there be?
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Bankruptcy Game [O’Neill, 1982]
A small company goes bankrupt owing money to three
creditors. It owns $10, 000 to creditor A, $20, 000 to creditor
B and $30, 000 to creditor C . If the company has a total of
$36, 000 in assets, how should the money be divided?
v (A) = 0 v (B) = 0 v (C ) = 6
v (A, B) = 6 v (A, C ) = 16 v (B, C ) = 26 and v (A, B, C ) = 36
x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
.
What additional requirements should there be?
x1 + x2 ≥ 6
x1 + x3 ≥ 16 x2 + x3 ≥ 26
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Bankruptcy Game [O’Neill, 1982]
A small company goes bankrupt owing money to three
creditors. It owns $10, 000 to creditor A, $20, 000 to creditor
B and $30, 000 to creditor C . If the company has a total of
$36, 000 in assets, how should the money be divided?
v (A) = 0 v (B) = 0 v (C ) = 6
v (A, B) = 6 v (A, C ) = 16 v (B, C ) = 26 and v (A, B, C ) = 36
x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
.
What additional requirements should there be?
x1 + x2 ≥ 6
x1 + x3 ≥ 16 x2 + x3 ≥ 26
x3 ≤ 30 x2 ≤ 20
x1 ≤ 10
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Graphing on the Simplex
The 2-dimensional simplex
{(x1 , x2 , x3 ) | x1 + x2 + x3 = 1, x1 ≥ 0, x2 ≥ 0, and x3 ≥ 0.}.
..
...
x3 ....
....
....
....
..
0, 1)
...•...........(0,
..
.... .... ........
.....
.... ....
.....
.....
.... ....
.....
.....
... ...
... .......................................................•..............................................
.. ....
........ (0, 1, 0) x2
.
.
.
.
.
.
.
.... ......
.
..........
.......... .....................
..... .......
(1, 0, 0) ...•.........
.
....
x1 ......
.
.
....
Jennifer Wilson
Outline
Introduction
(0, 0, 1)
.•...
... ....
... .....
...
...
...
...
...
...
.
.
..
.. .......
... ................. x1 .....
......... ......... ...
..
x
.
2
.
.•..... ..
..
...
....
.. ..
...
.
.
.
.
...
x
.
.
3 ..
..
...
...
...
.
•...........................................................................................•.
(1, 0, 0)
(0, 1, 0)
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Graphing on the Simplex
The 2-dimensional simplex
{(x1 , x2 , x3 ) | x1 + x2 + x3 = 1, x1 ≥ 0, x2 ≥ 0, and x3 ≥ 0.}.
..
...
x3 ....
....
....
....
..
0, 1)
...•...........(0,
..
.... .... ........
.....
.... ....
.....
.....
.... ....
.....
.....
... ...
... .......................................................•..............................................
.. ....
........ (0, 1, 0) x2
.
.
.
.
.
.
.
.... ......
.
..........
.......... .....................
..... .......
(1, 0, 0) ...•.........
.
....
x1 ......
.
.
....
Jennifer Wilson
Outline
Introduction
(0, 0, 1)
.•...
... ....
... .....
...
...
...
...
...
...
.
.
..
.. .......
... ................. x1 .....
......... ......... ...
..
x
.
2
.
.•..... ..
..
...
....
.. ..
...
.
.
.
.
...
x
.
.
3 ..
..
...
...
...
.
•...........................................................................................•.
(1, 0, 0)
(0, 1, 0)
Possible “solutions” of the Bankrupcy Game.
....
... ...
... ...
x3 = 30/36.............................
. . . . . . ..
............................
................................
x1 = 10/36...................................................
. .. . . . . . . . . ..
... ..........................................................
..................................
...
.
.
...................................
.
... ................. ....x2 = 20/36
...
.............................. ...
...
... .............. ...
...
..................... ...
...
...............
.
...
.
.........
...
...
......
.
..
.
.................................................................................................................. x3 = 6/36
.
.
...
... ......
...
.
.
.
.
.
...........................................................................................................................
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Imputations
Cooperative Game
Theory
Jennifer Wilson
Definition
Given a cooperative game ν on n players, an imputation is a
payoff vector
x1 , x2 , . . . , xn
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Imputations
Jennifer Wilson
Outline
Definition
Given a cooperative game ν on n players, an imputation is a
payoff vector
x1 , x2 , . . . , xn
such that
xi ≥ ν(i)
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Imputations
Jennifer Wilson
Outline
Definition
Given a cooperative game ν on n players, an imputation is a
payoff vector
x1 , x2 , . . . , xn
such that
xi ≥ ν(i) individual rationality
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Imputations
Jennifer Wilson
Outline
Definition
Given a cooperative game ν on n players, an imputation is a
payoff vector
x1 , x2 , . . . , xn
such that
xi ≥ ν(i) individual rationality
and
X
i
xi = ν(N)
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Imputations
Jennifer Wilson
Outline
Definition
Given a cooperative game ν on n players, an imputation is a
payoff vector
x1 , x2 , . . . , xn
such that
xi ≥ ν(i) individual rationality
and
X
i
xi = ν(N)
collective rationality.
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Imputations
Jennifer Wilson
Outline
Definition
Given a cooperative game ν on n players, an imputation is a
payoff vector
x1 , x2 , . . . , xn
such that
xi ≥ ν(i) individual rationality
and
X
xi = ν(N)
collective rationality.
i
If
P
i
ν(i) = ν(N), then xi = ν(i).
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Imputations
Jennifer Wilson
Outline
Definition
Given a cooperative game ν on n players, an imputation is a
payoff vector
x1 , x2 , . . . , xn
such that
xi ≥ ν(i) individual rationality
and
X
xi = ν(N)
collective rationality.
i
If
P
i ν(i) = ν(N), then xi = ν(i).
These games are called inessential.
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Imputations
Jennifer Wilson
Outline
Definition
Given a cooperative game ν on n players, an imputation is a
payoff vector
x1 , x2 , . . . , xn
such that
xi ≥ ν(i) individual rationality
and
X
xi = ν(N)
collective rationality.
i
If
P
i ν(i) = ν(N), then xi = ν(i).
These games are called inessential.
We will assume ν is essential.
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
The Core
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
Definition
The core is the set of imputations such that
X
xi ≥ v (S) for every S ⊂ N.
i∈S
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Glove Market
Cooperative Game
Theory
Jennifer Wilson
Outline
Example
A market consists of people with left-handed glove and
people with right-handed gloves (but not both). The value of
a coalition is the number of complete pairs of gloves it has.
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Glove Market
Cooperative Game
Theory
Jennifer Wilson
Outline
Example
A market consists of people with left-handed glove and
people with right-handed gloves (but not both). The value of
a coalition is the number of complete pairs of gloves it has.
We will look at the case when A1 and A2 have left-handed
glove and B1 , B2 and B3 have right-handed gloves.
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Glove Market
Cooperative Game
Theory
Jennifer Wilson
Outline
Example
A market consists of people with left-handed glove and
people with right-handed gloves (but not both). The value of
a coalition is the number of complete pairs of gloves it has.
We will look at the case when A1 and A2 have left-handed
glove and B1 , B2 and B3 have right-handed gloves.
ν(Ai ) = ν(Bj ) = 0 ν(A1 , A2 ) = ν(Bi , Bj ) = 0 ν(Ai , Bi ) = 1
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
The Glove Market
Jennifer Wilson
Outline
Example
Introduction
A market consists of people with left-handed glove and
people with right-handed gloves (but not both). The value of
a coalition is the number of complete pairs of gloves it has.
We will look at the case when A1 and A2 have left-handed
glove and B1 , B2 and B3 have right-handed gloves.
