Preserving coalitional rationality for non-balanced games St ephane - - PowerPoint PPT Presentation

Preserving coalitional rationality for non-balanced games

St´ ephane GONZALEZ & Michel GRABISCH Paris School of Economics, University of Paris I, France

Centre d’´ economie de la Sorbonne Universit´ e Paris I

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 1 / 27

Introduction

A central problem of cooperative TU games is to propose an allocation of gains obtained by a set N of players.

◮ Typically, this total amount is divided among the individual players. ◮ This vision has two unfortunate consequences:

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 2 / 27

Introduction

A central problem of cooperative TU games is to propose an allocation of gains obtained by a set N of players.

◮ Typically, this total amount is divided among the individual players. ◮ This vision has two unfortunate consequences: 1

In many situations, decision makers do not make a sharing among individuals, but often give to groups (associations, companies, families, etc.)

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 2 / 27

Introduction

A central problem of cooperative TU games is to propose an allocation of gains obtained by a set N of players.

◮ Typically, this total amount is divided among the individual players. ◮ This vision has two unfortunate consequences: 1

In many situations, decision makers do not make a sharing among individuals, but often give to groups (associations, companies, families, etc.)

2

In many cases, the core of the game is empty.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 2 / 27

Introduction

A central problem of cooperative TU games is to propose an allocation of gains obtained by a set N of players.

◮ Typically, this total amount is divided among the individual players. ◮ This vision has two unfortunate consequences: 1

In many situations, decision makers do not make a sharing among individuals, but often give to groups (associations, companies, families, etc.)

2

In many cases, the core of the game is empty.

We propose the use of general solutions, that is, to distribute the total worth of the game among groups rather than among individuals. We propose a new way to preserve coalitional stability for non-balanced games.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 2 / 27

Introduction

Example We consider a game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 3 / 27

Introduction

Example S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 A general payoff vector will be able to give: 2 to the player 1. 2 to the player 2. 3 to the player 3. −1 to the coalition 12. −1 to the coalition 13. −1 to the coalition 23. −2 to the coalition 123. 2 + 2 + 3 − 1 − 1 − 1 − 2 = 2,

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 3 / 27

Introduction

Example S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 A general payoff vector will be able to give: 2 to the player 1. 2 to the player 2. 3 to the player 3. −1 to the coalition 12. −1 to the coalition 13. −1 to the coalition 23. −2 to the coalition 123. 2 + 2 + 3 − 1 − 1 − 1 − 2 = 2, we have an alternative sharing of the value

• f N.
• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 3 / 27

Introduction

Interest of this point of view?

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 4 / 27

Introduction

Interest of this point of view? → It is possible to preserve the notion of coalitional rationality coming from the core for every games.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 4 / 27

Introduction

Example S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 5 / 27

Introduction

Example S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 The core of v is empty.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 5 / 27

Introduction

Example S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 A general efficient payoff vector will be able to give: 2 to the player 1. 2 to the player 2. 3 to the player 3. −1 to the coalition 12. −1 to the coalition 13. −1 to the coalition 23. −2 to the coalition 123.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 5 / 27

Introduction

Example S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 A general efficient payoff vector will be able to give: 2 to the player 1. 2 to the player 2. 3 to the player 3. −1 to the coalition 12. → 2 + 2 − 1 ≥ 1.5 −1 to the coalition 13. −1 to the coalition 23. −2 to the coalition 123.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 5 / 27

Introduction

Example S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 A general efficient payoff vector will be able to give: 2 to the player 1. 2 to the player 2. 3 to the player 3. −1 to the coalition 12. −1 to the coalition 13.→ 2 + 3 − 1 ≥ 1.5 −1 to the coalition 23. −2 to the coalition 123.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 5 / 27

Introduction

Example S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 A general efficient payoff vector will be able to give: 2 to the player 1. 2 to the player 2. 3 to the player 3. −1 to the coalition 12. −1 to the coalition 13. −1 to the coalition 23.→ 2 + 3 − 1 ≥ 1.5 −2 to the coalition 123.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 5 / 27

Introduction

Example S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 A general efficient payoff vector will be able to give: 2 to the player 1. 2 to the player 2. 3 to the player 3. −1 to the coalition 12. −1 to the coalition 13. −1 to the coalition 23. −2 to the coalition 123.→ 2 + 2 + 3 − 1 − 1 − 1 − 2 = 2

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 5 / 27

General Solution

A general efficient payoff vector of v is a vector x ∈ R2N\∅ that assigns to a coalition S ⊆ N a payoff xS such that

• ∅=S⊆N

xS = v(N).

