In this paper, we are concerned with the preorderings (SS) and (BC) induced in the set of players of a simple game by the Shapley–Shubik and the Banzhaf–Coleman's indices, respectively. Our main result is a generalization of Tomiyama's 1987 result on ordinal power equivalence in simple games; more precisely, we obtain a characterization of the simple games for which the (SS) and the (BC) preorderings coincide with the desirability preordering (T), a concept introduced by Isbell (1958), and recently reconsidered by (...) Taylor (1995): this happens if and only if the game is swap robust, a concept introduced by Taylor and Zwicker (1993). Since any weighted majority game is swap robust, our result is therefore a generalization of Tomiyama's. Other results obtained in this paper say that the desirability relation keeps itself in all the veto-holder extensions of any simple game, and so does the (SS) preordering in all the veto-holder extensions of any swap robust simple game. (shrink)

In this paper we discuss the meaning of Owen's coalitional extension of the Banzhaf index in the context of voting situations. It is discussed the possibility of accommodating this index within the following model: in order to evaluate the likelihood of a voter to be crucial in making a decision by means of a voting rule a second input (apart from the rule itself) is necessary: an estimate of the probability of different vote configurations. It is shown how Owen's coalitional (...) extension can be seen as three different normative variations of this model. (shrink)