Joint-action-can-make-a-difference-Measures-of-voting-power-generalized.pdf
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1、Joint action can make a diff erence: Measures of voting power generalized Claus Beisbart Institute for Philosophy, Faculty 14, Technische Universit at Dortmund, D-44221 Dortmund, Germany LOFT 2008; a longer version of this paper is available in the LSE Philosophy Papers series 2008, http:/www.lse.ac
2、.uk/collections/CPNSS/CPNSS- DPS/LSEPhilosopyPapers.htm Abstract The political infl uence that a voter has under a specifi c decision rule is often measured in terms of the probability that that voters vote is critical. If it is calculated on the basis of the Bernoulli model, the popular Banzhaf mea
3、sure of (a priori) voting power is obtained. But the probability of being critical can also be calculated for alternative probability models. If they are constrained by empirical data, measures of a posteriori voting power arise. In the recent literature, it has been argued that the probability of c
4、riti- cality does not provide a suitable measure of a posteriori voting power. As an example due to G. Wilmers shows, this measure will assign zero voting power to every voter, if voting profi les under which at least one voter is critical have zero probability. It seems odd that nobody has any powe
5、r whatsoever, though. I therefore propose to go beyond the probability of being critical for measuring voting power. The probability of criticality quantifi es the extent to which a voter has the opportunity to make a diff erence as to whether a bill passes or not. Likewise, one can calculate the ex
6、tent to which a voter has the opportunity to fi nd other voters in order to form a group that makes a diff erence as to whether a bill passes or not. Put diff erently, for each voter, we look at the opportunities for group actions that involve her. I proceed in two steps. I fi rst defi ne criticalit
7、y for a group. Roughly, given a specifi c coalition, a group is critical, iff the following is true: There is some way in which the group could have voted diff erently such that the outcome of the vote would have been diff erent. The second step introduces criticality of higher ranks for individual
8、voters. Roughly, a voter is critical 1 2 of rank , iff there is a group of votes including a and with cardinality such that the group is critical. The proposed new measures are then the probabilities that a voter is critical of a fi xed rank. In order to avoid double-counting, I introduce a diff ere
9、ntial counting. As a result, for each voter there is a hierarchy of measures with ranks ranging from 1 to the cardinality of the assembly. The powers of the diff erent ranks add up to 1 for each voter. Roughly, you have overall more voting power than I have, if your measures start growing for smalle
10、r ranks than mine do. For rank 1 and the Bernoulli model, the new measure coincides with the Banzhaf measure. For higher ranks, additional information is provided. 1Introduction Voting rules assign voters the power to aff ect the outcome of collective deci- sions. This is the starting point for a re
11、search program at the borderline between political science, social choice theory, economics and political philosophy. The research program aims at measuring the voting power of each voter, i.e. the ex- tent to which her vote can aff ect the outcome of a collective decision (Felsenthal under probabil
12、ity models, e.g., that were fi tted to empirical data from past votes (see Good Machover 2007, p. 3). Example 1.1 Suppose that fi ve voters vote following simple majority voting. There are 32 voting profi les possible. Assume that the profi les with exactly two or exactly three yes-votes have probab
13、ility zero, each, and that the other profi les have a probability of 1/12, each. Consider an arbitrary voter. The probability of her being critical is zero, because all profi les under which a voter is critical have zero probability. Thus, everybody has zero voting power. But this assignment of a po
14、steriori power seems strange, to say the least. What is particularly off ensive is the claim that nobody has voting power. As Machover (2007), p. 3 puts it, “it would be absurd to claim that every voter here is powerless, in the sense of having no infl uence over the outcome .”. He concludes that, a
15、s a measure of voting power, the probability of being critical “behaves in a strange way . At least, . it doesnt tell the whole story about that infl uence a voters infl uence.” According to his diagnosis (Machover 2007, pp. 23), the counterintuitive assignment of voting power measures arises, becau
16、se the votes in the example are stochastically dependent. Wilmers example also points to a second, slightly diff erent problem. Proba- bility models can be specifi ed such that the probability of being critical is zero for each voter even under alternative voting rules for the same electorate. Accor
17、d- ingly, the related measure of a posteriori voting power assigns every voter zero power for each of the alternative rules. Thus, we can not distinguish between the alternative voting rules in terms of power. This is dissatisfying. Is it really true that the alternative voting rules are completely
18、on par as far as the powers to aff ect the outcome are concerned? The probability of being critical has not the discriminatory power that we wish it to have. Even the Banzhaf measure suff ers from a similar problem.Although the Bernoulli model assigns each voting profi le a non-zero probability, the
19、 following is possible under the Bernoulli model. Two voting rules are diff erent there is at least one voting profi le under which one rule yields acceptance, whereas the other rule yields rejection but for each voter, the probability of being critical, i.e. the Banzhaf measure is identical under b
20、oth rules. An example and qualifi cations 1 Of course, it is possible that empirical data favor the Bernoulli model for some specifi c setting. Thus, measures of a posteriori voting power may use the Bernoulli model as well. 4 will be given below in Subsec. 4.3. The question is again whether the alt
21、ernative voting rules are really completely on par, as far as voting power is concerned. This paper proposes to go beyond the probability of criticality in order to measure voting power. I look for measures of voting power that fulfi ll the following requirements: R1 When calculated under the assump
22、tion of the Bernoulli model, the measures partly coincide with the Banzhaf measure of voting power. R2 The measures are conceptually tied to the notion of criticality. R3 The measures concern individual voters. R4 The contraintuitive results for Wilmers example are avoided under the measures. These
23、requirements are not beyond criticism. In particular, R2 might be given up. For instance, in order to measure power, one might start from the Shapley- Shubik index (Shapley group criticality is defi ned in Sec. 3. In Sec. 4, I provide the defi nition of my measure. A few math- ematical results are p
24、roven, and applications are discussed. Finally, discussion points are given in Sec. 5. 2Voting power In order to introduce the main idea of this paper, I will fi rst consider the standard notion of voting power, or I-power, more specifi cally. Let me start with the notion of power. According to Morr
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