ν(Ai ) = ν(Bj ) = 0 ν(A1 , A2 ) = ν(Bi , Bj ) = 0 ν(Ai , Bi ) = 1
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
ν(Ai , Bi1 , Bi2 ) = 1
...
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Glove Market
Let (x1 , x2 , y1 , y2 , y3 ) be the payoff vector.
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
The Glove Market
Let (x1 , x2 , y1 , y2 , y3 ) be the payoff vector.
It is an imputation if
xi ≥ 0,
yj ≥ 0
Jennifer Wilson
Outline
Introduction
y2 ≥ 0
x1 + x2 + y1 + y2 + y3 = 2.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
The Glove Market
Jennifer Wilson
Let (x1 , x2 , y1 , y2 , y3 ) be the payoff vector.
Outline
It is an imputation if
xi ≥ 0,
yj ≥ 0
Introduction
y2 ≥ 0
x1 + x2 + y1 + y2 + y3 = 2.
A Survey of
Different Solution
Concepts
The payoff vector is in the core if
x1 + y1 ≥ 1 x2 + y2 ≥ 1
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
...
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
The Glove Market
Jennifer Wilson
Let (x1 , x2 , y1 , y2 , y3 ) be the payoff vector.
Outline
It is an imputation if
xi ≥ 0,
yj ≥ 0
Introduction
y2 ≥ 0
x1 + x2 + y1 + y2 + y3 = 2.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
The payoff vector is in the core if
x1 + y1 ≥ 1 x2 + y2 ≥ 1
...
But
(x1 + y1 ) + (x2 + y2 ) ≥ 1 + 1 = 2.
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Hence
(x1 + y1 ) = (x2 + y2 ) = 1
and y3 = 0.
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
The Glove Market
Jennifer Wilson
Let (x1 , x2 , y1 , y2 , y3 ) be the payoff vector.
Outline
It is an imputation if
xi ≥ 0,
yj ≥ 0
Introduction
y2 ≥ 0
x1 + x2 + y1 + y2 + y3 = 2.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
The payoff vector is in the core if
x1 + y1 ≥ 1 x2 + y2 ≥ 1
...
But
(x1 + y1 ) + (x2 + y2 ) ≥ 1 + 1 = 2.
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Hence
(x1 + y1 ) = (x2 + y2 ) = 1
and y3 = 0.
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
So y1 = y2 = y3 = 0, and x1 = x2 = 1.
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
The Glove Market
Jennifer Wilson
Let (x1 , x2 , y1 , y2 , y3 ) be the payoff vector.
Outline
It is an imputation if
xi ≥ 0,
yj ≥ 0
Introduction
y2 ≥ 0
x1 + x2 + y1 + y2 + y3 = 2.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
The payoff vector is in the core if
x1 + y1 ≥ 1 x2 + y2 ≥ 1
...
But
(x1 + y1 ) + (x2 + y2 ) ≥ 1 + 1 = 2.
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Hence
(x1 + y1 ) = (x2 + y2 ) = 1
and y3 = 0.
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
So y1 = y2 = y3 = 0, and x1 = x2 = 1.
The core consists of the single point (1, 1, 0, 0, 0).
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Divide the Dollar
[ Von Neumann & Morgenstern, 1947]
Example
Three players are given a dollar to divide amongst them.
The decision is to be made by majority rule.
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Divide the Dollar
[ Von Neumann & Morgenstern, 1947]
Cooperative Game
Theory
Jennifer Wilson
Outline
Example
Introduction
Three players are given a dollar to divide amongst them.
The decision is to be made by majority rule.
ν(1) = ν(2) = ν(3) = 0.
ν(1, 2) = ν(1, 3) = ν(2, 3) = ν(1, 2, 3) = 1.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Divide the Dollar
[ Von Neumann & Morgenstern, 1947]
Jennifer Wilson
Outline
Example
Introduction
Three players are given a dollar to divide amongst them.
The decision is to be made by majority rule.
ν(1) = ν(2) = ν(3) = 0.
ν(1, 2) = ν(1, 3) = ν(2, 3) = ν(1, 2, 3) = 1.
x1 ≥ 0
Cooperative Game
Theory
x2 ≥ 0
x3 ≥ 0 x1 + x2 + x3 = 1
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Divide the Dollar
[ Von Neumann & Morgenstern, 1947]
Cooperative Game
Theory
Jennifer Wilson
Outline
Example
Introduction
Three players are given a dollar to divide amongst them.
The decision is to be made by majority rule.
ν(1) = ν(2) = ν(3) = 0.
ν(1, 2) = ν(1, 3) = ν(2, 3) = ν(1, 2, 3) = 1.
x1 ≥ 0
x2 ≥ 0
x1 + x2 ≥ 1
x3 ≥ 0 x1 + x2 + x3 = 1
x1 + x3 ≥ 1
x2 + x3 ≥ 1.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Divide the Dollar
[ Von Neumann & Morgenstern, 1947]
Cooperative Game
Theory
Jennifer Wilson
Outline
Example
Introduction
Three players are given a dollar to divide amongst them.
The decision is to be made by majority rule.
ν(1) = ν(2) = ν(3) = 0.
ν(1, 2) = ν(1, 3) = ν(2, 3) = ν(1, 2, 3) = 1.
x1 ≥ 0
x2 ≥ 0
x1 + x2 ≥ 1
This is impossible.
x3 ≥ 0 x1 + x2 + x3 = 1
x1 + x3 ≥ 1
x2 + x3 ≥ 1.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Divide the Dollar
[ Von Neumann & Morgenstern, 1947]
Cooperative Game
Theory
Jennifer Wilson
Outline
Example
Introduction
Three players are given a dollar to divide amongst them.
The decision is to be made by majority rule.
ν(1) = ν(2) = ν(3) = 0.
ν(1, 2) = ν(1, 3) = ν(2, 3) = ν(1, 2, 3) = 1.
x1 ≥ 0
x2 ≥ 0
x1 + x2 ≥ 1
x3 ≥ 0 x1 + x2 + x3 = 1
x1 + x3 ≥ 1
This is impossible.
Hence the core is empty.
x2 + x3 ≥ 1.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Divide the Dollar
[ Von Neumann & Morgenstern, 1947]
Cooperative Game
Theory
Jennifer Wilson
Outline
Example
Introduction
Three players are given a dollar to divide amongst them.
The decision is to be made by majority rule.
ν(1) = ν(2) = ν(3) = 0.
ν(1, 2) = ν(1, 3) = ν(2, 3) = ν(1, 2, 3) = 1.
x1 ≥ 0
x2 ≥ 0
x1 + x2 ≥ 1
x3 ≥ 0 x1 + x2 + x3 = 1
x1 + x3 ≥ 1
x2 + x3 ≥ 1.
This is impossible.
Hence the core is empty.
All constant-sum games have empty cores.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Divide the Dollar
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
Problem: If Players 1 and 2 agree to a (0.50, 0.50, 0) split,
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Divide the Dollar
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
Problem: If Players 1 and 2 agree to a (0.50, 0.50, 0) split,
Player 3 can offer Player 1 a (0.60, 0, 0.40) split.
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Divide the Dollar
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
Problem: If Players 1 and 2 agree to a (0.50, 0.50, 0) split,
Player 3 can offer Player 1 a (0.60, 0, 0.40) split.
Player 2 can retaliate by offering Player 3 a (0, 0.50, 0.50)
split.
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Divide the Dollar
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
Problem: If Players 1 and 2 agree to a (0.50, 0.50, 0) split,
Player 3 can offer Player 1 a (0.60, 0, 0.40) split.
