Shapley-Shubik and Banzhaf Indices Revisited

Abstract

We provide a new axiomatization of the Shapley-Shubik and the Banzhaf power indices in the domain of simple superadditive games by means of transparent axioms. Only anonymity is shared with the former characterizations in the literature. The rest of the axioms are substituted by more transparent ones in terms of power in collective decision-making procedures. In particular, a clear restatement and a weaker alternative for the transfer axiom are proposed. Only one axiom differentiates the characterization of either index, and these differentiating axioms provide a new point of comparison. In a first step both indices are characterized up to a zero and a unit of scale. Then both indices are singled out by simple normalizing axioms.

Symmetric Coalitional Binomial Semivalues

“…In the papers by Straffin [53], Laruelle [34] and Laruelle and Valenciano [35], the Banzhaf value β is suggested as a power measure more suitable than the Shapley value. The natural generalization to semivalues has been argued by Laruelle and Valenciano [36], Carreras and Freixas [17], and Carreras, Freixas and Puente [20].…”

Power indices and the veil of ignorance

“…It is in these terms that all axioms should be interpreted. Then the following assumptions are the result of translating into the present framework the axioms reviewed in Section 2, used by us in Laruelle and Valenciano (1999) to characterize the Shapley-Shubik and Banzhaf indices up to a zero and a unit in SG n : Namely, in what follows all assumptions refer to a preference ordering ¹ on L(SG n £ N) (for we will not use the zero-game). In order to make it clearer this correspondence we use the same names, full or abbreviated, just adding one asterisk.…”

“…In Section 2 the basic game theoretical background is given along with a summary of our characterization of the Shapley-Shubik and Banzhaf indices in Laruelle and Valenciano (1999). In Section 3 Roth's (1977b) setting and characterization are brie°y reviewed.…”

Political influence in multi-choice institutions: cyclicity, anonymity, and transitivity

“…It consists of a binary relation called "replacement relation" or "in ‡uence relation", which ranks players according to their a priori in ‡uence in a vote. 1 The two approaches have been essentially developed in the basic framework of binary voting games, where a voter may only vote "yes" or "no" (see, e.g., Laruelle and Valenciano 2001;Carreras and Freixas 2005;Freixas and Pons 2008).…”