Voting Power and Voting Blocs

Dennis Leech and Robert Leech

November 2004

We investigate the method of power indices to study voting power of members of a

legislature that has voting blocs. Our analysis is theoretical, intended to contribute to a theory

of positive political science in which social actors are motivated by the pursuit of power as

measured by objective power indices. Our starting points are the papers by Riker

(Behavioural Science, 1959, “A test of the adequacy of the power index”) and Coleman

(American Sociological Review, 1973, “Loss of Power”). We apply the Penrose index (the

absolute Banzhaf index) to a hypothetical voting body with 100 members. We show how the

power indices of individual bloc members can be used to study the implications of the

formation of blocs and how voting power varies as bloc size varies. We argue that the

Shapley-Shubik index is inapplicable to this context and moreover that the Shapley-Shubik

index per head – which has been used in some studies - is inappropriate. We briefly consider

incentives to migrate between blocs. This technique of analysis has many real world

applications to legislatures and international bodies. It can be generalised in many ways: our

analysis is a priori, assuming formal voting and ignoring actual voting behaviour, but can be

made empirical with voting data reflecting behaviour; the paper examines the consequences

of two blocs but can easily be extended to more.

Keywords: Power Index; Penrose Index; Banzhaf Index; Shapley-Shubik Index; Voting

Power.