有数字在身边的人更强大吗?

IF 2.9 1区 哲学 Q1 ETHICS Journal of Political Philosophy Pub Date : 2023-08-02 DOI:10.1111/jopp.12307
Sean Ingham, Niko Kolodny
{"title":"有数字在身边的人更强大吗?","authors":"Sean Ingham,&nbsp;Niko Kolodny","doi":"10.1111/jopp.12307","DOIUrl":null,"url":null,"abstract":"<p>Suppose that in a jurisdiction there are 2,000,001 white voters and 1,000,000 black voters, all of whom enjoy equally weighted votes. The question of white supremacy is routinely put to a majority-rule plebiscite. In each such plebiscite, all white voters vote <i>yes</i> for white supremacy and all black voters vote <i>no</i>. This has been going on as long as anyone can remember, and it will continue for as long as anyone can foresee. This is a paradigm of a persistent minority, to which, intuitively, each black voter has an objection.</p><p>What is their objection? One answer is that black voters don't get their preferences satisfied. Another answer is that black voters are oppressed by the eventuating policies of white supremacy. Yet another answer considers the white majority <i>as a group</i>. As a group, they have greater power to determine the outcome than have the blacks as a group. Indeed, the white majority as a group is always decisive. In the last plebiscite, all the whites voted for white supremacy and it passed; and if all the whites had voted against, it would have failed. By contrast, the black minority as a group is never decisive. They voted against and it passed; and if any assemblage of them had voted <i>yes</i>, it would still have passed.</p><p>In this article, we raise a number of doubts about Abizadeh's suggestion that the power-of-numbers thesis can vindicate the thought that members of the minority as individuals have less voting power and thereby account for their objection to belonging to a persistent minority. Perhaps the most serious doubt is that while Abizadeh correctly holds that voting power must be assessed in part by counterfactualizing on votes—by asking what would have happened if a voter had voted otherwise than he in fact did—he does not counterfactualize in the right way.</p><p>We cannot vindicate the thought that members of the minority have less voting power if we assume an a priori measure of voting power that abstracts from information about the distribution of political preferences and its causes. An example is the Banzhaf (or Penrose–Banzhaf) measure of voting power, according to which a voter's power is her probability of casting a decisive vote if all other voters vote independently and with equal probability for either alternative.4 (A voter's vote in favor of (or against) a measure is <i>decisive</i> if the measure passes (fails), but would have failed (passed) if the voter had instead voted against (in favor).) An a priori measure of voting power will not register any differences between members of the persistent minority and anyone else, because it ignores, by construction, the facts about the distribution of preferences and its causes, in virtue of which some voters qualify as members of a persistent minority.</p><p>Abizadeh therefore seeks to vindicate the power-of-numbers thesis with an a posteriori measure: power is calculated on the basis of, rather than in abstraction from, information about how social structure influences the distribution of political preferences. This measure of power also differs from the Penrose–Banzhaf measure in making voting power a function of the degree to which the voter can expect her actions to be “efficacious,” where even non-decisive, redundant votes for the winning alternative count as partially efficacious.</p><p>So let us consider voting power in the a posteriori context, in which we take into account information about how voters are likely to vote. The information on the basis of which power is measured may be more or less predictive of voting behavior. At the limit, knowledge of a person's structural position, together with knowledge of how they will vote, fully predicts how everyone else will vote. For the sake of simplicity, we start by assuming that such fully predictive information is available and is the appropriate basis for measuring voting power. This assumption is obviously unrealistic, but it simplifies the discussion by relieving us of the need to calculate probabilities, and it is an innocent simplification because it does not prejudice the assessment of the power-of-numbers thesis. It would be bizarre to argue that a member of a persistent minority has an objection only when there is genuine uncertainty about the future distribution of votes, but not when the future distribution, and her status as a political minority, is certain. We will, in any case, relax this assumption in due course.