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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.

# Chapter 2: Groundwork of the Theory

## Dan S. Felsenthal and Moshé Machover

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

## Extract

2.1 Simple Voting Games We begin by deﬁning the most general class of mathematical structures commonly used to model voting decision rules. This, then, is the basic deﬁnition of the theory. 2.1.1 Deﬁnition A simple voting game — brieﬂy, SVG — is a collection W of subsets of a ﬁnite set N , satisfying the following three conditions: (1) N ∈ W; (2) ∅ ∈ W; (3) Monotonicity: whenever X ⊆ Y ⊆ N and X ∈ W then also Y ∈ W. W is said to be a proper SVG if, in addition, it satisﬁes the condition (4) Whenever X ∈ W and Y ∈ W then X ∩ Y = ∅. Otherwise, W is said to be improper. We shall refer to N , the largest set in W, as the latter’s assembly. The members of N are the voters of W. A set of voters (that is, a subset of N ) is called a coalition of W. A coalition S is said to be a winning or losing coalition, according as S ∈ W or S ∈ W. 2.1.2 Remarks (i) Apart from some inessential modiﬁcations, this deﬁnition is the same as that given by Shapley in [95] for what he calls ‘simple game’. He attributes the concept to von 11 12 2. Groundwork of the Theory Neumann and Morgenstern [108], but as a matter of fact his class of simple games is considerably wider than that admitted by [108]. To prevent confusion with the latter, we use the term ‘simple voting game’ for the broader concept. In...