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 6 / 27

General Solution

A general efficient payoff vector of v is a vector x ∈ R2N\∅ that assigns to a coalition S ⊆ N a payoff xS such that

• ∅=S⊆N

xS = v(N). In fact, xS can be seen as the value of the M¨

• bius transform (or

Harsanyi dividends) mφ(S) on the coalition S, where φ is the game defined by: φ(S) =

• T⊆S

xT. and mφ(S) :=

• T⊆S

(−1)|S\T|φ(T), ∀S ⊆ N.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 6 / 27

General Solution

Definition A general solution on the set of games G(N) is a mapping σ : G(N) → 2(R2n−1) such that: ∀v ∈ G(N), σ(v) is a set of general payoffs efficient for the game v.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 7 / 27

General Solution

Definition A general solution on the set of games G(N) is a mapping σ : G(N) → 2(R2n−1) such that: ∀v ∈ G(N), σ(v) is a set of general payoffs efficient for the game v. Example The core C, the Shapley value Sh etc., are general solutions

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 7 / 27

General Solution

Definition A general solution on the set of games G(N) is a mapping σ : G(N) → 2(R2n−1) such that: ∀v ∈ G(N), σ(v) is a set of general payoffs efficient for the game v. Example The core C, the Shapley value Sh etc., are general solutions The M¨

• bius transform is a general solution.
• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 7 / 27

The k-additive core

Idea: Generalisation of the core by generalisation of the additivity:

◮ A game v is additive if its M¨

• bius transform vanishes for any S ⊆ N

such that |S| > 1.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 8 / 27

The k-additive core

Idea: Generalisation of the core by generalisation of the additivity:

◮ A game v is additive if its M¨

• bius transform vanishes for any S ⊆ N

such that |S| > 1.

◮ A game v is said to be at most k-additive if its M¨

• bius transform

vanishes for any S ⊆ N such that |S| > k.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 8 / 27

The k-additive core

Idea: Generalisation of the core by generalisation of the additivity:

◮ A game v is additive if its M¨

• bius transform vanishes for any S ⊆ N

such that |S| > 1.

◮ A game v is said to be at most k-additive if its M¨

• bius transform

vanishes for any S ⊆ N such that |S| > k.

Definition Let v be a game. The k-additive core of v, denoted by Ck(v), is defined by: Ck(v) = {φ at most k-additive , φ(S) ≥ v(S), ∀S ⊆ N, φ(N) = v(N)} (Vassil’ev 1978 ; Grabisch & Miranda 2008)

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 8 / 27

The k-additive core

Idea: Generalisation of the core by generalisation of the additivity:

◮ A game v is additive if its M¨

• bius transform vanishes for any S ⊆ N

such that |S| > 1.

◮ A game v is said to be at most k-additive if its M¨

• bius transform

vanishes for any S ⊆ N such that |S| > k.

Definition Let v be a game. The k-additive core of v, denoted by Ck(v), is defined by: Ck(v) = {φ at most k-additive , φ(S) ≥ v(S), ∀S ⊆ N, φ(N) = v(N)} (Vassil’ev 1978 ; Grabisch & Miranda 2008)

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 8 / 27

The k-additive core

Idea: Generalisation of the core by generalisation of the additivity:

◮ A game v is additive if its M¨

• bius transform vanishes for any S ⊆ N

such that |S| > 1.

◮ A game v is said to be at most k-additive if its M¨

• bius transform

vanishes for any S ⊆ N such that |S| > k.

Definition Let v be a game. The k-additive core of v, denoted by Ck(v), is defined by: Ck(v) = {φ at most k-additive , φ(S) ≥ v(S), ∀S ⊆ N, φ(N) = v(N)} (Vassil’ev 1978 ; Grabisch & Miranda 2008)

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 8 / 27

The k-additive core

Idea: Generalisation of the core by generalisation of the additivity:

◮ A game v is additive if its M¨

• bius transform vanishes for any S ⊆ N

such that |S| > 1.

◮ A game v is said to be at most k-additive if its M¨

• bius transform

vanishes for any S ⊆ N such that |S| > k.

