Cooperative Game TheoryCooperative Game Theory Jennifer Wilson Outline Introduction Relationship...

Cooperative GameTheory

Jennifer Wilson

Outline

Introduction

Relationship betweenNon-cooperative andCooperative Games

CooperativeGameTheory

A Survey ofDifferent SolutionConcepts

A Small Market

Imputations and theCore

The Glove Market

Divide the Dollar

Dominance Relations

Other SolutionConcepts

Shapley Value

Definition

River Cleanup

Shapley-Shubik PowerIndex

UN Security Council

Multichoice Games

Extensions ofCooperative GameTheory

Definitions

Examples

Extensions of theShapley Value

Cooperative Game Theory

Jennifer Wilson

Department of Natural Sciences and MathematicsEugene Lang College The New School for Liberal Arts

August 6, 2008

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

IntroductionRelationship between Non-cooperative and CooperativeGamesCooperative GameTheory

A Survey of Different Solution ConceptsA Small MarketImputations and the CoreThe Glove MarketDivide the DollarDominance RelationsOther Solution Concepts

Shapley ValueDefinitionRiver CleanupShapley-Shubik Power IndexUN Security Council

Multichoice GamesExtensions of Cooperative Game TheoryDefinitionsExamplesExtensions of the Shapley Value

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

A Three-Person Zero-Sum Game[Rapoport, 1970]

Example

zero-sum game.pdf

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ........

........

.............................................................................................................................................................................................................................................................................................................

Player 3: B

Player 2

Player 1

B (−6, 12,−6)

(−2,−1, 3) (2,−4, 2)

(3,−1,−2)

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ........

........

.............................................................................................................................................................................................................................................................................................................

Player 3: A

Player 2

Player 1

B (3, 1,−4)

(1,−2, 1) (−4, 1, 3)

(−5, 10,−5)

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ........

........

.............................................................................................................................................................................................................................................................................................................

Player 3: B

Player 2

Player 1

B (−6, 12,−6)

(−2,−1,3) (2,−4, 2)

(3,−1,−2)

←−x−→

y.....................................................................................................................................................................................................................................................................................................................

........

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .........................................................................................................................................................................................................

.............................................................................................................................................................................................................................................................................................................

Player 3: A

Player 2

Player 1

B (3, 1,−4)

(1,−2, 1) (−4,1,3)

(−5, 10,−5)

−→ x−→

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

zero-sum game.pdf

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ........

........

.............................................................................................................................................................................................................................................................................................................

Player 3: B

Player 2

Player 1

B (−6, 12,−6)

(−2,−1, 3) (2,−4, 2)

(3,−1,−2)

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ ........

........

.............................................................................................................................................................................................................................................................................................................

Player 3: A

Player 2

Player 1

B (3, 1,−4)

(1,−2, 1) (−4, 1, 3)

(−5, 10,−5)

vs 2and3.pdf

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

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Player 2 & 3

Player 1

AA AB BA BB

(1,−1)

(3,−3)

(−2, 2)

(−6, 6)

(−4, 4)

(−5, 5)

(2,−2)

(3,−3)

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

vs 2and3 a.pdf

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 2 & 3

Player 1

AA AB BA BB

(1,−1)

(3,−3)

(−2, 2)

(−6, 6)

(−4, 4)

(−5, 5)

(2,−2)

(3,−3)

................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 2 & 3

Player 1

AA AB BA BB

(1,−1)

(3,−3)

(−2, 2)

(−6, 6)

(−4, 4)

(−5, 5)

(2,−2)

(3,−3)

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 2 & 3

Player 1

AA AB BA BB

(1,−1)

(3,−3)

(−2, 2)

(−6, 6)

(−4, 4)

(−5, 5)

(2,−2)

(3,−3)

................................................................................................................................................................................................................................

.......................................................................

..............................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 2 & 3

Player 1

AA AB BA BB

(1,−1)

(3,−3)

(−2, 2)

(−6, 6)

(−4, 4)

(−5, 5)

(2,−2)

(3,−3)

................................................................................................................................................................................................................................

