Previous work by Diffo Lambo and Moulen [Theory and Decision 53, 313–325 (2002)] and Felsenthal and Machover [The Measurement of Voting Power, Edward Elgar Publishing Limited (1998)], shows that all swap preserving measures of voting power are ordinally equivalent on any swap robust simple voting game. Swap preserving measures include the Banzhaf, the Shapley–Shubik and other commonly used measures of a priori voting power. In this paper, we completely characterize the achievable hierarchies for any such measure on a swap robust simple voting game. Each possible hierarchy can be induced by a weighted voting game and we provide a constructive proof of this result. In particular, the strict hierarchy is always achievable as long as there are at least five players. Copyright Springer 2006desirability relation, ordinal equivalence, power indices, swap robust, voting games,

hierarchy, ordinal equivalence, power indices, simple majority, voting games,

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Consider the collection of all integer partitions whose part sizes lie in a given set. Such a set is called monotone if the generating function has weakly increasing coefficients. The monotone subsets are classified, assuming an open conjecture.