Definition Let v be a game. The k-additive core of v, denoted by Ck(v), is defined by: Ck(v) = {φ at most k-additive , φ(S) ≥ v(S), ∀S ⊆ N, φ(N) = v(N)} (Vassil’ev 1978 ; Grabisch & Miranda 2008)

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 8 / 27

The k-additive core

Idea: Generalisation of the core by generalisation of the additivity:

◮ A game v is additive if its M¨

• bius transform vanishes for any S ⊆ N

such that |S| > 1.

◮ A game v is said to be at most k-additive if its M¨

• bius transform

vanishes for any S ⊆ N such that |S| > k.

Definition Let v be a game. The k-additive core of v, denoted by Ck(v), is defined by: Ck(v) = {φ at most k-additive , φ(S) ≥ v(S), ∀S ⊆ N, φ(N) = v(N)} (Vassil’ev 1978 ; Grabisch & Miranda 2008) m ◦ Ck is a general solution

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 8 / 27

The k-additive core

Example We consider a game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 9 / 27

The k-additive core

Example We consider a game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 Let φ be a game defined by S 1 2 3 12 13 23 123 φ(S) 2 2 3 3 3 3 2

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 9 / 27

The k-additive core

Example We consider a game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 Let φ be a game defined by S 1 2 3 12 13 23 123 φ(S) 2 ≥ 1 2 ≥ 1 3≥ 1 3≥ 1.5 3≥ 1.5 3≥ 1.5 2= 2

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 9 / 27

The k-additive core

Example We consider a game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 Let φ be a game defined by S 1 2 3 12 13 23 123 φ(S) 2 ≥ 1 2 ≥ 1 3≥ 1 3≥ 1.5 3≥ 1.5 3≥ 1.5 2= 2 S 1 2 3 12 13 23 123 mφ(S) 2 2 3

• 1
• 2
• 2
• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 9 / 27

The k-additive core

Example We consider a game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 Let φ be a game defined by S 1 2 3 12 13 23 123 φ(S) 2 ≥ 1 2 ≥ 1 3≥ 1 3≥ 1.5 3≥ 1.5 3≥ 1.5 2= 2 Therefore φ ∈ C2(v) and the corresponding payoff vector is S 1 2 3 12 13 23 123 mφ(S) 2 2 3

• 1
• 2
• 2
• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 9 / 27

The k-additive core

C(v) ≡ C1(v) ⊆ C2(v) ⊆ · · · ⊆ Cn(v)

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 10 / 27

The k-additive core

C(v) ≡ C1(v) ⊆ C2(v) ⊆ · · · ⊆ Cn(v) The k-additive core is a convex polyhedron, nonempty for k ≥ 2.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 10 / 27

The k-additive core

C(v) ≡ C1(v) ⊆ C2(v) ⊆ · · · ⊆ Cn(v) The k-additive core is a convex polyhedron, nonempty for k ≥ 2. For all k ∈ {1, . . . , n}, we have: Ck ◦ Ck = Ck

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 10 / 27

The k-additive core

C(v) ≡ C1(v) ⊆ C2(v) ⊆ · · · ⊆ Cn(v) The k-additive core is a convex polyhedron, nonempty for k ≥ 2. For all k ∈ {1, . . . , n}, we have: Ck ◦ Ck = Ck The n-additive core is the largest general solution satisfying coalitional rationality.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 10 / 27

The k-additive core

Problem: ∀k ≥ 2, the k-additive core is unbounded.

◮ Sh(C2(v)) = { efficient payoffs vectors }

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 11 / 27

The k-additive core

Problem: ∀k ≥ 2, the k-additive core is unbounded.

◮ Sh(C2(v)) = { efficient payoffs vectors }

→ We must make a choice of a pertinent φ ∈ Ck(v).