.......................................................................

..............................................................................................................................................

.......................................................................

So ν(1) = −4 and ν(2, 3) = 4.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

vs 2and3 a.pdf

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 2 & 3

Player 1

AA AB BA BB

(1,−1)

(3,−3)

(−2, 2)

(−6, 6)

(−4, 4)

(−5, 5)

(2,−2)

(3,−3)

................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 2 & 3

Player 1

AA AB BA BB

(1,−1)

(3,−3)

(−2, 2)

(−6, 6)

(−4, 4)

(−5, 5)

(2,−2)

(3,−3)

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 2 & 3

Player 1

AA AB BA BB

(1,−1)

(3,−3)

(−2, 2)

(−6, 6)

(−4, 4)

(−5, 5)

(2,−2)

(3,−3)

................................................................................................................................................................................................................................

.......................................................................

..............................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 2 & 3

Player 1

AA AB BA BB

(1,−1)

(3,−3)

(−2, 2)

(−6, 6)

(−4, 4)

(−5, 5)

(2,−2)

(3,−3)

................................................................................................................................................................................................................................

.......................................................................

..............................................................................................................................................

.......................................................................

So ν(1) = −4 and ν(2, 3) = 4.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

vs 1and3 a.pdf

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

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Player 1 & 3

Player 2

AB1− q

B 1− p

(2,−2)

(1,−1)

(−1, 1)

(−4, 4)

(−1, 1)

(10,−10)

(12,−12)

(−1, 1)

................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 1 & 3

Player 2

AA AB BA BB

(2,−2)

(1,−1)

(−1, 1)

(−4, 4)

(−1, 1)

(10,−10)

(12,−12)

(−1, 1)

Solving, gives p = 56 and q = 1

2 .So ν(2) = −1.5 and ν(1, 3) = 1.5.Similarly ν(3) = −4.48 and ν(1, 2) = 4.48.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

vs 1and3 a.pdf

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 1 & 3

Player 2

AB1− q

B 1− p

(2,−2)

(1,−1)

(−1, 1)

(−4, 4)

(−1, 1)

(10,−10)

(12,−12)

(−1, 1)

................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 1 & 3

Player 2

AA AB BA BB

(2,−2)

(1,−1)

(−1, 1)

(−4, 4)

(−1, 1)

(10,−10)

(12,−12)

(−1, 1)

2 .So ν(2) = −1.5 and ν(1, 3) = 1.5.

Similarly ν(3) = −4.48 and ν(1, 2) = 4.48.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

vs 1and3 a.pdf

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 1 & 3

Player 2

AB1− q

B 1− p

(2,−2)

(1,−1)

(−1, 1)

(−4, 4)

(−1, 1)

(10,−10)

(12,−12)

(−1, 1)

................................................................................................................................................................................................................................

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... ........

........

.................................................................................................................................................................................................................................................................................................................................................................................................................

Player 1 & 3

Player 2

AA AB BA BB

(2,−2)

(1,−1)

(−1, 1)

(−4, 4)

(−1, 1)

(10,−10)

(12,−12)

(−1, 1)

2 .So ν(2) = −1.5 and ν(1, 3) = 1.5.Similarly ν(3) = −4.48 and ν(1, 2) = 4.48.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

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)

What should each player receive from the game?

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

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)

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Assumptions

I Players may communicate and form coalitions with oneanother.

I Players may make sidepayments to each other.

I Utility is transferable between players.I Players’ utility values are comparable.

We will look at transferable utitility(TU) games.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Assumptions

I Players may make sidepayments to each other.I Utility is transferable between players.

