A Yes-No combinatorial look on the Shapley-Shubik index

Présentation de Nicolas Andjiga (University of Yaounde)

Co-auteurs : Issofa Moyouwou, Fabrice Valognes

Résumé : Man and Shapley (1964), and Felsenthal and Machover (1996) have presented and analyzed voting contexts where each voter can cast a “Yes” or “No” opinion in a roll-call. And given a simple voting game, a pivotal voter in a roll-call refers to the voter whose vote completely determines the final outcome no matter the votes of his/her successors. Using an axiomatic approach, it has been indirectly shown that the Shapley-Shubik index of a voter is the probability of that voter to be pivotal in a roll-call. A complete combinatorial analysis of the Shapley-Shubik index of voting power is provided by a detailed investigation of the pivotality in all possible roll-call. By so doing, we address the challenging combinatorial problem raised in Felsenthal and Machover (1996) by identifying occurrences of positive pivotality (for adoption) and also those of negative pivotality (for rejection). As result, we obtain an instructive decomposition of the Shapey-Shubik index into a convex combination of two new indices. Some appealing properties of the two new indices are then explored.