Brams’ paradox of new members and Shenoy’s paradox of smaller coalitions are, in a sense, equivalent. They are both implied by the monotonicity of a power index: while the first is exhibited on every simple game that is not strong, the latter can be observed on every simple game in which players are not almost symmetric. For the Shapley–Shubik index, this symmetry condition is not only necessary but also sufficient to avoid the paradox of smaller coalitions.

Monotonicity is commonly considered an essential requirement for power measures; violation of local monotonicity or related postulates supposedly disqualifies an index as a valid yardstick for measuring power. This paper questions if such claims are really warranted. In the light of features of real-world collective decision making such as coalition formation processes, ideological affinities, a priori unions, and strategic interaction, standard notions of monotonicity are too narrowly defined. A power measure should be able to indicate that power is non-monotonic in (...) a given dimension of players' resources if – given a decision environment and plausible assumptions about behaviour – itis non-monotonic. (shrink)

We analyse and evaluate the qualified majority (QM) decision rules for the Council of Ministers of the EU that are included in the Draft Constitution for Europe proposed by the European Convention [5]. We use a method similar to the one we used in [9] for the QM prescriptions made in the Treaty of Nice.

In this account, we explain the meaning of a priori voting power and outline how it is measured. We distinguish two intuitive notions as to what voting power means, leading to two approaches to measuring it. We discuss some philosophical and pragmatic objections, according to which a priori (as distinct from actual) voting power is worthless or inapplicable.