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 11 / 27

The extended core

Given v ∈ G(N), t ≥ 0, the t-expansion v t of v is the game v t ∈ G(N) defined by v t(S) := v(S) for all S ⊂ N, and v t(N) := v(N) + t ¯ t(v) := min{t ≥ 0 | C(v t) = ∅}, the minimum amount to be given to the grand coalition to ensure balancedness Bejan and Gomez (2009) have introduced the extended core (EC(v)) as the set of efficient payoff vectors x for which ∃y ∈ RN s.t. (x + y)(S) ≥ v(S), ∀S ⊆ N, and y(N) = ¯ t(v)

• r

(x + y) ∈ C(v¯

t(v))

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 12 / 27

The G-extended core

We can redefine the extended core as a general solution concept. The G-extended core is defined by GEC(v) = {φ ∈ G(N) | (φ(i))i∈N ∈ C(v¯

t(v)),

mφ(S) = 0, ∀S, 1 < |S| < n, and mφ(N) = −¯ t(v)}.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 13 / 27

The G-extended core

We can redefine the extended core as a general solution concept. The G-extended core is defined by GEC(v) = {φ ∈ G(N) | (φ(i))i∈N ∈ C(v¯

t(v)),

mφ(S) = 0, ∀S, 1 < |S| < n, and mφ(N) = −¯ t(v)}. As for the k-additive core, for any φ ∈ GEC(v), mφ is the associated general payoff vector. Therefore m ◦ GEC is a general solution, ∀φ ∈ GEC(v) we have m ◦ φ = (m ◦ φ(1), . . . , m ◦ φ(n)

• ∈C(v¯

t(v))

, 0, . . . , 0

• 2n−2

, −¯ t(v))

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 13 / 27

The G-extended core

We can redefine the extended core as a general solution concept. The G-extended core is defined by GEC(v) = {φ ∈ G(N) | (φ(i))i∈N ∈ C(v¯

t(v)),

mφ(S) = 0, ∀S, 1 < |S| < n, and mφ(N) = −¯ t(v)}. As for the k-additive core, for any φ ∈ GEC(v), mφ is the associated general payoff vector. Therefore m ◦ GEC is a general solution, ∀φ ∈ GEC(v) we have m ◦ φ = (m ◦ φ(1), . . . , m ◦ φ(n)

• ∈C(v¯

t(v))

, 0, . . . , 0

• 2n−2

, −¯ t(v)) Main properties:

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 13 / 27

The G-extended core

We can redefine the extended core as a general solution concept. The G-extended core is defined by GEC(v) = {φ ∈ G(N) | (φ(i))i∈N ∈ C(v¯

t(v)),

mφ(S) = 0, ∀S, 1 < |S| < n, and mφ(N) = −¯ t(v)}. As for the k-additive core, for any φ ∈ GEC(v), mφ is the associated general payoff vector. Therefore m ◦ GEC is a general solution, ∀φ ∈ GEC(v) we have m ◦ φ = (m ◦ φ(1), . . . , m ◦ φ(n)

• ∈C(v¯

t(v))

, 0, . . . , 0

• 2n−2

, −¯ t(v)) Main properties:

◮ GEC(v) is a nonempty convex polyhedron, coinciding with the core if

the latter is nonempty

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 13 / 27

The G-extended core

We can redefine the extended core as a general solution concept. The G-extended core is defined by GEC(v) = {φ ∈ G(N) | (φ(i))i∈N ∈ C(v¯

t(v)),

mφ(S) = 0, ∀S, 1 < |S| < n, and mφ(N) = −¯ t(v)}. As for the k-additive core, for any φ ∈ GEC(v), mφ is the associated general payoff vector. Therefore m ◦ GEC is a general solution, ∀φ ∈ GEC(v) we have m ◦ φ = (m ◦ φ(1), . . . , m ◦ φ(n)

• ∈C(v¯

t(v))

, 0, . . . , 0

• 2n−2

, −¯ t(v)) Main properties:

◮ GEC(v) is a nonempty convex polyhedron, coinciding with the core if

the latter is nonempty

◮ GEC(v) ⊆ Cn(v) for all v ∈ G(N)

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 13 / 27

The G-extended core

We can redefine the extended core as a general solution concept. The G-extended core is defined by GEC(v) = {φ ∈ G(N) | (φ(i))i∈N ∈ C(v¯

t(v)),

mφ(S) = 0, ∀S, 1 < |S| < n, and mφ(N) = −¯ t(v)}. As for the k-additive core, for any φ ∈ GEC(v), mφ is the associated general payoff vector. Therefore m ◦ GEC is a general solution, ∀φ ∈ GEC(v) we have m ◦ φ = (m ◦ φ(1), . . . , m ◦ φ(n)

• ∈C(v¯

t(v))

, 0, . . . , 0

• 2n−2

, −¯ t(v)) Main properties:

