An axiomatic characterization for regular semivalues

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Abstract

Semivalues form a wide family of solutions for cooperative games. The payoff that a semivalue allocates to each player in a given game is a weighted sum of the marginal contributions of the player in the game, and the weighting coefficients depend only on the coalition size. In this paper we provide an axiomatic characterization for each semivalue the weighting coefficients of which are all positive (regular semivalue).

Introduction

The Shapley value (Shapley, 1953) and the Banzhaf value (Banzhaf, 1965, Owen, 1975) are two of the most important solution concepts for cooperative games. A generalization of both solutions is formed by the family of semivalues (Dubey et al., 1981), allocation rules that assign to each player in a given game a weighted sum of the marginal contributions of the player in the game. The weight system depends only on the coalition sizes and all weighting coefficients are nonnegative. We speak of regular semivalues (Carreras and Freixas, 1999) if all weighting coefficients are positive. In this case, all marginal contributions of each player matter when we consider the allocation by such a semivalue. The Shapley value and the Banzhaf value are regular semivalues.

Many properties of the Shapley value extend, first, to the Banzhaf value and, later, to all semivalues. For instance, the potential notion, introduced by Hart and Mas-Colell (1989) for the Shapley value and later defined for the Banzhaf value (Dragan, 1996) and for all semivalue (Dragan, 1999). And also the computation of allocations by means of the multilinear extension, first obtained for the Shapley value (Owen, 1972) and the Banzhaf value (Owen, 1975) and extended by Amer and Giménez (2003) to all semivalues.

Besides, different authors have provided systems of properties in order to axiomatically characterize the Shapley value or the Banzhaf value. For instance, Shapley (1953), Hart and Mas-Colell (1989) or Feltkamp (1995) for the Shapley value, and Lehrer (1988) or Feltkamp (1995) for the Banzhaf value.

In the characterization of the Banzhaf value in Lehrer (1988), a cooperative game essential to the axiomatic system is introduced: the quotient game in the case where only two players join together. The procedure followed here is a generalization of Lehrer's method. It requires the introduction of a particular game for each regular semivalue.

The aim of the present paper is to state an axiomatic characterization for each regular semivalue. The paper is then organized as follows. After a preliminary Section 2, we introduce in Section 3 the family of delegation games. In Section 4 we provide an axiomatic system for each regular semivalue based on linearity, the dummy player property, symmetry, and a specific property concerning delegation games. Finally, some particular cases are analyzed in Section 5.

Section snippets

Preliminaries

A cooperative game with transferable utility is a pair (N, v), where N is a finite set of players and v:2N→R is the so-called characteristic function, which assigns to every coalition S ⊆ N a real number v (S), the gain or worth of coalition S, and satisfies the natural condition v (∅) = 0. By GN we denote the set of all cooperative games on N. For a given set of players N, we identify each game (N, v) with its characteristic function v.

With the usual operations of addition, given by (v1 + v2)(S) = v1 (S) +

The family of delegation games

Definition 2

Let N be a finite player set with |N| ≥ 2. Given a game v ∈ GN, distinct players i, j ∈ N and a pair of families of nonnegative real numbers {as}s = 1n − 1, {bs}s = 1n − 1, we define the family of delegation games from player j to player i by:vi⁢◃⁢j{as}s=1n−1,{bs}s=1n−1(S)={v⁡(S)+as[v⁡(S∪{j})−v⁡(S)]ifi∈S,v⁡(S)−bs[v⁡(S)−v⁡(S⁢\{j})]ifi∉S,where s = |S| and S ⊆ N.

For short, we write game vi⁢◃⁢j{as}s=1n−1,{bs}s=1n−1 as vi⁢◃⁢j{as},{bs} having in mind that both families of numbers have n − 1 elements.

In utility terms,

An axiomatic system for regular semivalues

Let GN be the vector space of all cooperative games on a finite set N, with |N| ≥ 2. For allocation rules X:GN→RN we consider the following properties:

P1

Linearity. X⁡[u+v]=X⁡[u]+X⁡[ν]andX⁡[λ⁢u]=λ⁢X⁡[u]∀u,v∈GN,∀λ∈R.

P2

Dummy player. If i ∈ N is a dummy player in v ∈ GN, then Xi [v] = v ({i}).

P3

Symmetry. If v (S ⋃ {i}) = v (S ⋃ {j}) ∀S ⊆ N \ {i, j}, then Xi [v] = Xj [v].

P4

Delegation transfer. Given two families of nonnegative real numbers {as}s = 1n − 1 and {bs}s = 1n − 1, if i, j ∈ N, then Xi[vi⁢◃⁢j{as},{bs}]=Xi⁡[v]+Xj⁡[v].

Properties P1, P2

Concluding remarks and examples

(i) The family of delegation games from player j to player i forms a set of games that are obtained by modifying an initial game v ∈ GN. It is possible to determine as many delegation games as families of nonnegative real numbers {as}s = 1n − 1 and {bs}s = 1n − 1 that define the levels of increase or loss of marginal contribution according to either the presence or the absence of player i in each coalition.

Proposition 5 exactly determines the delegation games that satisfy the transfer property for

Acknowledgement

Research partially supported by Grant BFM 2003-01314 of the Spanish Ministry of Science and Technology and the European Regional Development Fund.

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