Player 2 can retaliate by offering Player 3 a (0, 0.50, 0.50)
split.
Player 1 can offer....
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Dominance Relations
[Von Neumann and Morgenstern, 1947]
Definition
Imputation x is dominated by imputation y through coalition
S if
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Dominance Relations
[Von Neumann and Morgenstern, 1947]
Definition
Imputation x is dominated by imputation y through coalition
S if
I
I
yi > xi for every i ∈ S
P
ν(S) ≥ i∈S yi .
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Dominance Relations
[Von Neumann and Morgenstern, 1947]
Introduction
Imputation x is dominated by imputation y through coalition
S if
I
yi > xi for every i ∈ S
P
ν(S) ≥ i∈S yi .
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
Consider the two imputations:
x = (0.50, 0.50, 0)
Jennifer Wilson
Outline
Definition
I
Cooperative Game
Theory
and y = (0.60, 0, 0.40).
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Dominance Relations
[Von Neumann and Morgenstern, 1947]
Introduction
Imputation x is dominated by imputation y through coalition
S if
I
yi > xi for every i ∈ S
P
ν(S) ≥ i∈S yi .
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
Consider the two imputations:
x = (0.50, 0.50, 0)
Jennifer Wilson
Outline
Definition
I
Cooperative Game
Theory
and y = (0.60, 0, 0.40).
Players 1 and 3 can ’force’ player 2 to accept y.
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Dominance Relations
[Von Neumann and Morgenstern, 1947]
Introduction
Imputation x is dominated by imputation y through coalition
S if
I
yi > xi for every i ∈ S
P
ν(S) ≥ i∈S yi .
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
Consider the two imputations:
x = (0.50, 0.50, 0)
Jennifer Wilson
Outline
Definition
I
Cooperative Game
Theory
and y = (0.60, 0, 0.40).
Players 1 and 3 can ’force’ player 2 to accept y.
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
So (0.50, 0, 50, 0) is dominated by (0.60, 0, 0.40) through
S = {1, 3}
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Dominance Relations
[Von Neumann and Morgenstern, 1947]
Introduction
Imputation x is dominated by imputation y through coalition
S if
I
yi > xi for every i ∈ S
P
ν(S) ≥ i∈S yi .
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
Consider the two imputations:
x = (0.50, 0.50, 0)
Jennifer Wilson
Outline
Definition
I
Cooperative Game
Theory
and y = (0.60, 0, 0.40).
Players 1 and 3 can ’force’ player 2 to accept y.
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
So (0.50, 0, 50, 0) is dominated by (0.60, 0, 0.40) through
S = {1, 3} which is dominated by (0, 0.50, 0.50) through
coalition S = {2, 3} which is dominated by ...
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Dominance Relations
Cooperative Game
Theory
Jennifer Wilson
The core consists of all undominated imputations.
Outline
Proof.
Suppose x is dominated by y through the coalition S.
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Dominance Relations
Jennifer Wilson
The core consists of all undominated imputations.
Outline
Proof.
Introduction
Suppose x is dominated by y through the coalition S. Then
X
X
xi <
yi ≤ ν(S)
i∈S
i∈S
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Dominance Relations
Jennifer Wilson
The core consists of all undominated imputations.
Outline
Proof.
Introduction
Suppose x is dominated by y through the coalition S. Then
X
X
xi <
yi ≤ ν(S)
i∈S
hence x is not in the core.
i∈S
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Dominance Relations
Jennifer Wilson
The core consists of all undominated imputations.
Outline
Proof.
Introduction
Suppose x is dominated by y through the coalition S. Then
X
X
xi <
yi ≤ ν(S)
i∈S
i∈S
hence x is not in the core.
Suppose
x is an imputation that is not in the core and
P
x
<
ν(S).
i
i∈S
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Dominance Relations
Jennifer Wilson
The core consists of all undominated imputations.
Outline
Proof.
Introduction
Suppose x is dominated by y through the coalition S. Then
X
X
xi <
yi ≤ ν(S)
i∈S
i∈S
hence x is not in the core.
Suppose
x is an imputation that is not in the core and
P
x
<
ν(S).
i
i∈S
Let
P
1
xi + |S|
[ν(S) − i∈S xi ]
if i ∈ S
P
yi =
1
if i ∈
/ S.
ν(i) + |N\S| [ν(N) − (ν(S) + i ∈S
/ ν(i))]
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Dominance Relations
Jennifer Wilson
The core consists of all undominated imputations.
Outline
Proof.
Introduction
Suppose x is dominated by y through the coalition S. Then
X
X
xi <
yi ≤ ν(S)
i∈S
i∈S
hence x is not in the core.
Suppose
x is an imputation that is not in the core and
P
x
<
ν(S).
i
i∈S
Let
P
1
xi + |S|
[ν(S) − i∈S xi ]
if i ∈ S
P
yi =
1
if i ∈
/ S.
ν(i) + |N\S| [ν(N) − (ν(S) + i ∈S
/ ν(i))]
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Dominance Relations
Jennifer Wilson
Outline
Introduction
yi =
P
1
xi + |S|
[ν(S) − i∈S xi ]
1
[ν(N) − (ν(S)
ν(i) + |N\S|
Claim:
I
yi > xi for all i ∈ S
+
P
i ∈S
/ ν(i))]
if i ∈ S
if i ∈
/ S.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Dominance Relations
Jennifer Wilson
Outline
Introduction
yi =
P
1
xi + |S|
[ν(S) − i∈S xi ]
1
[ν(N) − (ν(S)
ν(i) + |N\S|
Claim:
I
I
yi > xi for all i ∈ S
P
ν(S) ≥ i∈S yi
+
P
i ∈S
/ ν(i))]
if i ∈ S
if i ∈
/ S.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Dominance Relations
Jennifer Wilson
Outline
Introduction
yi =
P
1
xi + |S|
[ν(S) − i∈S xi ]
1
[ν(N) − (ν(S)
ν(i) + |N\S|
Claim:
I
I
I
yi > xi for all i ∈ S
P
ν(S) ≥ i∈S yi
y is an imputation
+
P
i ∈S
/ ν(i))]
if i ∈ S
if i ∈
/ S.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Dominance Relations
Jennifer Wilson
Outline
Introduction
yi =
P
1
xi + |S|
[ν(S) − i∈S xi ]
1
[ν(N) − (ν(S)
ν(i) + |N\S|
Claim:
I
I
I
yi > xi for all i ∈ S
P
ν(S) ≥ i∈S yi
y is an imputation
I
yi ≥ ν(i) for all i
+
P
i ∈S
/ ν(i))]
if i ∈ S
if i ∈
/ S.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Dominance Relations
Jennifer Wilson
Outline
Introduction
yi =
P
1
xi + |S|
[ν(S) − i∈S xi ]
1
[ν(N) − (ν(S)
ν(i) + |N\S|
Claim:
I
I
I
yi > xi for all i ∈ S
P
ν(S) ≥ i∈S yi
y is an imputation
I
I
yP
i ≥ ν(i) for all i
i yi = ν(N)
+
P
i ∈S
/ ν(i))]
if i ∈ S
if i ∈
/ S.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Dominance Relations
Jennifer Wilson
Outline
Introduction
yi =
P
1
xi + |S|
[ν(S) − i∈S xi ]
1
[ν(N) − (ν(S)
ν(i) + |N\S|
+
P
i ∈S
/ ν(i))]
Claim:
I
I
I
yi > xi for all i ∈ S
P
ν(S) ≥ i∈S yi
y is an imputation
I
I
yP
i ≥ ν(i) for all i
i yi = ν(N)
Hence y is an imputation that dominates x.
if i ∈ S
if i ∈
/ S.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Stable Sets
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Definition
A stable set is a set I of imputations such that
I
no imputation in I is dominated by any other
imputation in I
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Stable Sets
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Definition
A stable set is a set I of imputations such that
I
no imputation in I is dominated by any other
imputation in I internally stable
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Stable Sets
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Definition
A stable set is a set I of imputations such that
I
no imputation in I is dominated by any other
imputation in I internally stable
I
every imputation not in I is dominated by some
imputation in I .