</p><p>We assume, in particular, that the relevant information perfectly predicts the scenario described above: in every plebiscite, the 2,000,001 white voters are certain to vote in favor of white supremacy and the 1,000,000 black voters are certain to vote against. Our question is whether a given white voter can be said to have significantly more power than a given black voter when voting power is measured on the basis of this information.</p><p>In this situation, everyone's probability of casting a decisive vote is zero. Thus, if a voter's power corresponds to the probability of casting a decisive vote (an assumption behind the Penrose–Banzhaf measure), then every voter, white or black, has the same amount of power.</p><p>What if, following Abizadeh, we grant that a member of the winning side can enjoy partial efficacy, even if they are not decisive? When everyone casts equally weighted votes, as will be true in all our examples, we can assume the degree of partial efficacy is a function of the size of the winning coalition (relative to the size of the electorate). At one end of the spectrum is the case of someone who votes for the winning side along with every other member of the electorate; their vote has some partial efficacy, but not as much as it would have if the relative size of the winning coalition were smaller. At the other end is the case of the fully decisive voter who votes for the winning side and is part of a minimal decisive coalition (for example, a bare majority under simple majority rule).</p><p>In our example, when a white voter votes with the winning coalition, in favor of white supremacy, then his vote has some partial efficacy, namely the degree of partial efficacy that corresponds to voting with 2,000,000 other voters, in an electorate of 3,000,001 voters, for the winning alternative. If the white voter were to vote against white supremacy, the efficacy of his vote would be zero. If a black voter were to vote with the winning coalition, in favor of white supremacy, then her vote would have some partial efficacy, namely the degree of partial efficacy that corresponds to voting with 2,000,001 other voters, in an electorate of 3,000,001 voters, for the winning alternative. When the black voter votes against white supremacy, the efficacy of her vote would be zero. Their partial efficacy scores are virtually identical.</p><p>These claims rest on the tacit assumption that if a voter were to vote differently from how they will actually vote, everyone else would still vote the same as they are going to vote in actual fact. One must reject this assumption if one wishes to argue for the power-of-numbers thesis.</p><p>Some might object to the tacit assumption on the following grounds. Given our stipulations about the case, all white voters are <i>certain</i> to vote in the same way. So, it follows that if a given white voter were to vote <i>against</i> white supremacy rather than <i>for</i>, then all other white voters would also vote against, too. But that does <i>not</i> follow. Suppose newspapers A and B always report the same events; when the scandal breaks, they are each certain to report it. It does not follow that if A were to refrain from reporting the scandal, then B would refrain as well. Probabilistic dependence does not imply counterfactual dependence.5</p><p>One would reject the tacit assumption with good reason if one voter's decision about how to vote causally influenced other voters' decisions. But let us assume that no one's vote has any causal influence over anyone else's vote. If you like, assume everyone votes secretly and simultaneously. The power-of-numbers thesis is not supposed to rest on voters' abilities to influence each other.</p><p>A different reason one might reject the assumption is if one thought that measurements of voting power ought to reflect counterfactualizing not only the voter's action, but also the past events that causally influence the voter's action as well as other voters' behavior. One might reason as follows. The example assumes that all white voters always vote as a block and all black voters always vote as a block. This pattern of correlation can only hold in virtue of underlying social-structural causes that have one (fully determining) impact on white voters' preferences and an opposite (fully determining) effect on black voters' preferences. If a given white voter were to vote against the proposed ballot measure, it would have to be because the underlying social-structural influences caused him and all other white voters to be opposed. Thus if a given white voter were to vote against, they would find themselves on the winning side and would enjoy some partial efficacy, just as they enjoy some partial efficacy when they vote in favor; by contrast, if a given black voter were to vote in favor, she would still be on the losing side and her vote would still be inefficacious, just as her actual vote against is inefficacious. Thus the white voter enjoys greater power than the black voter.