I Players’ utility values are comparable.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Assumptions

I Players may make sidepayments to each other.I Utility is transferable between players.I Players’ utility values are comparable.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Assumptions

I Players may make sidepayments to each other.I Utility is transferable between players.I Players’ utility values are comparable.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Characteristic Function Form

The characteristic function ν of a game is a function thatassigns 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

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

I ν(∅) = 0

I ν(N) = K

I ν(S) + ν(N \ S) = K

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

I ν(∅) = 0

I ν(N) = K

I ν(S) + ν(N \ S) = K

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

I ν(∅) = 0

I ν(N) = K

I ν(S) + ν(N \ S) = K

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Cooperative Games

DefinitionA cooperative game is a set N of players, with a functionν : 2N → < such that ν(∅) = 0.

DefinitionA cooperative game is called superadditive if

ν(S ∪ T ) ≥ ν(S) + ν(T ) for all S ∩ T = ∅.

Every normal form game can be put in characteristic(cooperative) form.

There are cooperative games that do not correspond to anynormal form games.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Cooperative Games

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Cooperative Games

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Cooperative Games

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 playtheir 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.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Cooperative Games

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 playtheir 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.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Bankruptcy Game [O’Neill, 1982]

Example

A small company goes bankrupt owing money to threecreditors. It owns $10, 000 to creditor A, $20, 000 to creditorB 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

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

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

v(A) = 0 v(B) = 0 v(C ) = 6

x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

v(A) = 0 v(B) = 0 v(C ) = 6

x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

v(A) = 0 v(B) = 0 v(C ) = 6

x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

v(A) = 0 v(B) = 0 v(C ) = 6

x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

v(A) = 0 v(B) = 0 v(C ) = 6

x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Bankruptcy Game [O’Neill, 1982]A small company goes bankrupt owing money to threecreditors. It owns $10, 000 to creditor A, $20, 000 to creditorB 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

x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36

What additional requirements should there be?

x1 + x2 ≥ 6 x1 + x3 ≥ 16 x2 + x3 ≥ 26

x3 ≤ 30 x2 ≤ 20 x1 ≤ 10

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

v(A) = 0 v(B) = 0 v(C ) = 6

x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36

x1 + x2 ≥ 6 x1 + x3 ≥ 16 x2 + x3 ≥ 26

x3 ≤ 30 x2 ≤ 20 x1 ≤ 10

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

v(A) = 0 v(B) = 0 v(C ) = 6

x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36

x1 + x2 ≥ 6 x1 + x3 ≥ 16 x2 + x3 ≥ 26

x3 ≤ 30 x2 ≤ 20 x1 ≤ 10

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

v(A) = 0 v(B) = 0 v(C ) = 6

x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36

x1 + x2 ≥ 6 x1 + x3 ≥ 16 x2 + x3 ≥ 26

x3 ≤ 30 x2 ≤ 20 x1 ≤ 10

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

v(A) = 0 v(B) = 0 v(C ) = 6

x1 ≥ 0 x2 ≥ 0 x3 ≥ 6 and x1 + x2 + x3 = 36

x1 + x2 ≥ 6 x1 + x3 ≥ 16 x2 + x3 ≥ 26

x3 ≤ 30 x2 ≤ 20 x1 ≤ 10

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Graphing on the SimplexThe 2-dimensional simplex{(x1, x2, x3) | x1 + x2 + x3 = 1, x1 ≥ 0, x2 ≥ 0, and x3 ≥ 0.}.

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................

............................................

......................................................................................................................................................................................................................................................................................................................................

..............................................................................

...............................................................

(1, 0, 0)

(0, 1, 0)

(0, 0, 1)

....................................................................................................................................................................................................................................................................................................................

....................................

....................................................................................................................................................................................................................................................................................................• •

(1, 0, 0) (0, 1, 0)

(0, 0, 1)

................

..........................................x1

..............................................................

..................................................

Possible “solutions” of the Bankrupcy Game.

Bankrupcy.pdf

..............................................................................................................................................................................................................................................................................................................................................................................................................

....................................

............................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................

x1 = 10/36

x3 = 30/36

x3 = 6/36

x2 = 20/36

... .. . .. . . .. . .. . . .. . . . .. . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . . . . .. . . . . .

