Theory and Decision 44 (1):83-116 (1998)
|Abstract||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 â if it is to be used as a tool for analysing abstract, âuninhabitedâ decision rules â should satisfy this postulate. We show that, among known indices, only the Banzhaf measure does so. Moreover, the ShapleyâShubik, DeeganâPackel and Johnston indices sometimes witness a reversal under these circumstances, with voter x âless powerfulâ than y when measured in the simple voting game G1 , but âmore powerfulâ than y when G1 is âbicamerally joinedâ 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 â the price monotonicity condition. We discuss implications of these findings, in light of recent work showing that only the ShapleyâShubik 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)|
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