</p><p>This reasoning involves “backtracking” counterfactuals: the assumption is that when one counterfactualizes over a white voter's action and looks for the closest possible world in which he voted no, one should include worlds with <i>different histories up until the time of that vote</i>.6 This allows one to say that the white voter would be in the majority even if he were to vote against the proposal, provided that the closest possible world in which he votes <i>no</i> is a world with a different history, one in which some common cause led all the other whites to prefer <i>no</i> and so to vote <i>no</i> as well.</p><p>In the next section, we will argue that the counterfactuals that enter into the measurement of power should not be interpreted in a way that permits backtracking.7 But even if we allow backtracking counterfactuals, it may not help the power-of-numbers thesis. Suppose that the closest world in which a given white voter votes no is a world in which the normal link between social-structural causes and political preferences is broken for him alone, but not for other white voters (he regularly converses about racial justice with a colleague, and the closest possible world in which he votes against white supremacy is one in which these conversations induce a moral epiphany, severing the causal link that continues to make other white voters' political preferences a deterministic function of their position in the racial hierarchy). Then the crucial counterfactual would be false: if he were to vote against white supremacy, the other whites would still vote in favor, and his situation is not appreciably different from the situation of any black voter. Nor would it help much if the closest world is one in which he and, say, a thousand others had racial justice epiphanies.</p><p>One worry about the power-of-numbers thesis is that, offhand, it's not clear why the closest possible world in which a given white voter votes no (even among those worlds with different histories) is one in which the social-structural causes that influence all other white voters' preferences are different. So even if we permit backtracking, it is not clear why, in our example, it would be true that if a given white voter were to vote against white supremacy, then all other white voters (or at least a subset that constitutes a majority) would also vote <i>no</i>. Another worry about the power-of-numbers thesis, as a diagnosis of our intuitive objection to a paradigm case of a persistent minority, is that the intuitive objection does not seem to depend on which of these backtracking counterfactuals is correct, but the power-of-numbers thesis does.</p><p>In any event, this backtracking counterfactual isn't the relevant conditional for assessing agential power generally or voting power in particular. If we are measuring the power of a given white voter on the day of the election, we should ask what would happen if he were to vote <i>no</i>, <i>holding fixed the actual history of the world up until the time he votes</i>. The following cases illustrate the importance of excluding backtracking from the analysis of agential power.</p><p>Jones sometimes prays for rain after eating breakfast. To be precise, he prays on all and only those mornings when, while eating breakfast and reading the newspaper, he notices that the weather forecast predicts rain. Thus, conditional on praying for rain in the morning, the probability that it will rain in the afternoon is high. And conditional on not praying, the probability that it will rain is low (because, conditional on not praying, there was no rain in the forecast, and forecasting the weather in his area of southern California is not hard). Jones does not have the power to make it rain, of course. We can explain this fact with normal, non-backtracking counterfactuals, but not with backtracking counterfactuals. On any given morning when Jones prays, the probability of rain is high, but would remain high even if he were not to pray for rain; and on any given morning when he does not pray, the probability of rain is low, but would remain low even if he were to pray. If agential power is measured with backtracking counterfactuals, however, Jones does have the power to make it rain. With backtracking, we have: on any given morning, if Jones were to pray, then (it would be because rain was in the forecast and so) it would probably rain, and if he were not to pray, then (it would be because rain was not in the forecast and so) it would probably not rain.