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 3: Power as Influence

Dan S. Felsenthal and Moshé Machover


3. Power as Influence 3.1 I-Power: A Probabilistic Notion In § 1.1 we said, by way of informal explanation, that the voting power of a member of a board, under a given decision rule, is the extent to which the member is able to control the outcome of a division of the board. We should now like to make this rough explanation more precise. In the voting-power literature there are in fact two different interpretations of a voter’s ‘ability to control the outcome of a division’, corresponding to the two motivations of voting behaviour discussed in Com. 2.2.2. However, these two interpretations are seldom clearly distinguished; more often than not they are conflated with each other.1 From the policy-seeking viewpoint on voting, the outcome of a division is simply the passage or failure of the bill in question. A bill having been proposed, each member forms a position — for or against it — and votes accordingly.2 Rational voters presumably form their positions by comparing the expected payoff of the bill’s passage with that of its failure (here ‘payoff’ must of course be understood in a very broad sense). These payoffs vary from member to member and from bill to bill; but — for a given member and a given bill — they are quite independent of the decision rule operated by the board. 1 A notable exception is [20], where the distinction is made quite explicitly (see passages quoted in Com. 2.2.2). This distinction is amplified in [36...

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