Voting Power and Voting Blocs

Abstract

Abstract. 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 argue against the Shapley-Shubik index and show that anyway the Shapley-Shubik index per head is inappropriate for voting blocs. 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 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; it examines the consequences of two blocs but can easily be extended to more.

Voting Power and Procedures

“…Again the shape of curves is very similar for both indices. The phenomenon of non-monotonicity of BC index of a party member treated as a function of the own party's size in composite games was also examined in the paper of Leech and Leech (2006), where it was interpreted …”

“…In case of theˇc index the power of a voter decomposes into two factors-one is the individual power in the internal voting and second is the power of a party as a whole. Relations between these two factors were examined in Leech and Leech (2006). On the other hand we can interpret the index O BC as a measure of power of a member of a party where there is no party whip, assuming that in all other parties voters follow the party discipline.…”

The exact bias of the Banzhaf measure of power when votes are neither equiprobable nor independent

“…In this case, pairwise correlation between an outsider and a bloc is equivalent to pairwise correlation between the outsider and a hypothetical member holding the total weight of the bloc in votes. The voting blocs typically discussed in the literature are deterministic (e.g., Leech and Leech (2004)). …”

Does everyone have equal voting power?

“…Holler (1985), for example, introduces a randomized voting rule as a solution to the problem of achieving equality between voter's weight and his voting power and analyses the properties of such rules. Leech and Leech (2006) look at the sensitivity of voting indices when people form blocs. They consider the Penrose voting index and investigate how voting power varies as bloc size varies.…”