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 c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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: c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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: In many situations, decision makers do not make a sharing among 1 individuals, but often give to groups (associations, companies, families, etc.) c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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: In many situations, decision makers do not make a sharing among 1 individuals, but often give to groups (associations, companies, families, etc.) In many cases, the core of the game is empty. 2 c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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: In many situations, decision makers do not make a sharing among 1 individuals, but often give to groups (associations, companies, families, etc.) In many cases, the core of the game is empty. 2 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. c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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 c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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, c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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 of N . c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 3 / 27

Introduction Interest of this point of view? c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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. c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 4 / 27

Introduction Example S 1 2 3 12 13 23 123 v(S) 1 1 1 1.5 1.5 1.5 2 c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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. c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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. c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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. c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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. c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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. c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 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 c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 5 / 27

General Solution A general efficient payoff vector of v is a vector x ∈ R 2 N \∅ that � assigns to a coalition S ⊆ N a payoff x S such that x S = v ( N ) . ∅� = S ⊆ N c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 6 / 27

General Solution A general efficient payoff vector of v is a vector x ∈ R 2 N \∅ that � assigns to a coalition S ⊆ N a payoff x S such that x S = v ( N ) . ∅� = S ⊆ N In fact, x S can be seen as the value of the M¨ obius transform (or Harsanyi dividends ) m φ ( S ) on the coalition S , where φ is the game defined by: � φ ( S ) = x T . T ⊆ S and � ( − 1 ) | S \ T | φ ( T ) , m φ ( S ) := ∀ S ⊆ N . T ⊆ S c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 6 / 27

General Solution Definition A general solution on the set of games G ( N ) is a mapping σ : G ( N ) → 2 ( R 2 n − 1 ) such that: ∀ v ∈ G ( N ) , σ ( v ) is a set of general payoffs efficient for the game v . c S. Gonzalez & M. Grabisch Preserving coalitional rationality for non-balanced games � 2012 (Centre d’´ economie de la Sorbonne Universit´ e Paris I) 7 / 27