Theory and Decision 53 (4):313-325 (2002)

Abstract
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
Keywords Voting games  Power indices  Desirability relation  Veto-holder extension  Swap robust
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DOI 10.1023/A:1024158301610
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Hierarchies Achievable in Simple Games.Josep Freixas & Montserrat Pons - 2010 - Theory and Decision 68 (4):393-404.

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