◮ GEC(v) is a nonempty convex polyhedron, coinciding with the core if

the latter is nonempty

◮ GEC(v) ⊆ Cn(v) for all v ∈ G(N) ◮ S(GEC(v)) = EC(v)

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 13 / 27

The G-extended core

Example we consider the game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 5 7 1 20 1 1 1

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 14 / 27

The G-extended core

Example we consider the game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 5 7 1 20 1 1 1 then the game φ defined by its Moebius transform S 1 2 3 12 13 23 123 mφ(S) 13 7 1

• 20
• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 14 / 27

The G-extended core

Example we consider the game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 5 7 1 20 1 1 1 then the game φ defined by its Moebius transform S 1 2 3 12 13 23 123 mφ(S) 13≥ 5 7≥ 7 1≥ 1

• 20
• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 14 / 27

The G-extended core

Example we consider the game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 5 7 1 20 1 1 1 then the game φ defined by its Moebius transform S 1 2 3 12 13 23 123 mφ(S) 13 7 1 13 + 7 + 0 ≥ 20

• 20
• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 14 / 27

The G-extended core

Example we consider the game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 5 7 1 20 1 1 1 then the game φ defined by its Moebius transform S 1 2 3 12 13 23 123 mφ(S) 13 7 1 13 + 1 + 0 ≥ 1

• 20
• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 14 / 27

The G-extended core

Example we consider the game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 5 7 1 20 1 1 1 then the game φ defined by its Moebius transform S 1 2 3 12 13 23 123 mφ(S) 13 7 1 7 + 1 + 0 ≥ 1

• 20
• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 14 / 27

The G-extended core

Example we consider the game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 5 7 1 20 1 1 1 then the game φ defined by its Moebius transform S 1 2 3 12 13 23 123 mφ(S) 13 7 1 13 + 7 + 1 + 0 + 0 + 0 − 20 = 1

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 14 / 27

The G-extended core

Example we consider the game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 5 7 1 20 1 1 1 then the game φ defined by its Moebius transform S 1 2 3 12 13 23 123 mφ(S) 13 7 1

• 20

belongs to GEC(v).

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 14 / 27

The minimum bargaining set

General philosophy:

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c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 15 / 27

The minimum bargaining set

General philosophy:

◮ Select a bounded subset of the k-additive core,

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 15 / 27

The minimum bargaining set

General philosophy:

◮ Select a bounded subset of the k-additive core, ◮ which coincides with the core if the latter is nonempty;

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 15 / 27

The minimum bargaining set

General philosophy:

◮ Select a bounded subset of the k-additive core, ◮ which coincides with the core if the latter is nonempty; ◮ Since payoffs given to coalitions imply a bargaining among the members

• f those coalitions, minimize the “total amount” given to coalitions
• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 15 / 27

The minimum bargaining set

General philosophy:

◮ Select a bounded subset of the k-additive core, ◮ which coincides with the core if the latter is nonempty; ◮ Since payoffs given to coalitions imply a bargaining among the members

• f those coalitions, minimize the “total amount” given to coalitions

This is expressed by the following nonlinear program Minimize B(φ) := (mφ(S))S⊆N

|S|≥2

• subject to φ ∈ Ck(v)

(for a given norm · and fixed 2 ≤ k ≤ n)

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 15 / 27

The minimum bargaining set

General philosophy:

◮ Select a bounded subset of the k-additive core, ◮ which coincides with the core if the latter is nonempty; ◮ Since payoffs given to coalitions imply a bargaining among the members

• f those coalitions, minimize the “total amount” given to coalitions

This is expressed by the following nonlinear program Minimize B(φ) := (mφ(S))S⊆N

|S|≥2

• subject to φ ∈ Ck(v)

(for a given norm · and fixed 2 ≤ k ≤ n) Call B(φ) the bargaining level of φ. The set of optimal solutions is called the minimum bargaining set, and is denoted by Ik(v) ⊆ Ck(v).

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 15 / 27

The minimum bargaining set

Theorem Let v ∈ G(N), 2 ≤ k ≤ n. We have:

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c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 16 / 27

The minimum bargaining set

Theorem Let v ∈ G(N), 2 ≤ k ≤ n. We have: Ik(v) is a nonempty convex and compact set, equal to C(v) if C(v) = ∅,

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c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 16 / 27

The minimum bargaining set

Theorem Let v ∈ G(N), 2 ≤ k ≤ n. We have: Ik(v) is a nonempty convex and compact set, equal to C(v) if C(v) = ∅, Otherwise ∃ǫ > 0 such that Ik(v) = {φ ∈ Ck(v), B(φ) = ǫ}.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 16 / 27

The minimum bargaining set

Important fact: For all φ ∈ Ik(v), mφ(S) ≤ 0, ∀S ⊆ N such that |S| ≥ 2.

