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 6: Power as a Prize

Dan S. Felsenthal and Moshé Machover


6.1 P-Power: A Game-Theoretic Notion The notion of voting power studied in Ch. 3 was that held by Penrose [78], Banzhaf [5] and Coleman [20]; this notion is what we have called ‘I-power’ (see § 3.1). It assumes a policy-seeking motivation of voting behaviour: the way a voter votes on a given bill is determined by his or her attitude to the bill — an attitude which the voter presumably forms by comparing the expected payoff (to him or her) of the bill’s passage with that of its failure. These payoffs are independent of the decision rule and exogenous to it. Thus an SVG W by itself provides no information whatever as to how any voter might vote on an unspecified bill. This state of total a priori ignorance was encapsulated in the Bernoulli model BN , with which W must be supplemented. Notice that one and the same BN is shared by all SVGs with the same assembly N . In this chapter we shall examine an alternative notion of voting power, which we have termed P-power, first adopted by Shapley and Shubik [97]. This posits an office-seeking motivation of voting behaviour. We begin by sketching the intended meaning of P-power, which we shall then amplify and clarify in a series of comments. 6.1.1 Sketch The basic idea is that a division of a board is a play of a game whose rules are given by the decision rule operated by the board. If the outcome of the division is...

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