</p><p>Backtracking counterfactuals are no more plausible a basis for ascriptions of power when we move from the single-agent example to a voting situation. A boss threatens to fire anyone who votes for a proposal to unionize. As a result, each worker knows that if she votes for the proposal to unionize and it fails, she will lose her job; otherwise she will keep it. Needless to say, no one sticks her neck out for the union. It seems false to say, and a cruel joke in the mouth of the boss, that each worker has the power to effect, with the others, the passage of the unionization proposal.8 And yet it is plausibly true if we assess a worker's power using backtracking counterfactuals. The closest world in which she votes for the union might be one in which the boss never made the threat and no one faces the prospect of losing their job for voting for it, in which case everyone else would also vote for the union and the unionization proposal would pass. Thus if she were to vote in favor of the unionization proposal, then—backtracking and counterfactualizing the boss's past behavior—it would pass, in which case she would have effected, with others, the passage of the proposal.</p><p>Or suppose an American football coach is deciding whether to go for a touchdown on fourth down or instead have his kicker kick a field goal. One consideration when he makes such decisions is the wind: if the wind is too strong, the kicker will not have the power to make the field goal, and so in such circumstances the coach decides to go for the touchdown. Suppose in actual fact the winds are strong, so he decides to go for the touchdown. After they fail, his critics say: “Coach made the wrong call. The kicker <i>did</i> have the power to make a field goal. In the closest possible world in which he kicked, he kicked because coach told him to, and coach told him to because the wind conditions were favorable, and since they were favorable, kicker made the field goal.” We have taken Monday-morning quarterbacking to a whole new level.</p><p>We have so far considered an example in which the measurement of voting power is made on the basis of information that is fully predictive of everyone's voting behavior. Under this assumption, the differences in voting power between an individual member of the persistent majority and an individual member of the persistent minority were negligible. If one relaxes this assumption and measures power instead on the basis of information that is only partially predictive of the distribution of votes, there are cases in which members of a structural minority have <i>more</i> power in expectation than members of a structural majority—provided, again, that one does not measure power with backtracking counterfactuals.</p><p>Voter 1 belongs to a structural minority, voters 2, 3, 4, and 5 to a structural majority. For each plebiscite <i>t</i>, there is a social-structural variable <math>\n <semantics>\n <mrow>\n <msub>\n <mi>S</mi>\n <mi>t</mi>\n </msub>\n </mrow>\n </semantics></math>  that influences the voting behavior of all five voters. Either <math>\n <semantics>\n <mrow>\n <msub>\n <mi>S</mi>\n <mi>t</mi>\n </msub>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n </semantics></math>   or <math>\n <semantics>\n <mrow>\n <msub>\n <mi>S</mi>\n <mi>t</mi>\n </msub>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n </semantics></math>. If  <math>\n <semantics>\n <mrow>\n <msub>\n <mi>S</mi>\n <mi>t</mi>\n </msub>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n </semantics></math>, then the probability that voters 1, 2, and 3 will support the measure and voters 4 and 5 will oppose it is <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n </mrow>\n </semantics></math>; and the probability that voter 1 supports it, but voters 2, 3, 4, and 5 oppose it is <math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>−</mo>\n <mi>p</mi>\n </mrow>\n </semantics></math>. If  <math>\n <semantics>\n <mrow>\n <msub>\n <mi>S</mi>\n <mi>t</mi>\n </msub>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n </semantics></math>, then the probability that voters 1, 2, and 3 will oppose the measure and voters 4 and 5 will support it is <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n </mrow>\n </semantics></math>; and the probability that voter 1 opposes it, but voters 2, 3, 4, and 5 support it is <math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>−</mo>\n <mi>p</mi>\n </mrow>\n </semantics></math>.9 Assume <math>\n <semantics>\n <mrow>\n <mn>0</mn>\n <mo>&lt;</mo>\n <mi>p</mi>\n <mo>&lt;</mo>\n <mn>1</mn>\n <mo>/</mo>\n <mn>2</mn>\n </mrow>\n </semantics></math>. Most of the time (a fraction <math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>−</mo>\n <mi>p</mi>\n <mo>&gt;</mo>\n <mn>1</mn>\n <mo>/</mo>\n <mn>2</mn>\n </mrow>\n </semantics></math> of the time), voter 1 finds herself in the minority and voters 2, 3, 4, and 5 find themselves in the majority.