A note on the ordinal equivalence of power indices in games with coalition structure

The desirability relation was introduced by Isbell (A class of simple games 25: 423–439, <CitationRef CitationID="CR8">1958</CitationRef>) to qualitatively compare the a priori influence of voters in a simple game. In this paper, we extend this desirability relation to simple games with coalition structure. In these games, players organize themselves into a priori disjoint coalitions. It appears that the desirability relation defined in this paper is a complete preorder in the class of swap-robust games. We also compare our desirability relation with the preorders induced by the generalizations to games with coalition structure of the Shapley–Shubik and Banzhaf–Coleman power indices (Owen, Lecture notes in economics and mathematical systems: Essays in honor of Oskar Morgenstern. In: Hernn R, Moschlin O (eds.) Values of games with a priori unions, <CitationRef CitationID="CR11">1977</CitationRef>; Owen, Power, voting and voting power. In: Holler MJ (ed.) Modification of the Banzhaf–Coleman index for games with a priori unions, <CitationRef CitationID="CR12">1981</CitationRef>). It happens that in general they are different even if one considers the subclass of weighed voting games. However, if structural coalitions have equal size then both Owen–Banzhaf and the desirability preordering coincide. Copyright Springer Science+Business Media New York 2015