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Graphing on the SimplexThe 2-dimensional simplex{(x1, x2, x3) | x1 + x2 + x3 = 1, x1 ≥ 0, x2 ≥ 0, and x3 ≥ 0.}.

.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................

............................................

......................................................................................................................................................................................................................................................................................................................................

..............................................................................

...............................................................

(1, 0, 0)

(0, 1, 0)

(0, 0, 1)

....................................................................................................................................................................................................................................................................................................................

....................................

....................................................................................................................................................................................................................................................................................................• •

(1, 0, 0) (0, 1, 0)

(0, 0, 1)

................

..........................................x1

..............................................................

..................................................

Possible “solutions” of the Bankrupcy Game.

Bankrupcy.pdf

..............................................................................................................................................................................................................................................................................................................................................................................................................

....................................

............................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................

..................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................

x1 = 10/36

x3 = 30/36

x3 = 6/36

x2 = 20/36

... .. . .. . . .. . .. . . .. . . . .. . . . . .. . . . . . .. . . . . .. . . . . . .. . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . . .. . . . . . . . . .. . . . . . . . .. . . . . . . .. . . . . . .. . . . . . . .. . . . . . .. . . . . .

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Imputations

DefinitionGiven a cooperative game ν on n players, an imputation is apayoff vector

x1, x2, . . . , xn

such thatxi ≥ ν(i) individual rationality

and ∑i

xi = ν(N) collective rationality.

i ν(i) = ν(N), then xi = ν(i).These games are called inessential.

We will assume ν is essential.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Imputations

x1, x2, . . . , xn

such thatxi ≥ ν(i)

individual rationality

and ∑i

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Imputations

x1, x2, . . . , xn

and ∑i

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Imputations

x1, x2, . . . , xn

and ∑i

xi = ν(N)

collective rationality.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Imputations

x1, x2, . . . , xn

and ∑i

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Imputations

x1, x2, . . . , xn

and ∑i

i ν(i) = ν(N), then xi = ν(i).

These games are called inessential.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Imputations

x1, x2, . . . , xn

and ∑i

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Imputations

x1, x2, . . . , xn

and ∑i

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Core

DefinitionThe core is the set of imputations such that∑

xi ≥ v(S) for every S ⊂ N.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Glove Market

Example

A market consists of people with left-handed glove andpeople with right-handed gloves (but not both). The value ofa coalition is the number of complete pairs of gloves it has.

We will look at the case when A1 and A2 have left-handedglove 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

ν(Ai ,Bi1 ,Bi2) = 1 . . .

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Glove Market

Example

ν(Ai ,Bi1 ,Bi2) = 1 . . .

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Glove Market

Example

ν(Ai ,Bi1 ,Bi2) = 1 . . .

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Glove Market

Example

ν(Ai ,Bi1 ,Bi2) = 1 . . .

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Glove Market

Let (x1, x2, y1, y2, y3) be the payoff vector.

It is an imputation if

xi ≥ 0, yj ≥ 0 y2 ≥ 0 x1 + x2 + y1 + y2 + y3 = 2.

The payoff vector is in the core if

x1 + y1 ≥ 1 x2 + y2 ≥ 1 . . .

But(x1 + y1) + (x2 + y2) ≥ 1 + 1 = 2.

(x1 + y1) = (x2 + y2) = 1 and y3 = 0.

So y1 = y2 = y3 = 0, and x1 = x2 = 1.The core consists of the single point (1, 1, 0, 0, 0).

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Glove Market

xi ≥ 0, yj ≥ 0 y2 ≥ 0 x1 + x2 + y1 + y2 + y3 = 2.

x1 + y1 ≥ 1 x2 + y2 ≥ 1 . . .

But(x1 + y1) + (x2 + y2) ≥ 1 + 1 = 2.

