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

The desirability relation was introduced by Isbell (1958) 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 Banzahf-Coleman power indices (Owen, 1977, 1981). 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.