Aspects of Power Overlooked by Power Indices

Abstract

The a priori voting power indices concentrate on actor resource distributions and decision rules to determine the potential influence over outcomes by various actors. That these indices sometimes seem to be at odds with the intuitive distribution of real power in voting bodies follows naturally from their a priori nature. Indices based on actor preferences address this by equating an actor’s voting power with the proximity of voting outcomes to his/her ideal point. It is, however, shown that in some cases the preference-based indices are just as questionable as the classic ones. The main aim of this paper is to delineate the proper scope of power indices. In the pursuit of this aim we try to show that the procedures resorted to in making collective decisions are as important—if not more so—as the actor resource distribution. We review some results on agenda-systems to drive home this point. The proper role of power indices then turns out to be in the study of actor influences over outcomes when the actors are on the same level of aggregation (individuals, groups, states) and “comparable” in the sense of having similar sets of strategies at their disposal and preferences are not taken into consideration, e.g. because a veil of ignorance applies.

This paper was presented at The Leverhulme Trust sponsored Voting Power in Practice Symposium at the London School of Economics, 20–22 March 2011. Comments of the participants, in general, and of Dan S. Felsenthal, in particular, are gratefully acknowledged. We would also like to thank an anonymous referee for several perceptive and constructive suggestions that we have tried to accommodate in the present version.

Appendix

The Shapley-Shubik index value of player i is:

$$\displaystyle{\phi _{i} = \Sigma _{S\subseteq N}\frac{(s - 1)!(n - s)!} {n!} [v(S) - v(S\setminus \{i\})].}$$

Here s denotes the number of members of coalition S and n! is defined as the product $n \cdot (n - 1) \cdot (n - 2) \cdot \ldots 2 \cdot 1.$ The expression in square brackets differs from zero just in case S is winning but S∖{i} is not. In this case, then, i is a decisive member in S. In other words, i has a swing in S. Indeed, the Shapley-Shubik index value of i indicates the expected share of i’s swings in all swings assuming that coalitions are formed sequentially.

Player i’s PGI value H _i is computed as follows:

$$\displaystyle{H_{i} = \frac{\Sigma _{S{\ast}\subseteq N}[v(S{\ast}) - v(S {\ast}\setminus \{ i\})]} {\Sigma _{j\in N}\Sigma _{S{\ast}\subseteq N}[v(S{\ast}) - v(S {\ast}\setminus \{ j\})]}.}$$

Here S∗ is a minimal winning coalition, i.e. every proper subset of S∗ is a losing coalition.

The Deegan-Packel index value of player i, denoted DP _i , in turn, is obtained as follows:

$$\displaystyle{DP_{i} = \frac{\Sigma _{S{\ast}\subseteq N}1/s[v(S{\ast}) - v(S {\ast}\setminus \{ i\})]} {\Sigma _{j\in N}\Sigma _{S{\ast}\subseteq N}1/s[v(S{\ast}) - v(S {\ast}\setminus \{ j\})]}.}$$

The standardized Banzhaf index value of i is defined as:

$$\displaystyle{\bar{\beta }_{i} = \frac{\Sigma _{S\subseteq N}[v(S) - v(S\setminus \{i\})]} {\Sigma _{j\in N}\Sigma _{S\subseteq N}[v(S) - v(S\setminus \{j\})]}.}$$

The absolute Penrose-Banzhaf index (Penrose 1946; Banzhaf 1965), in turn, is defined as:

$$\displaystyle{\beta _{i} = \frac{\Sigma _{S\subseteq N}[v(S) - v(S\setminus \{i\})]} {2^{n-1}}.}$$