(x1 + y1) = (x2 + y2) = 1 and y3 = 0.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Glove Market

xi ≥ 0, yj ≥ 0 y2 ≥ 0 x1 + x2 + y1 + y2 + y3 = 2.

x1 + y1 ≥ 1 x2 + y2 ≥ 1 . . .

But(x1 + y1) + (x2 + y2) ≥ 1 + 1 = 2.

(x1 + y1) = (x2 + y2) = 1 and y3 = 0.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Glove Market

xi ≥ 0, yj ≥ 0 y2 ≥ 0 x1 + x2 + y1 + y2 + y3 = 2.

x1 + y1 ≥ 1 x2 + y2 ≥ 1 . . .

But(x1 + y1) + (x2 + y2) ≥ 1 + 1 = 2.

(x1 + y1) = (x2 + y2) = 1 and y3 = 0.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Glove Market

xi ≥ 0, yj ≥ 0 y2 ≥ 0 x1 + x2 + y1 + y2 + y3 = 2.

x1 + y1 ≥ 1 x2 + y2 ≥ 1 . . .

But(x1 + y1) + (x2 + y2) ≥ 1 + 1 = 2.

(x1 + y1) = (x2 + y2) = 1 and y3 = 0.

So y1 = y2 = y3 = 0, and x1 = x2 = 1.

The core consists of the single point (1, 1, 0, 0, 0).

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

The Glove Market

xi ≥ 0, yj ≥ 0 y2 ≥ 0 x1 + x2 + y1 + y2 + y3 = 2.

x1 + y1 ≥ 1 x2 + y2 ≥ 1 . . .

But(x1 + y1) + (x2 + y2) ≥ 1 + 1 = 2.

(x1 + y1) = (x2 + y2) = 1 and y3 = 0.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

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.

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 x3 ≥ 0 x1 + x2 + x3 = 1

x1 + x2 ≥ 1 x1 + x3 ≥ 1 x2 + x3 ≥ 1.

This is impossible.

Hence the core is empty.All constant-sum games have empty cores.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

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 x3 ≥ 0 x1 + x2 + x3 = 1

x1 + x2 ≥ 1 x1 + x3 ≥ 1 x2 + x3 ≥ 1.

This is impossible.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

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 x3 ≥ 0 x1 + x2 + x3 = 1

x1 + x2 ≥ 1 x1 + x3 ≥ 1 x2 + x3 ≥ 1.

This is impossible.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

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 x3 ≥ 0 x1 + x2 + x3 = 1

x1 + x2 ≥ 1 x1 + x3 ≥ 1 x2 + x3 ≥ 1.

This is impossible.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

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 x3 ≥ 0 x1 + x2 + x3 = 1

x1 + x2 ≥ 1 x1 + x3 ≥ 1 x2 + x3 ≥ 1.

This is impossible.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

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 x3 ≥ 0 x1 + x2 + x3 = 1

x1 + x2 ≥ 1 x1 + x3 ≥ 1 x2 + x3 ≥ 1.

This is impossible.

Hence the core is empty.

All constant-sum games have empty cores.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Example

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 x3 ≥ 0 x1 + x2 + x3 = 1

x1 + x2 ≥ 1 x1 + x3 ≥ 1 x2 + x3 ≥ 1.

This is impossible.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Divide the Dollar

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....

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Divide the Dollar

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....

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Divide the Dollar

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....

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Divide the Dollar

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....

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations[Von Neumann and Morgenstern, 1947]

DefinitionImputation x is dominated by imputation y through coalitionS if

I yi > xi for every i ∈ S

I ν(S) ≥∑i∈S yi .

Consider the two imputations:

x = (0.50, 0.50, 0) and y = (0.60, 0, 0.40).

Players 1 and 3 can ’force’ player 2 to accept y.

So (0.50, 0, 50, 0) is dominated by (0.60, 0, 0.40) throughS = {1, 3} which is dominated by (0, 0.50, 0.50) throughcoalition S = {2, 3} which is dominated by ...

