The Measurement of Voting Power
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The Measurement of Voting Power Theory and Practice, Problems and Paradoxes

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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Appendix B: Axiomatic Characterizations

Dan S. Felsenthal and Moshé Machover


B. Appendix: Axiomatic Characterizations B.1 Convention In this Appendix we shall use the terminology and notation of § 6.2. By ‘CF’ we shall always mean CF of some SVG (rather than a more general cooperative game). We shall attribute various properties of SVGs to their respective CFs. Thus, for example, when we say that a CF is proper we mean that its SVG is proper.1 Similarly, we shall attribute various pieces of furniture of an SVG to its CF. Thus, for example, by the assembly of a CF we mean the assembly of its SVG. We shall always denote a CF and its SVG by corresponding letters, possibly with affixes. So, for example, if we denote a CF by ‘wM ’, the reader should take it for granted that ‘WM ’ denotes its SVG. The characterization Thms. 6.2.14 and 6.2.15 fail if we replace the class of all games (or all superadditive games) on N by the class of CFs (or proper CFs) with assembly N . We shall illustrate this for the case n = 3, although counter-examples can be found for any n > 2. B.2 Example Let G be the class of all CFs with assembly I3 = {1, 2, 3}. Let ξ be any value assignment such that ξa (w) = βa [W] for all w ∈ G. 1 As we noted in Rem. 6.2.2(ii), this amounts to saying that the CF is superadditive. 299 300 B. Axiomatic Characterizations It is easy to see that ξ is iso-invariant, efficient and vanishes for...

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