The Bicameral Postulates and Indices of a Priori Voting Power

If K is an index of relative voting power for simple voting games, the bicameral postulate requires that the distribution of K -power within a voting assembly, as measured by the ratios of the powers of the voters, be independent of whether the assembly is viewed as a separate legislature or as one chamber of a bicameral system, provided that there are no voters common to both chambers. We argue that a reasonable index – if it is to be used as a tool for analysing abstract, uninhabited decision rules – should satisfy this postulate. We show that, among known indices, only the Banzhaf measure does so. Moreover, the Shapley–Shubik, Deegan–Packel and Johnston indices sometimes witness a reversal under these circumstances, with voter x less powerful than y when measured in the simple voting game G₁ , but more powerful than y when G₁ is bicamerally joined with a second chamber G₂ . Thus these three indices violate a weaker, and correspondingly more compelling, form of the bicameral postulate. It is also shown that these indices are not always co-monotonic with the Banzhaf index and that as a result they infringe another intuitively plausible condition – the price monotonicity condition. We discuss implications of these findings, in light of recent work showing that only the Shapley–Shubik index, among known measures, satisfies another compelling principle known as the bloc postulate. We also propose a distinction between two separate aspects of voting power: power as share in a fixed purse (P-power) and power as influence (I-power).     

Class locations,consumption locations,and the geography of voting in England

Much ecological analysis of voting patterns in England has been concerned to show (a) the importance of occupational class as a determinant of electoral behavior, and (b) that deviations from a predicted pattern based on class variables are consistent with the neighborhood effect hypothesis. Recently, Dunleavy has criticized both these orientations, replacing them with the concept of consumption locations. The present paper shows, using a new procedure to estimate voting at the constituency level, that there still remain spatial variations to be accounted for.