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

x = (0.50, 0.50, 0) and y = (0.60, 0, 0.40).

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

x = (0.50, 0.50, 0) and y = (0.60, 0, 0.40).

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

x = (0.50, 0.50, 0) and y = (0.60, 0, 0.40).

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

x = (0.50, 0.50, 0) and y = (0.60, 0, 0.40).

So (0.50, 0, 50, 0) is dominated by (0.60, 0, 0.40) throughS = {1, 3}

which is dominated by (0, 0.50, 0.50) throughcoalition S = {2, 3} which is dominated by ...

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

x = (0.50, 0.50, 0) and y = (0.60, 0, 0.40).

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations

The core consists of all undominated imputations.

Proof.Suppose x is dominated by y through the coalition S .

Then∑i∈S

xi <∑i∈S

yi ≤ ν(S)

hence x is not in the core.

Suppose x is an imputation that is not in the core and∑i∈S xi < ν(S).

yi ={ xi + 1

|S| [ν(S)−∑i∈S xi ] if i ∈ S

ν(i) + 1|N\S| [ν(N)− (ν(S) +

∑i /∈S ν(i))] if i /∈ S .

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations

Proof.Suppose x is dominated by y through the coalition S . Then∑

xi <∑i∈S

yi ≤ ν(S)

yi ={ xi + 1

|S| [ν(S)−∑i∈S xi ] if i ∈ S

ν(i) + 1|N\S| [ν(N)− (ν(S) +

∑i /∈S ν(i))] if i /∈ S .

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations

xi <∑i∈S

yi ≤ ν(S)

yi ={ xi + 1

|S| [ν(S)−∑i∈S xi ] if i ∈ S

ν(i) + 1|N\S| [ν(N)− (ν(S) +

∑i /∈S ν(i))] if i /∈ S .

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations

xi <∑i∈S

yi ≤ ν(S)

yi ={ xi + 1

|S| [ν(S)−∑i∈S xi ] if i ∈ S

ν(i) + 1|N\S| [ν(N)− (ν(S) +

∑i /∈S ν(i))] if i /∈ S .

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations

xi <∑i∈S

yi ≤ ν(S)

yi ={ xi + 1

|S| [ν(S)−∑i∈S xi ] if i ∈ S

ν(i) + 1|N\S| [ν(N)− (ν(S) +

∑i /∈S ν(i))] if i /∈ S .

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations

xi <∑i∈S

yi ≤ ν(S)

yi ={ xi + 1

|S| [ν(S)−∑i∈S xi ] if i ∈ S

ν(i) + 1|N\S| [ν(N)− (ν(S) +

∑i /∈S ν(i))] if i /∈ S .

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations

yi ={ xi + 1

|S| [ν(S)−∑i∈S xi ] if i ∈ S

ν(i) + 1|N\S| [ν(N)− (ν(S) +

∑i /∈S ν(i))] if i /∈ S .

Claim:

I yi > xi for all i ∈ S

I ν(S) ≥∑i∈S yi

I y is an imputationI yi ≥ ν(i) for all iI∑

i yi = ν(N)

Hence y is an imputation that dominates x.

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations

yi ={ xi + 1

|S| [ν(S)−∑i∈S xi ] if i ∈ S

ν(i) + 1|N\S| [ν(N)− (ν(S) +

∑i /∈S ν(i))] if i /∈ S .

Claim:

i yi = ν(N)

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations

yi ={ xi + 1

|S| [ν(S)−∑i∈S xi ] if i ∈ S

ν(i) + 1|N\S| [ν(N)− (ν(S) +

∑i /∈S ν(i))] if i /∈ S .

Claim:

I y is an imputation

I yi ≥ ν(i) for all iI∑

i yi = ν(N)

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations

yi ={ xi + 1

|S| [ν(S)−∑i∈S xi ] if i ∈ S

ν(i) + 1|N\S| [ν(N)− (ν(S) +

∑i /∈S ν(i))] if i /∈ S .