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Stable Sets
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Definition
A stable set is a set I of imputations such that
I
no imputation in I is dominated by any other
imputation in I internally stable
I
every imputation not in I is dominated by some
imputation in I . externally stable
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Stable Sets and Divide the Dollar
Examples:
I1 = {(0.50, 0.50, 0), (0.50, 0, 0.50), (0, 0.50, 0.50)}
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Stable Sets and Divide the Dollar
Examples:
Cooperative Game
Theory
Jennifer Wilson
I1 = {(0.50, 0.50, 0), (0.50, 0, 0.50), (0, 0.50, 0.50)}
Outline
Introduction
I2 = {0.7, x, 0.3 − x | 0 ≤ x ≤ 0.7}
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Stable Sets and Divide the Dollar
Examples:
Jennifer Wilson
I1 = {(0.50, 0.50, 0), (0.50, 0, 0.50), (0, 0.50, 0.50)}
Outline
Introduction
I2 = {0.7, x, 0.3 − x | 0 ≤ x ≤ 0.7}
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
C=(0,0,1)
.
...•....
... ...
... ......
...
...
...
...
...
...
.
.
...
..
...
...
...
...
...
.
.
.•.. (0,1/2,1/2)
..
(1/2,0,1/2) ..•
...
.
...
...
...
.
.
...
...
...
.
.
...
...
...
.
.
...
...
...
.
.
...
...
.
...
.
.
•............................................................•.............................................................•.
A=(1,0,0)
(1/2,1/2,0)
B=(0,1,0)
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
C=(0,0,1)
.
...•....
... ...
... ......
...
...
...
...
...
...
.
...
.
..
...
...
...
...
...
.
.
...
...
...
.
.
...
...
......
.
.
...
... ......
...
.
.
...
...
...
...
.
.
.
.
...
...
...
...
...
.
.
...
...
..
...
...
.....
•.........................................................................................................................•.
A=(1,0,0)
{0.7, x2 , 0.3 − x2 }
B=(0,1,0)
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Other Solution concepts
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
I
Bargaining Set
I
Kernel
I
Nucleolus
I
Shapley Value
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Other Solution concepts
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
I
Bargaining Set
I
Kernel
I
Nucleolus
I
Shapley Value
Several solution concepts are based on the idea of the
“excess” of an imputation.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Other Solution concepts
Jennifer Wilson
Outline
Introduction
I
Bargaining Set
I
Kernel
I
Nucleolus
I
Shapley Value
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
Several solution concepts are based on the idea of the
“excess” of an imputation.
X
e(x, S) = ν(S) −
xi
i∈S
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value
[Shapley, 1953]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Given a cooperative game ν on N players, there exists a
unique map φ : N → < which assigns to each player i a
value φi (v ) satisfying
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value
[Shapley, 1953]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Given a cooperative game ν on N players, there exists a
unique map φ : N → < which assigns to each player i a
value φi (v ) satisfying
I
If π is a permutation of the players then
φi (νπ ) = φπ(i) (ν).
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value
[Shapley, 1953]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Given a cooperative game ν on N players, there exists a
unique map φ : N → < which assigns to each player i a
value φi (v ) satisfying
I
If π is a permutation of the players then
φi (νπ ) = φπ(i) (ν). Symmetry
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value
[Shapley, 1953]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Given a cooperative game ν on N players, there exists a
unique map φ : N → < which assigns to each player i a
value φi (v ) satisfying
I
If π is a permutation of the players then
φi (νπ ) = φπ(i) (ν). Symmetry
I
If i is a dummy player then φi (ν) = 0.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value
[Shapley, 1953]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Given a cooperative game ν on N players, there exists a
unique map φ : N → < which assigns to each player i a
value φi (v ) satisfying
I
If π is a permutation of the players then
φi (νπ ) = φπ(i) (ν). Symmetry
I
If i is a dummy player then φi (ν) = 0.
Dummy Property
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value
[Shapley, 1953]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Given a cooperative game ν on N players, there exists a
unique map φ : N → < which assigns to each player i a
value φi (v ) satisfying
I
If π is a permutation of the players then
φi (νπ ) = φπ(i) (ν). Symmetry
I
If i is a dummy player then φi (ν) = 0.
Dummy Property
(Player i is a dummy if ν(S ∪ i) = ν(S) for all S ⊂ N.)
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value
[Shapley, 1953]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Given a cooperative game ν on N players, there exists a
unique map φ : N → < which assigns to each player i a
value φi (v ) satisfying
I
If π is a permutation of the players then
φi (νπ ) = φπ(i) (ν). Symmetry
I
If i is a dummy player then φi (ν) = 0.
Dummy Property
I
(Player i is a dummy if ν(S ∪ i) = ν(S) for all S ⊂ N.)
P
i∈N φi (ν) = ν(N)
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value
[Shapley, 1953]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Given a cooperative game ν on N players, there exists a
unique map φ : N → < which assigns to each player i a
value φi (v ) satisfying
I
If π is a permutation of the players then
φi (νπ ) = φπ(i) (ν). Symmetry
I
If i is a dummy player then φi (ν) = 0.
Dummy Property
I
(Player i is a dummy if ν(S ∪ i) = ν(S) for all S ⊂ N.)
P
Efficiency
i∈N φi (ν) = ν(N)
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value
[Shapley, 1953]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Given a cooperative game ν on N players, there exists a
unique map φ : N → < which assigns to each player i a
value φi (v ) satisfying
I
If π is a permutation of the players then
φi (νπ ) = φπ(i) (ν). Symmetry
I
If i is a dummy player then φi (ν) = 0.
Dummy Property
I
(Player i is a dummy if ν(S ∪ i) = ν(S) for all S ⊂ N.)
P
Efficiency
i∈N φi (ν) = ν(N)
I
φi (ν + w ) = φi (ν) + φi (w ).
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
The Shapley Value
[Shapley, 1953]
Jennifer Wilson
Outline
Introduction
Given a cooperative game ν on N players, there exists a
unique map φ : N → < which assigns to each player i a
value φi (v ) satisfying
I
If π is a permutation of the players then
φi (νπ ) = φπ(i) (ν). Symmetry
I
If i is a dummy player then φi (ν) = 0.
Dummy Property
I
(Player i is a dummy if ν(S ∪ i) = ν(S) for all S ⊂ N.)
P
Efficiency
i∈N φi (ν) = ν(N)
I
φi (ν + w ) = φi (ν) + φi (w ).
Linearity
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Shapley Value
Outline of Proof:
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Shapley Value
Outline of Proof:
Let T ⊂ N, let νT be the cooperative game
1 if T ⊆ S
νT (S) =
0 if T * S.
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Shapley Value
Outline of Proof:
Let T ⊂ N, let νT be the cooperative game
1 if T ⊆ S
νT (S) =
0 if T * S.
Let
φi (νT ) =
1
|T |
0
if i ∈ T
if i ∈
/ T.
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Shapley Value
Outline of Proof:
Let T ⊂ N, let νT be the cooperative game
1 if T ⊆ S
νT (S) =
0 if T * S.
Let
φi (νT ) =
1
|T |
0
if i ∈ T
if i ∈
/ T.
