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 7: Paradoxes and Postulates

Dan S. Felsenthal and Moshé Machover


1 7.1 Preliminaries Voting power, however you measure it, seems to be a strange beast, often displaying behaviour that ranges from the slightly surprising to the bizarre. This chapter is devoted to the description, explanation and classification of the so-called paradoxes of voting power. By paradox we mean a true proposition that appears to be absurd.2 By extension, a real phenomenon that seems to be contrary to common sense is also referred to as a paradox. Paradoxicality is a matter of degree: a true proposition may be slightly surprising or barely believable. It is also largely subjective: an experienced well-informed prudent observer may be unimpressed by a phenomenon that astonishes a na¨ novice. ıve More often than not, authors on voting power are content to point at some apparently strange piece of behaviour of, say, the S-S and Bz indices, and declare it to be a voting-power paradox. But this begs the question as to whether the alleged paradox is inherent in the very notion of voting power or merely an artefact of 1 2 This chapter is largely based on [29], [36] and [33]. In his book Paradoxes [91, p. 1], Sainsbury defines paradox somewhat differently, as ‘an apparently unacceptable conclusion derived by apparently acceptable reasoning from apparently acceptable premises. Appearances have to deceive, since the acceptable cannot lead by acceptable steps to the unacceptable. So generally we have a choice: either the conclusion is not really unacceptable, or else the starting point, or the reasoning,...

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