Claim:

I y is an imputationI yi ≥ ν(i) for all i

i yi = ν(N)

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations

yi ={ xi + 1

|S| [ν(S)−∑i∈S xi ] if i ∈ S

ν(i) + 1|N\S| [ν(N)− (ν(S) +

∑i /∈S ν(i))] if i /∈ S .

Claim:

i yi = ν(N)

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Dominance Relations

yi ={ xi + 1

|S| [ν(S)−∑i∈S xi ] if i ∈ S

ν(i) + 1|N\S| [ν(N)− (ν(S) +

∑i /∈S ν(i))] if i /∈ S .

Claim:

i yi = ν(N)

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Stable Sets

DefinitionA stable set is a set I of imputations such that

I no imputation in I is dominated by any otherimputation in I

internally stable

I every imputation not in I is dominated by someimputation in I . externally stable

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Stable Sets

I no imputation in I is dominated by any otherimputation in I internally stable

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Stable Sets

I every imputation not in I is dominated by someimputation in I .

externally stable

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Stable Sets

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

Stable Sets and Divide the DollarExamples:

I1 = {(0.50, 0.50, 0), (0.50, 0, 0.50), (0, 0.50, 0.50)}

I2 = {0.7, x , 0.3− x | 0 ≤ x ≤ 0.7}

Set Divide the Dollar.pdf

..............................................................................................................................................................................................................................................................................................................................................................................................................

....................................

............................................................................................................................................................................................................................................................................................................................................................................

A=(1,0,0) B=(0,1,0)

C=(0,0,1)

• •

••

(1/2,1/2,0)

(1/2,0,1/2) (0,1/2,1/2)

Set Divide the Dollar 2.pdf

..............................................................................................................................................................................................................................................................................................................................................................................................................

....................................

............................................................................................................................................................................................................................................................................................................................................................................

A=(1,0,0) B=(0,1,0)

C=(0,0,1)

• •

.................................................................................................................................................

{0.7, x2, 0.3− x2}

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

I1 = {(0.50, 0.50, 0), (0.50, 0, 0.50), (0, 0.50, 0.50)}

I2 = {0.7, x , 0.3− x | 0 ≤ x ≤ 0.7}

..............................................................................................................................................................................................................................................................................................................................................................................................................

....................................

............................................................................................................................................................................................................................................................................................................................................................................

A=(1,0,0) B=(0,1,0)

C=(0,0,1)

• •

••

(1/2,1/2,0)

(1/2,0,1/2) (0,1/2,1/2)

..............................................................................................................................................................................................................................................................................................................................................................................................................

....................................

............................................................................................................................................................................................................................................................................................................................................................................

A=(1,0,0) B=(0,1,0)

C=(0,0,1)

• •

.................................................................................................................................................

{0.7, x2, 0.3− x2}

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

I1 = {(0.50, 0.50, 0), (0.50, 0, 0.50), (0, 0.50, 0.50)}

I2 = {0.7, x , 0.3− x | 0 ≤ x ≤ 0.7}

..............................................................................................................................................................................................................................................................................................................................................................................................................

....................................

............................................................................................................................................................................................................................................................................................................................................................................

A=(1,0,0) B=(0,1,0)

C=(0,0,1)

• •

••

(1/2,1/2,0)

(1/2,0,1/2) (0,1/2,1/2)

..............................................................................................................................................................................................................................................................................................................................................................................................................

....................................

............................................................................................................................................................................................................................................................................................................................................................................

A=(1,0,0) B=(0,1,0)

C=(0,0,1)

• •

.................................................................................................................................................

{0.7, x2, 0.3− x2}

Jennifer Wilson

Outline

Introduction

A Small Market

The Glove Market

Divide the Dollar

Dominance Relations

Shapley Value

Definition

River Cleanup

UN Security Council

Multichoice Games

Definitions

Examples

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