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c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 17 / 27

The minimum bargaining set

Important fact: For all φ ∈ Ik(v), mφ(S) ≤ 0, ∀S ⊆ N such that |S| ≥ 2. Corollary ∀φ ∈ Ik(v), φ is concave.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 17 / 27

The minimum bargaining set

m ◦ Ik is a general solution which satisfies: covariance under strategic equivalence (COV):

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 18 / 27

The minimum bargaining set

m ◦ Ik is a general solution which satisfies: covariance under strategic equivalence (COV):

◮ ∀v ∈ G(N), ∀α > 0, ∀β ∈ R2n−1:

Ik(αv + β) = αIk(v) + β.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 18 / 27

The minimum bargaining set

m ◦ Ik is a general solution which satisfies: covariance under strategic equivalence (COV):

◮ ∀v ∈ G(N), ∀α > 0, ∀β ∈ R2n−1:

Ik(αv + β) = αIk(v) + β.

idempotence (IDEM):

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 18 / 27

The minimum bargaining set

m ◦ Ik is a general solution which satisfies: covariance under strategic equivalence (COV):

◮ ∀v ∈ G(N), ∀α > 0, ∀β ∈ R2n−1:

Ik(αv + β) = αIk(v) + β.

idempotence (IDEM):

◮ ∀v ∈ G(N):

Ik ◦ Ik(v) = Ik(v).

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 18 / 27

The minimum bargaining set

m ◦ Ik is a general solution which satisfies: covariance under strategic equivalence (COV):

◮ ∀v ∈ G(N), ∀α > 0, ∀β ∈ R2n−1:

Ik(αv + β) = αIk(v) + β.

idempotence (IDEM):

◮ ∀v ∈ G(N):

Ik ◦ Ik(v) = Ik(v).

symmetry (SYM):

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 18 / 27

The minimum bargaining set

m ◦ Ik is a general solution which satisfies: covariance under strategic equivalence (COV):

◮ ∀v ∈ G(N), ∀α > 0, ∀β ∈ R2n−1:

Ik(αv + β) = αIk(v) + β.

idempotence (IDEM):

◮ ∀v ∈ G(N):

Ik ◦ Ik(v) = Ik(v).

symmetry (SYM):

◮ ∀v ∈ G(N), for all permutations π such that v(π(S)) = v(S) holds for

all S ⊆ N: Ik(π(v)) = Ik(v).

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 18 / 27

The minimum bargaining set

m ◦ Ik is a general solution which satisfies: covariance under strategic equivalence (COV):

◮ ∀v ∈ G(N), ∀α > 0, ∀β ∈ R2n−1:

Ik(αv + β) = αIk(v) + β.

idempotence (IDEM):

◮ ∀v ∈ G(N):

Ik ◦ Ik(v) = Ik(v).

symmetry (SYM):

◮ ∀v ∈ G(N), for all permutations π such that v(π(S)) = v(S) holds for

all S ⊆ N: Ik(π(v)) = Ik(v).

dummy player property (DPP):

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 18 / 27

The minimum bargaining set

m ◦ Ik is a general solution which satisfies: covariance under strategic equivalence (COV):

◮ ∀v ∈ G(N), ∀α > 0, ∀β ∈ R2n−1:

Ik(αv + β) = αIk(v) + β.

idempotence (IDEM):

◮ ∀v ∈ G(N):

Ik ◦ Ik(v) = Ik(v).

symmetry (SYM):

◮ ∀v ∈ G(N), for all permutations π such that v(π(S)) = v(S) holds for

all S ⊆ N: Ik(π(v)) = Ik(v).

dummy player property (DPP):

◮ ∀v ∈ G(N) such that i is dummy, we have:

∀x ∈ m ◦ Ik(v), xi = v(i).

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 18 / 27

The minimum bargaining set (Lp norm, p > 1)

Theorem for all Lp norms with p > 1, Ik(v) = C(v) if C(v) = ∅, otherwise Ik(v) is a singleton (The unique element of Ck(v) which minimizes the bargaining level).