The games νT form a basis for the space of cooperative
games on N,
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Shapley Value
Outline of Proof:
Let T ⊂ N, let νT be the cooperative game
1 if T ⊆ S
νT (S) =
0 if T * S.
Let
φi (νT ) =
1
|T |
0
if i ∈ T
if i ∈
/ T.
The games νT form a basis for the space of cooperative
games on N,
X
X
ν=
cT νT where cT =
(−1)t−s ν(S).
T ⊂N
S⊂T
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Shapley Value
Outline of Proof:
Let T ⊂ N, let νT be the cooperative game
1 if T ⊆ S
νT (S) =
0 if T * S.
Let
φi (νT ) =
1
|T |
0
if i ∈ T
if i ∈
/ T.
The games νT form a basis for the space of cooperative
games on N,
X
X
ν=
cT νT where cT =
(−1)t−s ν(S).
T ⊂N
φi (ν) =
S⊂T
X s!(n − 1 − s)!
[ν(S ∪ i) − ν(S)].
n!
S⊂N
i ∈S
/
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Suppose a river is divided into n segments numbered from
upstream to downstream, with one corporation/polluter in
each segment. Local environmental agencies dictate the cost
ci in the i th section of the river.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Suppose a river is divided into n segments numbered from
upstream to downstream, with one corporation/polluter in
each segment. Local environmental agencies dictate the cost
ci in the i th section of the river.
Under the LR (Local Responsibility) principle, each player
situated along the i th section of the river is responsible for
the cleanup of the i th section of the river.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Suppose a river is divided into n segments numbered from
upstream to downstream, with one corporation/polluter in
each segment. Local environmental agencies dictate the cost
ci in the i th section of the river.
Under the LR (Local Responsibility) principle, each player
situated along the i th section of the river is responsible for
the cleanup of the i th section of the river.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Under the DR (Downstream Responsibility) principle, each
player situated along the i th section of the river is responsible
for the cleanup of the i th through nth sections of the river.
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Under the LR (Local Responsibility) principle, each player
situated along the i th section of the river is responsible for
the cleanup of the i th section of the river.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Under the LR (Local Responsibility) principle, each player
situated along the i th section of the river is responsible for
the cleanup of the i th section of the river.
For each S ⊂ N, let
v C (S) =
X
i∈S
ci .
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Under the LR (Local Responsibility) principle, each player
situated along the i th section of the river is responsible for
the cleanup of the i th section of the river.
For each S ⊂ N, let
v C (S) =
X
ci .
i∈S
Claim
φi (v ) = ci .
C
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Jennifer Wilson
Outline
Proof.
Introduction
(LR)
φi (v C ) =
Cooperative Game
Theory
X s!(n − 1 − s)!
[v (S ∪ i) − v (S)]
n!
S⊂N
i ∈S
/
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Cooperative Game
Theory
Jennifer Wilson
Outline
Proof.
Introduction
(LR)
φi (v C ) =
=
X s!(n − 1 − s)!
[v (S ∪ i) − v (S)]
n!
S⊂N
i ∈S
/
n−1
X
s=0
s!(n − 1 − s)! X
ci
n!
S⊂N\i
|S|=s
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Jennifer Wilson
Outline
Proof.
Introduction
(LR)
φi (v C ) =
=
X s!(n − 1 − s)!
[v (S ∪ i) − v (S)]
n!
S⊂N
i ∈S
/
n−1
X
s=0
= ci
s!(n − 1 − s)! X
ci
n!
S⊂N\i
|S|=s
n−1
X
s!(n − 1 − s)!
s=0
n!
(n − 1)!
s!(n − 1 − s)!
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Jennifer Wilson
Outline
Proof.
Introduction
(LR)
φi (v C ) =
=
X s!(n − 1 − s)!
[v (S ∪ i) − v (S)]
n!
S⊂N
i ∈S
/
n−1
X
s=0
= ci
= ci
s!(n − 1 − s)! X
ci
n!
S⊂N\i
|S|=s
n−1
X
s!(n − 1 − s)!
s=0
n−1
X
s=0
n!
1
= ci
n
(n − 1)!
s!(n − 1 − s)!
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Jennifer Wilson
Outline
Proof.
Introduction
(LR)
φi (v C ) =
=
X s!(n − 1 − s)!
[v (S ∪ i) − v (S)]
n!
S⊂N
i ∈S
/
n−1
X
s=0
= ci
= ci
s!(n − 1 − s)! X
ci
n!
S⊂N\i
|S|=s
n−1
X
s!(n − 1 − s)!
s=0
n−1
X
s=0
n!
1
= ci
n
(n − 1)!
s!(n − 1 − s)!
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Under the DR (Downstream Responsibility) principle, each
player situated along the i th section of the river is responsible
for the cleanup of the i th through nth sections of the river.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Under the DR (Downstream Responsibility) principle, each
player situated along the i th section of the river is responsible
for the cleanup of the i th through nth sections of the river.
For each S ⊂ N, let
v C (S) =
n
X
i=minS
ci .
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Under the DR (Downstream Responsibility) principle, each
player situated along the i th section of the river is responsible
for the cleanup of the i th through nth sections of the river.
For each S ⊂ N, let
v C (S) =
n
X
ci .
i=minS
Claim
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
φi (v C ) =
X1
j≥i
1
1
1
cj = ci +
ci+1 · · · + cn .
j
i
i +1
n
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Proof.
(DR) LetCj = (0, . . . , 0, cj , 0, . . . , 0),
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
(DR) LetCj = (0, . . . , 0, cj , 0, . . . , 0), then
j
Jennifer Wilson
Outline
Proof.
v C (S) = 0 if minS > j
Cooperative Game
Theory
j
and v C (S) = cj if minS ≤ j.
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
(DR) LetCj = (0, . . . , 0, cj , 0, . . . , 0), then
j
Jennifer Wilson
Outline
Proof.
v C (S) = 0 if minS > j
Cooperative Game
Theory
j
and v C (S) = cj if minS ≤ j.
So player i is a dummy if i > j, and the game is symmetric
for all players i ≤ j.
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Cooperative Game
Theory
Jennifer Wilson
Outline
Proof.
(DR) LetCj = (0, . . . , 0, cj , 0, . . . , 0), then
j
v C (S) = 0 if minS > j
j
and v C (S) = cj if minS ≤ j.
So player i is a dummy if i > j, and the game is symmetric
for all players i ≤ j.
Hence
j
φi (v C ) = 0 if i > j
1
j
and φi (v C ) = cj if i ≤ j.
j
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Cooperative Game
Theory
Jennifer Wilson
Outline
Proof.
(DR) LetCj = (0, . . . , 0, cj , 0, . . . , 0), then
j
v C (S) = 0 if minS > j
j
and v C (S) = cj if minS ≤ j.
So player i is a dummy if i > j, and the game is symmetric
for all players i ≤ j.
Hence
j
φi (v C ) = 0 if i > j
Since
vC =
X
j
1
j
and φi (v C ) = cj if i ≤ j.
j
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
v
Cj
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Sharing Costs of River Cleanup
[Ni & Wang, 2007]
Jennifer Wilson
Outline
Proof.
Introduction
(DR) LetCj = (0, . . . , 0, cj , 0, . . . , 0), then
j
v C (S) = 0 if minS > j
j
and v C (S) = cj if minS ≤ j.
So player i is a dummy if i > j, and the game is symmetric
for all players i ≤ j.
Hence
j
φi (v C ) = 0 if i > j
1
j
and φi (v C ) = cj if i ≤ j.
j
Since
vC =
X
j
v
Cj
φi (v C ) =
X
j
Cj
φi (v ) =
X1
j≥i
j
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
cj .
