Joint-action-can-make-a-difference-Measures-of-voting-power-generalized.pdf

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.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 measure 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 criti- 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 power 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 extent 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 criticality 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 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 erential 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 smaller 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 research 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 probability 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 probability 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 posteriori 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, as 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, because 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. Accord- 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 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 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 both 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 alternative 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 assumption 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 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 proven, 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 Morriss (1987), pp. 32 35 power-over and power-to have to be distinguished. Voting power is a variety of power-to, but it is not the power to vote, but rather the power to aff ect the outcome of a collective decision by voting. Thus, the reason why it is called voting power is that the focus is on the power that a political agent has due to her vote (cf. Morriss 1987, p. 155). But what, then, does it mean to aff ect the outcome of a collective decision (the outcome of a vote, for short)? In the simple framework that is often adopted in voting theory, the outcome is either the passage or the failure of the proposal that is voted on (of the bill, for short; FM, p. 35). You aff ect the outcome of a collective decision, if you make a diff erence as to whether the bill passes or not. And you aff ect the outcome of a collective decision as a voter, if your vote makes a diff erence as to whether a bill passes or not. That is, whether the bill passes or not, depends on whether you vote yes or no. The recent literature on voting power focuses on the measurement of voting power. What is measured is the extent to which the vote of a political agent can make a diff erence as to whether a bill passes or not. What needs to be clarifi ed now is the “can”. Morriss (1987), p. 8083 has an- other helpful distinction. He diff erentiates between ability and ableness. Whereas ability is roughly about what a person could do, if the circumstances were appro- priate, assignments of ableness additionally take into account the opportunity to exercise ones ability. For instance, John might have the ability to swim hundred meters in less than one minute. But if there is no suitable pool, he lacks the ableness to do so. Whether John has the ableness and thus the opportunity to swim depends on the circumstances and not just on John. According to Morriss (1987), p. 83, ableness rather than ability is what polit- ical philosophers are usually interested in it is the opportunities of the people that are investigated. Let us therefore focus on ableness.2Essential part of what is to be measured, then, is the extent to which a voter has opportunity to aff ect the outcome of a collective decision by voting. Indeed, it is the only part of what 2Morriss (1987), Ch. 22, particularly pp. 157160, also claims that Banzhaf voting power provides a measure of ability, but this will not be important in what follows. 6 is measured, because there is nothing interesting about skills here. Thus, what is to be quantifi ed for a voter a is the extent E to which as vote has the opportunity to make a diff erence as to whether a bill passes or not. But what is the opportunity in question? A very natural answer indeed the answer taken by most theorists is this: We take the other votes as given and ask whether they leave the opportunity that the outcome of the collective decision depends on as vote. If and only if the confi guration of the other votes is such that, whether a bill passes or not, depends on as vote, then a has the opportunity to make a diff erence. Indeed, a will make a diff erence then.3 Altogether, what is to be measured is the extent to which the other votes form a confi guration in which, whether the bill passes, depends on as vote. The only way to measure this extent seems to be to calculate the respective probability. Thus, we arrive at the following measure of voting power for a voter: It is the probability that the confi guration of the other votes is such that, whether the bill passes or not, depends on her vote. In more technical terms: It is the probability for a coalition wrt which her vote is critical (see FM, Def. 2.1.1 on p. 11 and Def. 2.3.4 on p. 24). This line of thought substantiates the statement that the probability of being critical is “arguably the only reasonable way” of explicating the notion of I-power in mathematical terms (FM, p. 36). However, as Wilmers example shows, just calculating the probability of being critical can lead to counterintuitive results according to which nobody has power. The challenge is thus to modify this line of thought slightly in order to have additional measurements of voting power. These measures should enable us to say something interesting in Wilmers example. I follow a lead of Machover (2007), p. 4, who suggests that “the very concept of power that is used in the a priori mode is in some sense too individualistic”. In more detail, I propose to proceed in two steps. As a fi rst step, I suggest to shift the focus from single voters to groups of voters for a while. Consider an arbitrary subset of the voters. It may be asked: What is the voting power of this group? To what extent does the group have the opportunity to make a diff erence as to whether the bill passes or not? As a second step, I suggest to get back to one voter and to ask: To what extent does she have opportunity to form groups that will make a diff erence as 3 This is diff erent from the example with Johns swim. John may have the skill and the opportunity to swim hundred meters in less than one minute, but still not do so. For instance, he may decide not to swim the hundred meters at all. On the contrary, I cannot decide not to make a diff erence with my vote. If the confi guration of the other votes is suitable, then I will always make a diff erence independently what I do (this is true independently on whether abstention is possible or not). “To make a diff erence regarding X” does not describe an action, but rather compares the consequences that diff erent options for acting have on X. The power to swim hundred meters in less than a minute and voting power do not completely parallel in this respect. 7 to whether a bill passes? For instance, to what extent will she together with one other voter jointly make a diff erence as to whether the bill passes? More generally, what will be quantifi ed for a voter a is the extent Eto which the following is true: There are ( 1) other voters such that there is opportunity for the votes of a and of these other voters to jointly make a diff erence as to whether the bill passes. Obviously, the extent E1coincides with extent E. Note, also, that the extent E provides information specifi cally about a fi xed voter (a) and not just for groups. As before E, the extents Ek will be quantifi ed in terms of probabilities. The idea is thus to calculate a hierarchy of probabilities for each voter and for each possible group size . Each probability in the hierarchy will tell us about the opportunity to enter a group that makes a diff erence.These probabilities, I suggest, provide a very natural extension of the probability of being critical. They tell us something about how important a voter is for whether a bill passes or not. And they will also provide us with some non-zero measures for Wilmers example. The move that I suggest is very natural.Suppose that I want to buy a particular house. Unfortunately, I lack the opportunity to do so, because I dont have enough money. A natural way out is to look for someone else such that we two have the opportunity (the money, as it were) to buy the house jointly. Suppose, for instance, that I have many friends F such that F and I can jointly buy the house. Then I have some power to aff ect the fate of the house. To be sure, the extent E and the extents E are about diff erent things for 2. E2 etc. are not just about as vote making a diff erence. Correspondingly, whereas, under E, the other votes are taken as given, under E2etc., the other votes are not all given. For E, a is given freedom, so to speak, to pick (1) other votes and to command these votes together with her own vote. The question is whether the confi guration of votes leaves the opportunity to pi