Shapley-Shubik and Banzhaf indices revisited

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.