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value and Simple Games
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Example
A local county board consists of 4 representatives of different
townships with voting weight equal to 3, 1, 2 and 5
respectively. A motion is passed if it receives at least 6 votes.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value and Simple Games
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Example
A local county board consists of 4 representatives of different
townships with voting weight equal to 3, 1, 2 and 5
respectively. A motion is passed if it receives at least 6 votes.
Denote the representatives a, b, c and d.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value and Simple Games
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Example
A local county board consists of 4 representatives of different
townships with voting weight equal to 3, 1, 2 and 5
respectively. A motion is passed if it receives at least 6 votes.
Denote the representatives a, b, c and d.
Winning coalitions: {a, d}, {b, d}, {c, d}, {a, b, c},
{a, b, d}, {a, c, d},{b, c, d} and {a, b, c, d}.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value and Simple Games
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Example
A local county board consists of 4 representatives of different
townships with voting weight equal to 3, 1, 2 and 5
respectively. A motion is passed if it receives at least 6 votes.
Denote the representatives a, b, c and d.
Winning coalitions: {a, d}, {b, d}, {c, d}, {a, b, c},
{a, b, d}, {a, c, d},{b, c, d} and {a, b, c, d}.
1 if S is winning
ν(S) =
0 else
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value and Simple Games
Winning coalitions: {a, d}, {b, d}, {c, d}, {a, b, c},
{a, b, d}, {a, c, d},{b, c, d} and {a, b, c, d}.
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value and Simple Games
Winning coalitions: {a, d}, {b, d}, {c, d}, {a, b, c},
{a, b, d}, {a, c, d},{b, c, d} and {a, b, c, d}.
Applying the Shapley value to this game for Player a,
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value and Simple Games
Winning coalitions: {a, d}, {b, d}, {c, d}, {a, b, c},
{a, b, d}, {a, c, d},{b, c, d} and {a, b, c, d}.
Applying the Shapley value to this game for Player a,
X s!(n − 1 − s)!
φa (ν) =
[ν(S ∪ a) − ν(S)].
n!
S⊂N
a∈S
/
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value and Simple Games
Winning coalitions: {a, d}, {b, d}, {c, d}, {a, b, c},
{a, b, d}, {a, c, d},{b, c, d} and {a, b, c, d}.
Applying the Shapley value to this game for Player a,
X s!(n − 1 − s)!
φa (ν) =
[ν(S ∪ a) − ν(S)].
n!
S⊂N
a∈S
/
The only non-zero terms are those in which S ∪ a is winning
and S is not winning. (Player a is “critical.”)
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value and Simple Games
Winning coalitions: {a, d}, {b, d}, {c, d}, {a, b, c},
{a, b, d}, {a, c, d},{b, c, d} and {a, b, c, d}.
Applying the Shapley value to this game for Player a,
X s!(n − 1 − s)!
φa (ν) =
[ν(S ∪ a) − ν(S)].
n!
S⊂N
a∈S
/
The only non-zero terms are those in which S ∪ a is winning
and S is not winning. (Player a is “critical.”) S = {d} or
S = {b, c},
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value and Simple Games
Winning coalitions: {a, d}, {b, d}, {c, d}, {a, b, c},
{a, b, d}, {a, c, d},{b, c, d} and {a, b, c, d}.
Applying the Shapley value to this game for Player a,
X s!(n − 1 − s)!
φa (ν) =
[ν(S ∪ a) − ν(S)].
n!
S⊂N
a∈S
/
The only non-zero terms are those in which S ∪ a is winning
and S is not winning. (Player a is “critical.”) S = {d} or
S = {b, c},
φa (ν) =
1!2!
2!1!
1
[1 − 0] +
[1 − 0] =
24!
24!
6
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value and Simple Games
Winning coalitions: {a, d}, {b, d}, {c, d}, {a, b, c},
{a, b, d}, {a, c, d},{b, c, d} and {a, b, c, d}.
Applying the Shapley value to this game for Player a,
X s!(n − 1 − s)!
φa (ν) =
[ν(S ∪ a) − ν(S)].
n!
S⊂N
a∈S
/
The only non-zero terms are those in which S ∪ a is winning
and S is not winning. (Player a is “critical.”) S = {d} or
S = {b, c},
φa (ν) =
1!2!
2!1!
1
[1 − 0] +
[1 − 0] =
24!
24!
6
Similarly,
1
φb (ν) =
6
1
φc (ν) =
6
1
φd (ν) = .
3
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley Value and Simple Games
Winning coalitions: {a, d}, {b, d}, {c, d}, {a, b, c},
{a, b, d}, {a, c, d},{b, c, d} and {a, b, c, d}.
Applying the Shapley value to this game for Player a,
X s!(n − 1 − s)!
φa (ν) =
[ν(S ∪ a) − ν(S)].
n!
S⊂N
a∈S
/
The only non-zero terms are those in which S ∪ a is winning
and S is not winning. (Player a is “critical.”) S = {d} or
S = {b, c},
φa (ν) =
1!2!
2!1!
1
[1 − 0] +
[1 − 0] =
24!
24!
6
Similarly,
1
φb (ν) =
6
1
φc (ν) =
6
1
φd (ν) = .
3
This is known as the Shapley-Shubik Power Index.
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The United Nations Security Council
Cooperative Game
Theory
Jennifer Wilson
Example
United Nations Security Council consists of 5 permanent
members and 10 non-permanent members. Resolutions are
passed if they are supported by all 5 permanent members
and at least 4 of the non-permanent members.
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The United Nations Security Council
Cooperative Game
Theory
Jennifer Wilson
Example
United Nations Security Council consists of 5 permanent
members and 10 non-permanent members. Resolutions are
passed if they are supported by all 5 permanent members
and at least 4 of the non-permanent members.
Let P be the set of permanent members and T be the set of
non-permanent members.
1 if | S ∩ P |= 5 and | S ∩ T |≥ 4
ν(S) =
0 else
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The United Nations Security Council
Cooperative Game
Theory
Jennifer Wilson
Outline
Example
United Nations Security Council consists of 5 permanent
members and 10 non-permanent members. Resolutions are
passed if they are supported by all 5 permanent members
and at least 4 of the non-permanent members.
Let P be the set of permanent members and T be the set of
non-permanent members.
1 if | S ∩ P |= 5 and | S ∩ T |≥ 4
ν(S) =
0 else
After much combinatorics,
φP (ν) = 0.1974
φT (ν) = 0.0022.
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley-Shubik Power Index
Proposition (Straffin, 1977)
The Shapley-Shubik value is equal to the probability that a
player’s vote will make a difference, given that each player
has an equal probability of voting ’yes’ or ’no’.
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley-Shubik Power Index
Proposition (Straffin, 1977)
The Shapley-Shubik value is equal to the probability that a
player’s vote will make a difference, given that each player
has an equal probability of voting ’yes’ or ’no’.
The probability that coalition S forms is equal to
Z 1
s!(n − s)!
P(S) =
p s (1 − p)n−s dp =
(n + 1)!
0
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
The Shapley-Shubik Power Index
Proposition (Straffin, 1977)
The Shapley-Shubik value is equal to the probability that a
player’s vote will make a difference, given that each player
has an equal probability of voting ’yes’ or ’no’.
The probability that coalition S forms is equal to
Z 1
s!(n − s)!
P(S) =
p s (1 − p)n−s dp =
(n + 1)!
0
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Hence player i will make a difference with probability
X
X s!(n − s)! (s + 1)!(n − 1 − s)! Shapley Value
P(S) + P(S ∪ i) =
+
Definition
(n + 1)!
