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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,...