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A comprehensive analysis of the Shapley-Shubik [13] index of voting power is provided through a detailed investigation of voters’ pivotality in all possible roll-calls, as described in Felsenthal and Machover [4]. In a simple voting game, a pivotal voter in a roll-call is the one whose vote completely determines the final outcome, regardless of the votes of their successors. The present paper sheds new light on the Shapley-Shubik index of voting power by distinguishing between positive pivotality (for adoption) and negative pivotality (for rejection) which considerably strengthens their original result. This distinction has been first proposed by Hu [6] for the Shapley-Shubik power index and the Banzhaf index [1] to the case of “blocking”. We go deeper into the analysis and show that the Shapley-Shubik index is a convex combination of two new indices we provide namely the positive pivotality index and the negative pivotality index. Finally, we present an axiomatic characterization for each of these two new indices by using two weighted versions of the classical transfer axiom developed by Dubey [3].