(n + 1)!
River Cleanup
Shapley-Shubik Power
X s!(n − 1 − s)!
Index
=
[(n − s) + (s + 1)] UN Security Council
(n + 1)!
Multichoice Games
Extensions of
X s!(n − 1 − s)!
Cooperative Game
Theory
=
Definitions
n!
Examples
Extensions of the
Shapley Value
Extensions of Cooperative Game Theory
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
I
Multichoice Games [Hsiao & Raghaven, 1993]
I
Ternary Voting Games [Felsenthal & Machover, 1997]
I
Fuzzy Games
I
(j, k) Games [Zwicker & Freixas, 2003]
I
Games over Lattices [Grabisch & Lange, 2007]
Models in Cooperative Game Theory, Branzei, Dimitrov &
TIjs, 2005
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Multichoice Games
Cooperative Game
Theory
Jennifer Wilson
Definition (Hsiao & Raghaven, 1993)
A multichoice game is a game in which n players
N = {1, 2, . . . , n} select among m + 1 levels of participation
M = {0, 1, . . . , m}, and v : M n → R is the characteristic or
value function satisfying v (0, 0, . . . , 0) = 0.
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Multichoice Games
Cooperative Game
Theory
Jennifer Wilson
Definition (Hsiao & Raghaven, 1993)
A multichoice game is a game in which n players
N = {1, 2, . . . , n} select among m + 1 levels of participation
M = {0, 1, . . . , m}, and v : M n → R is the characteristic or
value function satisfying v (0, 0, . . . , 0) = 0.
Comments:
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Multichoice Games
Cooperative Game
Theory
Jennifer Wilson
Definition (Hsiao & Raghaven, 1993)
A multichoice game is a game in which n players
N = {1, 2, . . . , n} select among m + 1 levels of participation
M = {0, 1, . . . , m}, and v : M n → R is the characteristic or
value function satisfying v (0, 0, . . . , 0) = 0.
Comments:
I
given x = (x1 , x2 , . . . , xn ) ∈ M n , write v (x1 , x2 , . . . , xn )
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Multichoice Games
Cooperative Game
Theory
Jennifer Wilson
Definition (Hsiao & Raghaven, 1993)
A multichoice game is a game in which n players
N = {1, 2, . . . , n} select among m + 1 levels of participation
M = {0, 1, . . . , m}, and v : M n → R is the characteristic or
value function satisfying v (0, 0, . . . , 0) = 0.
Comments:
I
given x = (x1 , x2 , . . . , xn ) ∈ M n , write v (x1 , x2 , . . . , xn )
I
each x = (x1 , x2 , . . . , xn ) corresponds to a partition of
x , where S x = {i ∈ N : x = j}
the players S0x , S1x , . . . , Sm
i
j
is the set of players acting at level j.
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Multichoice Games
Cooperative Game
Theory
Jennifer Wilson
Definition (Hsiao & Raghaven, 1993)
A multichoice game is a game in which n players
N = {1, 2, . . . , n} select among m + 1 levels of participation
M = {0, 1, . . . , m}, and v : M n → R is the characteristic or
value function satisfying v (0, 0, . . . , 0) = 0.
Comments:
I
given x = (x1 , x2 , . . . , xn ) ∈ M n , write v (x1 , x2 , . . . , xn )
I
each x = (x1 , x2 , . . . , xn ) corresponds to a partition of
x , where S x = {i ∈ N : x = j}
the players S0x , S1x , . . . , Sm
i
j
is the set of players acting at level j.
I
x ≥ y if and only if xi ≥ yi for all i
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Multichoice Games
Jennifer Wilson
Definition (Hsiao & Raghaven, 1993)
A multichoice game is a game in which n players
N = {1, 2, . . . , n} select among m + 1 levels of participation
M = {0, 1, . . . , m}, and v : M n → R is the characteristic or
value function satisfying v (0, 0, . . . , 0) = 0.
Comments:
I
given x = (x1 , x2 , . . . , xn ) ∈ M n , write v (x1 , x2 , . . . , xn )
I
each x = (x1 , x2 , . . . , xn ) corresponds to a partition of
x , where S x = {i ∈ N : x = j}
the players S0x , S1x , . . . , Sm
i
j
is the set of players acting at level j.
I
x ≥ y if and only if xi ≥ yi for all i
I
assume v monotonic
v (x) ≥ v (y) if x ≥ y
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Examples
Cooperative Game
Theory
Jennifer Wilson
Example
United Nations Security Council resolutions on
non-procedural issues are approved if nine members support
it and all permanent members ‘concur.’ (n = 15 and m = 2)
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Examples
Cooperative Game
Theory
Jennifer Wilson
Example
United Nations Security Council resolutions on
non-procedural issues are approved if nine members support
it and all permanent members ‘concur.’ (n = 15 and m = 2)
Example (Hsiao & Raghaven, 1993)
A mathematics department with 50 faculty including 10
distinguished professors must vote to promote a junior
colleague. The colleague will be promoted if
I
I
At least 40 faculty marginally or strongly support the
candidate and at least 2 distinguished professors attend
the meeting
At least 25 faculty strongly support the candidate
including at least 1 distinguished professor
(n = 50 and m = 3)
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Examples
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
I
using weights [Hsiao & Raghavan, 1993]
I
using maximal chains [Nouweland & Tijs, 1995]
I
using roll-calls [Freixas, 2005, Felshenthal & Machover,
1997]
I
using axiomatic characterizations [Peters & Zenk, 2005,
Grabisch & Lange, 2007]
I
using multilinear extensions [Jones & Wilson, preprint]
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Shapley Value [Hsiao & Raghavan, 1993]
I
based on weights 0 = w0 ≤ w1 ≤ · · · ≤ wm
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Shapley Value [Hsiao & Raghavan, 1993]
I
based on weights 0 = w0 ≤ w1 ≤ · · · ≤ wm
I
defined on unanimity games vy where vy (x) = 1 iff
x≥y
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Shapley Value [Hsiao & Raghavan, 1993]
I
based on weights 0 = w0 ≤ w1 ≤ · · · ≤ wm
I
defined on unanimity games vy where vy (x) = 1 iff
x≥y
( wy
P i
ifj ≥ yi
wy k
τi,j (vy ) =
0
ifj < yi
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Shapley Value [Hsiao & Raghavan, 1993]
I
based on weights 0 = w0 ≤ w1 ≤ · · · ≤ wm
I
defined on unanimity games vy where vy (x) = 1 iff
x≥y
( wy
P i
ifj ≥ yi
wy k
τi,j (vy ) =
0
ifj < yi
I
extend linearly
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Cooperative Game
Theory
Extensions of the Shapley Value
Jennifer Wilson
Shapley Value [Hsiao & Raghavan, 1993]
Outline
I
based on weights 0 = w0 ≤ w1 ≤ · · · ≤ wm
I
defined on unanimity games vy where vy (x) = 1 iff
x≥y
( wy
P i
ifj ≥ yi
wy k
τi,j (vy ) =
0
ifj < yi
I
extend linearly
X
τi,j (v ) =
X
X
(−1)t
k≤j xn−1 ∈M n−1 T ⊂Bi (x−i )
||(x−i , k)||w +
×[v (x−i , k) − v (x−i , k − 1)]
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
P
w
k Value
Shapley
[w (x
r ∈T
River Cleanupr
Definition
+1
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
where Bi (x−i ) = {i 0 ∈ N \ {i}|i 0 6= m} and t = |T |.
