The Measurement of Voting Power
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The Measurement of Voting Power

Theory and Practice, Problems and Paradoxes

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.
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Chapter 2: Groundwork of the Theory

Dan S. Felsenthal and Moshé Machover


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...

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