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2014 | OriginalPaper | Chapter

Voting Power and Probability

Author : Claus Beisbart

Published in: Voting Power and Procedures

Publisher: Springer International Publishing

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Abstract

Voting power is commonly measured using a probability. But what kind of probability is this? Is it a degree of belief or an objective chance or some other sort of probability? The aim of this paper is to answer this question. The answer depends on the use to which a measure of voting power is put. Some objectivist interpretations of probabilities are appropriate when we employ such a measure for descriptive purposes. By contrast, when voting power is used to normatively assess voting rules, the probabilities are best understood as classical probabilities, which count possibilities. This is so because, from a normative stance, voting power is most plausibly taken to concern rights and thus possibilities. The classical interpretation also underwrites the use of the Bernoulli model upon which the Penrose/Banzhaf measure is based.

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previous chapter Voting Power Techniques: What Do They Measure?

next chapter A Probabilistic Re-View on Felsenthal and Machover’s “The Measurement of Voting Power”

See e.g. (Felsenthal and Machover 2000, p. 17). In this paper, I will focus on I-power, which is about influence, while I bracket P-power, which is about prizes. See (Felsenthal and Machover 1998, p. 36) for the definition of both types of power.

See Felsenthal and Machover (1998), Definition 3.1.1 on p. 37 for this model.

There is no presumption that all probabilities should be understood in the same way; rather, each probabilistic statement calls for its own interpretation. My focus here is entirely on probabilities that arise in the measurement of voting power. See (Hájek 1997, pp. 210–211) and (Hájek 2010, Sect. 2) for useful methodological remarks about the interpretation of probabilities. See Gillies (2000) for a text book on the philosophy of probability and Eagle (2011) for an anthology.

See Kolmogorov (1956) for a famous version of the axioms.

I will here assume that probabilities are ascribed to events from an event space. An alternative option is to assign probabilities to propositions, see e.g. (Howson and Urbach 2006, pp. 13–14).

See (Felsenthal and Machover 1998, Chap. 6).

Felsenthal and Machover (1998, p. 210) deny that the Shapley-Shubik index is a priori because it is not based upon the Principle of Indifference. However, the principle has come under attack because it does not lead to unambiguous results in some cases (see Gillies 2000, pp. 37–49 for a textbook account of the principle and its problems). Further, it is arguable that the principle leads to the Shapley-Shubik index provided that the space of ultimate possibilities is re-defined (see below; see Mellor 2005, pp. 24–26 for ultimate possibilities).

See Felsenthal and Machover (1998), Definition 2.1.1 on p. 21 for the definition of monotonicity.

For example (Morriss 1987, p. 19).

See Fara (2009) for an introduction to dispositions.

Voting powers are sometimes ascribed to votes, and at other times to voters. This will not make a difference in what follows, and, for simplicity, I will always assign voting power to voters.

There is a philosophical debate about what exactly probabilities qua strengths of dispositions attach to. Some have suggested that, properly speaking, the related chance set-up includes the whole world. See (Gillies 2000, pp. 126–129) for an overview of corresponding positions.

See Felsenthal and Machover (1998), Definitions 2.3.4 and 2.3.6 on pp. 24–25.

This is not to reject the general distinction between power and influence, which is rightly stressed by Morriss (1987, Chap. 2). I only think that the distinction crumbles if we turn to voting power. There is an important difference to other sorts of powers at this point. I can have the power to play Beethoven’s Pathétique on the piano, but simply decide not to execute the power. But if I have the power to make a difference in a collective decision, I cannot decide not to execute this power.

See also (Eagle 2004, pp. 377–383) for a more refined classification of propensity theories.

In the terms of Eagle (2004), my discussion of a single-case propensity theory is restricted to what Eagle calls tendency accounts (p. 379). I thus bracket the distribution display account attributed to Mellor. This account is close to Lewis’s account, which will be discussed below.

See (Eagle 2004, pp. 384–385) for this criticism.

Some authors call probabilities objective iff a weaker condition is fulfilled, viz. that their values are uniquely fixed for rational persons. See e.g. (Uffink 2011, pp. 25–26) for this point. In this paper, I use the stronger notion of objectivity.

See Hájek (1997) for a discussion of this position.

(Hájek 2009, pp. 218–220).

(Hájek 2009, pp. 217–218).

See Hájek (2009) and Eagle (2004) for criticism of views that appeal to hypothetical frequencies.

See Lewis (1994) and consult Lewis (1980) for an important fore-runner. For recent appraisals see Loewer (2004), Hoefer (2007), Frigg and Hoefer (2009).

Cf. (Lewis 1994, p. 481).

See Ramsey (1931) and de Finetti (1931a), de Finetti (1931b), de Finetti (1937) for important original contributions, and (Gillies 2000, Chap. 5) and (Mellor 2005, Chap. 5) for textbook accounts.

Cf. (Morriss 1987, Chap. 6).

See (Felsenthal and Machover 2000, p. 13) for the expression of related worries.

See Gelman et al. (2004) for empirical work about the probability of pivotality.

When Laruelle and Valenciano (2005) explain what they mean by the “normative point of view” (183), they say that a related assessment is concerned with a voting situation or rule, “irrespective of what voters occupy the seats.” It may be objected that this is too narrow a conception of “normative”.

Consult Gillies (2000, Chap. 2) and Mellor (2005, Chap. 2) for this interpretation.

This is a contrast with some propensity views, see (Mellor 2005, p. 25) for some details.

See (Mellor 2005, pp. 25–26) for a discussion.

See (Mellor 2005, pp. 25–26) though.

See (Gillies 2000, pp. 37–49) again.

(Mellor 2005, p. 24).

See Felsenthal and Machover (1998), Remark 6.3.12(ii) on p. 207.

Consult Theorem 6.3.13 on p. 208 in Felsenthal and Machover (1998) for the mathematical basis of this argument.

This is not to deny that the Shapley-Shubik index may succeed as a measure of P-power. See (Felsenthal and Machover 1998, Chap. 6).

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Title: Voting Power and Probability
Author: Claus Beisbart
Publisher: Springer International Publishing
Book: Voting Power and Procedures
Print ISBN: 978-3-319-05157-4

Electronic ISBN: 978-3-319-05158-1

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-3-319-05158-1_6