Measuring disproportionality, volatility and malapportionment: Axiomatization and solutions

The measurement of disproportionality, volatility and malapportionment often employ similar indices. Yet the debate on the issue of adequate measurement has remained open. We offer a formal and rigorous list of properties that roughly subsume those of Taagepera and Grofman (Party Polit 9(6):659-677, 2003). One of these properties, Dalton's principle of transfers, is formalized in a manner that resolves the ambiguity associated with it in previous studies. We show that the cosine measure satisfies all the properties. We also show how the Gallagher index can be modified to satisfy all the properties. The cosine measure and the modified Gallagher measure are co-monotone.