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Case where · is the L1 norm and k = n

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Case where · is the L1 norm and k = n

We denote by In

1 the solution In for the L1 norm

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The minimum bargaining set ( · = L1 norm and k = n)

Definition Let σ be a general solution on G(N). We say that σ is a minimization of the global debt (MGD) if for all nonnegative games v, for all x ∈ σ(v)

• S⊆N

min(xS, 0)

• = ¯

t(v).

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 21 / 27

The minimum bargaining set ( · = L1 norm and k = n)

Definition Let σ be a general solution on G(N). We say that σ is a minimization of the global debt (MGD) if for all nonnegative games v, for all x ∈ σ(v)

• S⊆N

min(xS, 0)

• = ¯

t(v). → This axiom ensures that the grand coalition is not in debt more than necessary to ensure coalitional rationality.

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c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 21 / 27

The minimum bargaining set ( · = L1 norm and k = n)

Theorem Put MGDCR the set of general solutions which satisfy (MGD) and (CR). Then: m ◦ In

1 = ⊤(MGDCR).

→ m ◦ In

1 is the top element of the set of solutions which satisfy coalitional

rationality and which minimize the amount of the global debts. Remark We have the following inclusions: C(v) ⊆ GEC(v) ⊆ In

1(v) ⊆ Cn(v)

for all games v. It follows that GEC inherits the properties of In

1.

Remark S(In

1(v)) = S(GEC(v)) = EC(v)

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c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 22 / 27

The minimum bargaining set ( · = L1 norm and k = n)

Example We consider a game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 5 7 1 20 1 1 1

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The minimum bargaining set ( · = L1 norm and k = n)

Example We consider a game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 5 7 1 20 1 1 1 We have C(v) = ∅.

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c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 23 / 27

The minimum bargaining set ( · = L1 norm and k = n)

Example We consider a game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 5 7 1 20 1 1 1 We have C(v) = ∅. The game φ defined by: S 1 2 3 12 13 23 123 φ 10 10 1 20 6 6 1 belongs to In

1(v).

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 23 / 27

The minimum bargaining set ( · = L1 norm and k = n)

Example We consider a game v on N = {1, 2, 3} defined by: S 1 2 3 12 13 23 123 v(S) 5 7 1 20 1 1 1 We have C(v) = ∅. The game φ defined by: S 1 2 3 12 13 23 123 φ 10 10 1 20 6 6 1 belongs to In

1(v). Its M¨

• bius transform is

S 1 2 3 12 13 23 123 mφ(S) 10 10 1

• 5
• 5
• 10
• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 23 / 27

The minimum bargaining set ( · = L1 norm and k = n)

Definition We say that a coalition T ⊆ N is autonomous if for any payoff vector x of C(v¯

t(v)), it holds x(T) = v(T).

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c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 24 / 27

The minimum bargaining set ( · = L1 norm and k = n)

Definition We say that a coalition T ⊆ N is autonomous if for any payoff vector x of C(v¯

t(v)), it holds x(T) = v(T).

Proposition For any non-balanced game v, for any i ∈ N, there exists T⊂ N such that i ∈ T and T is autonomous.

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c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 24 / 27

The minimum bargaining set ( · = L1 norm and k = n)

Theorem Let v be a game on N, the following properties are equivalent:

1

S is autonomous

2

∀φ ∈ In

1(v), φ(S) = v(S).

→ the zeros of In

1(v) are the coalitions S such that |S| ≥ 2 and ∃T ⊇ S

such that T is autonomous.

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c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 25 / 27

Conclusion We have defined the concept of general solution that distributes among coalitions rather than among individuals. We have seen that the k-additive core is a general solution which preserves the general mind of the core, but we have proved that this solution is always unbounded. We have proposed a new general solution called minimal bargaining set, which is a nonempty compact and convex subset of the k-additive core, equal to the core if the core is nonempty, and that minimizes the amount given to coalitions in favour of individual players, for a given norm. For the Lp norm, p > 2, this solution is a singleton for nonbalanced games, and for the L1 norm, it is the top element of the set of solutions that satisfy coalitional rationality and that minimize the amount of the global debts.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 26 / 27

Thank you for your attention.

• S. Gonzalez & M. Grabisch

c 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) Preserving coalitional rationality for non-balanced games 27 / 27