Theory and Practice, Problems and Paradoxes
6.1 P-Power: A Game-Theoretic Notion The notion of voting power studied in Ch. 3 was that held by Penrose , Banzhaf  and Coleman ; 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 . 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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