THE BICAMERAL POSTULATES AND INDEXES 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 vote rs, be independent of whether the assembly is viewed as a separate leg islature or as one chamber of a bicameral system, provided that there are no voters common to both chambers. We argue that a reasonable inde x - 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 revers al under these circumstances, with voter x 'less powerful' than y when measured in the simple voting game G(1), but 'more powerful' than y w hen G(1) is 'bicamerally joined' with a second chamber G(2). Thus thes e 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 resul t they infringe another intuitively plausible condition - the price mo notonicity condition. We discuss implications of these findings, in li ght of recent work showing that only the Shapley-Shubik index, among k nown measures, satisfies another compelling principle known as the blo c postulate. We also propose a distinction between two separate aspect s of voting power: power as share in a fixed purse (P-power) and power as influence (I-power).