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Shapley Value [Hsiao & Raghavan, 1993]
The extension τi,j uniquely satisfies
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Shapley Value [Hsiao & Raghavan, 1993]
The extension τi,j uniquely satisfies
I
τi,j (vy ) is proportional to wyi
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Shapley Value [Hsiao & Raghavan, 1993]
The extension τi,j uniquely satisfies
I
I
τi,j (vy ) is proportional to wyi
P
i τi,m (v ) = v (m, m . . . , m)
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Shapley Value [Hsiao & Raghavan, 1993]
The extension τi,j uniquely satisfies
I
τi,j (vy ) is proportional to wyi
P
i τi,m (v ) = v (m, m . . . , m)
I
τi,j (v + w ) = τi,j (v ) + τi,j (w )
I
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Shapley Value [Hsiao & Raghavan, 1993]
The extension τi,j uniquely satisfies
I
τi,j (vy ) is proportional to wyi
P
i τi,m (v ) = v (m, m . . . , m)
I
τi,j (v + w ) = τi,j (v ) + τi,j (w )
I
If v (x) = 0 for all x y, then τi,j (v ) = 0 for all j < yi
I
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Shapley value over lattices [Grabisch & Lange, 2007]
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Shapley value over lattices [Grabisch & Lange, 2007]
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Shapley value over lattices [Grabisch & Lange, 2007]
Based on projection of games with levels of approval
M = {0, 1, . . . , m} to {0, m}
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Shapley value over lattices [Grabisch & Lange, 2007]
Based on projection of games with levels of approval
M = {0, 1, . . . , m} to {0, m}
ρi,m (v ) =
X
x−i ∈{0,m}n−1
(n − sm − 1)!sm !
n!
×[v (x−i , m) − v (x−i , 0)]
where sm =| Sm |.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Shapley value over lattices [Grabisch & Lange, 2007]
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Shapley value over lattices [Grabisch & Lange, 2007]
The extension ρi,m uniquely satisfies
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Shapley value over lattices [Grabisch & Lange, 2007]
The extension ρi,m uniquely satisfies
I
linearity
I
dummy property
I
monotonicity
I
symmetry
I
efficiency
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Shapley value over lattices [Grabisch & Lange, 2007]
The extension ρi,m uniquely satisfies
I
linearity
I
dummy property
I
monotonicity
I
symmetry
I
efficiency
I
invariance
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Shapley value over lattices [Grabisch & Lange, 2007]
The extension ρi,m uniquely satisfies
I
linearity
I
dummy property
I
monotonicity
I
symmetry
I
efficiency
I
invariance
If v1 (x−i , j) = v2 (x−i , j − 1) for j ≥ 1 and
v1 (x−i , 0) = v2 (x−i , 0), then
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Shapley value over lattices [Grabisch & Lange, 2007]
The extension ρi,m uniquely satisfies
I
linearity
I
dummy property
I
monotonicity
I
symmetry
I
efficiency
I
invariance
If v1 (x−i , j) = v2 (x−i , j − 1) for j ≥ 1 and
v1 (x−i , 0) = v2 (x−i , 0), then
ρi,j (v1 ) = ρi,j−1 (v2 ) for j ≥ 1
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Shapley-Shubik power index for simple multichoice games
[Jones & Wilson]
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Shapley-Shubik power index for simple multichoice games
[Jones & Wilson]
X
φi,j (v ) =
x−i
∈M n−1
where sj =| Sj |.
s0 !s1 ! . . . sm !m!
[v (x−i , j) − v (x−i , 0)].
(n + m − 1)!
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Extensions of the Shapley Value
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Shapley-Shubik power index for simple multichoice games
[Jones & Wilson]
X
φi,j (v ) =
x−i
∈M n−1
s0 !s1 ! . . . sm !m!
[v (x−i , j) − v (x−i , 0)].
(n + m − 1)!
where sj =| Sj |.
φi,m is the probability that player i will make a difference
given that each player has equal probability of participating
at each level.
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Comparison of Shapley Values
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Example
United Nations Security Council resolutions on
non-procedural issues are approved if nine members support
it and all permanent members ‘concur.’
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Comparison of Shapley Values
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Example
United Nations Security Council resolutions on
non-procedural issues are approved if nine members support
it and all permanent members ‘concur.’
I
τi,j values: 0.1963 permanent members and 0.0016
temporary members
I
φi,j values: 0.1329 permanent members and 0.0213
temporary members
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Advantages to Teaching (Cooperative) Game
Theory in Undergraduate Courses
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Advantages to Teaching (Cooperative) Game
Theory in Undergraduate Courses
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
I
Introduces students to mathematical applications in the
social sciences
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Advantages to Teaching (Cooperative) Game
Theory in Undergraduate Courses
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
I
Introduces students to mathematical applications in the
social sciences
A Survey of
Different Solution
Concepts
I
Exposes them to mathematical formalism and structure
of proofs in a context where no background knowledge
is required
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Advantages to Teaching (Cooperative) Game
Theory in Undergraduate Courses
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
I
Introduces students to mathematical applications in the
social sciences
A Survey of
Different Solution
Concepts
I
Exposes them to mathematical formalism and structure
of proofs in a context where no background knowledge
is required
I
Integrates formal, algebraic and geometric techniques
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
Advantages to Teaching (Cooperative) Game
Theory in Undergraduate Courses
Cooperative Game
Theory
Jennifer Wilson
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
I
Introduces students to mathematical applications in the
social sciences
A Survey of
Different Solution
Concepts
I
Exposes them to mathematical formalism and structure
of proofs in a context where no background knowledge
is required
I
Integrates formal, algebraic and geometric techniques
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
I
They like it
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
References
Cooperative Game
Theory
Jennifer Wilson
I
Branzei, R. et. al., ”A simple algorithm for the
mucleolus of airport surplus games”, 2004.
I
Branzei, R. et. al., Models in Cooperative Game
Theory, Springer, Berlin, 2005.
I
Felsenthal, D. and Machover, M. ”Ternary Voting
Games”, Int. J. of Game Theory (26), 335-351, 1997.
I
Felsenthal, D. and Machover, M., The Measurement of
Voting Power, Edward Elgar, Cheltenham, 1998.
I
Hsiao, C. and Raghaven, T., ”Shapley Value for
Multichoice Cooperative Games, I”, Games and Econ.
Beh. (5),240-256, 1993.
I
Littlechild S.C. and Owen, G., A, ”A simple expression
for the Shapley value in a special case”, Management
Science (20), 370-2, 1974.
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value
References
Cooperative Game
Theory
Jennifer Wilson
I
I
Littlechild S.C. and Owen, G. ”A further note on the
nucleolus of the airport game”, International Journal of
Game Theory, (5), 91-5, 1977.
Luce, R.D. and Raiffa, H., Games and Decisions, Wiley
and Sons, NY, 1957.
I
Ni, D. and Wang, Y., ”Sharing a polluted river”, Games
and Economic Behavior (60), 176-186, 2007.
I
Owen,G., Game Theory, Academic Press, New York,
1982.
I
Rapoport, A., N-Person Game Theory, Dover, 2001.
I
Straffin, P. ”Homogeneity, Independence, and Power
Indices”, Public Choice (30), 107-119, 1977.
I
Straffin, P., Game Theory and Strategy, Mathematical
Association of America, 2004.
Outline
Introduction
Relationship between
Non-cooperative and
Cooperative Games
Cooperative
GameTheory
A Survey of
Different Solution
Concepts
A Small Market
Imputations and the
Core
The Glove Market
Divide the Dollar
Dominance Relations
Other Solution
Concepts
Shapley Value
Definition
River Cleanup
Shapley-Shubik Power
Index
UN Security Council
Multichoice Games
Extensions of
Cooperative Game
Theory
Definitions
Examples
Extensions of the
Shapley Value