# The Measurement of Voting Power

## Theory and Practice, Problems and Paradoxes

## Dan S. Felsenthal and Moshé Machover

# Appendix A: Numerical Examples

## Dan S. Felsenthal and Moshé Machover

## Extract

In Ex. A.1 we illustrate the calculation of the Bz measure and Bz index, using a fairly simple but non-trivial WVG with four voters. In Ex. A.2 we illustrate the calculation of the Shapley value, using a game with four players. In Ex. A.3 we return to the WVG of Ex. A.1 and calculate the S-S index for it. In Ex. A.4 we calculate the D-P and Js indices for the same WVG. A.1 Example (Bz Measure and Index) Let W be a WVG with assembly N = {a, b, c, d} that is isomorphic, in alphabetic order, to [5; 3, 2, 1, 1]. To compute the Bz power of each of the voters, you must ﬁrst list the coalitions of W and their critical members. Fortunately, you do not have to list all 16 coalitions: a coalition need be listed only if it has at least one critical member. Such a coalition is said to be vulnerable. Here are the ﬁve vulnerable coalitions of W, listed in order of size, with their critical members underlined: {a, b}, {a, b, c}, {a, b, d}, {a, c, d}, {a, b, c, d}. By Def. 3.2.2, the Bz power, β x , of voter x is equal to the number of [vulnerable] coalitions in which x is critical, divided by 2n−1 . Here n = 4, so in our WVG we have: βa = 5, βb = 3, βc = βd = 1. 8 8 8 Note that you need not repeat the calculation separately for c and d: these...

## You are not authenticated to view the full text of this chapter or article.

**Elgar**online requires a subscription or purchase to access the full text of books or journals. Please login through your library system or with your personal username and password on the homepage.

Non-subscribers can freely search the site, view abstracts/ extracts and download selected front matter and introductory chapters for personal use.

Your library may not have purchased all subject areas. If you are authenticated and think you should have access to this title, please contact your librarian.