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# The Measurement of Voting Power

## Dan S. Felsenthal and Moshé Machover

This book is the first of its kind: a monograph devoted to a systematic critical examination and exposition of the theory of a priori voting power. This important branch of social-choice theory overlaps with game theory and is concerned with the ability of members in bodies that make yes or no decisions by vote to affect the outcome. The book includes, among other topics, a reasoned distinction between two fundamental types of voting power, the authors' discoveries on the paradoxes of voting power, and a novel analysis of decision rules that admit abstention.

# Appendix A: Numerical Examples

## Dan S. Felsenthal and Moshé Machover

Subjects: economics and finance, econometrics, game theory, public choice theory, politics and public policy, public choice

## 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...