</p><p>We now show that if <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>&gt;</mo>\n <mn>3</mn>\n <mo>/</mo>\n <mn>23</mn>\n </mrow>\n </semantics></math>, then in any given plebiscite, voter 1 has more power than voter 5, despite the fact that voter 5 has numbers on his side. This conclusion holds for every plebiscite <i>t</i>, whatever the history preceding the plebiscite and, in particular, whatever the realized value of <math>\n <semantics>\n <mrow>\n <msub>\n <mi>S</mi>\n <mi>t</mi>\n </msub>\n </mrow>\n </semantics></math> . The argument for the conclusion abstracts from all facts about the voters' situation other than (1) the voting rule and (2) the social structure and probabilistic dependence that it induces in voting behavior.</p><p>Our argument uses the measure of partial efficacy proposed by Abizadeh.10 According to this measure, in an electorate of five voters, where everyone casts equally weighted votes, a vote for the winning alternative has an efficacy score of 1 if it is one of three votes cast for the winning alternative, a score of 3/4 if it is one of four votes cast for the winning alternative, and an efficacy score of 3/5 if it is one of five votes cast for the winning alternative. A vote cast for the losing alternative has no efficacy. Perhaps a different measure of efficacy could be proposed. But so long as the measure holds that a successful voter's efficacy is a function <math>\n <semantics>\n <mrow>\n <mi>f</mi>\n <mo>(</mo>\n <mi>x</mi>\n <mo>)</mo>\n </mrow>\n </semantics></math> of the fraction <math>\n <semantics>\n <mrow>\n <mi>x</mi>\n </mrow>\n </semantics></math> of the electorate voting for the winning alternative, and (in the 5-voter case) the function satisfies <math>\n <semantics>\n <mrow>\n <mn>0</mn>\n <mo>&lt;</mo>\n <mi>f</mi>\n <mo>(</mo>\n <mn>5</mn>\n <mo>/</mo>\n <mn>5</mn>\n <mo>)</mo>\n <mo>&lt;</mo>\n <mi>f</mi>\n <mo>(</mo>\n <mn>4</mn>\n <mo>/</mo>\n <mn>5</mn>\n <mo>)</mo>\n <mo>&lt;</mo>\n <mi>f</mi>\n <mo>(</mo>\n <mn>3</mn>\n <mo>/</mo>\n <mn>5</mn>\n <mo>)</mo>\n </mrow>\n </semantics></math>, one can construct an example similar to the one below (see Appendix).</p><p>The critical assumption here is that each counterfactual must be assessed without backtracking. Conditional on its being the case that voter 1 supports the measure, voters 4 and 5 are sure to vote against, and voters 2 and 3 will vote for the measure with probability <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n </mrow>\n </semantics></math> and vote against it with probability <math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>-</mo>\n <mi>p</mi>\n </mrow>\n </semantics></math>. This describes the way in which the other voters' behavior is probabilistically dependent on voter 1's behavior. But it does not imply counterfactual dependence (unless one allows for backtracking counterfactuals). Their voting behavior is counterfactually independent of what voter 1 chooses to do, so even if voter 1 were to vote against the measure, it would still be the case that voters 4 and 5 are sure to vote against and voters 2 and 3 vote in favor with probability <i>p</i>.11</p><p>Again, the critical assumption is that the behavior of the other voters is counterfactually independent of what voter 5 chooses to do.</p><p>Now notice that for any nonzero value of <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n </mrow>\n </semantics></math>, voter 1's action of voting <i>yes</i> has a greater expected efficacy than voter 5's action of voting <i>yes</i>. And voter 1's action of voting <i>no</i> has a greater expected efficacy than voter 5's action of voting <i>no</i> so long as <math>\n <semantics>\n <mrow>\n <mn>3</mn>\n <mo>/</mo>\n <mn>5</mn>\n <mo>+</mo>\n <mn>2</mn>\n <mo>/</mo>\n <mn>5</mn>\n <mi>p</mi>\n <mo>&gt;</mo>\n <mn>3</mn>\n <mo>/</mo>\n <mn>4</mn>\n <mo>(</mo>\n <mn>1</mn>\n <mo>-</mo>\n <mi>p</mi>\n <mo>)</mo>\n </mrow>\n </semantics></math>, which is equivalent to  <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>&gt;</mo>\n <mn>3</mn>\n <mo>/</mo>\n <mn>23</mn>\n </mrow>\n </semantics></math>. Thus, provided <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>&gt;</mo>\n <mn>3</mn>\n <mo>/</mo>\n <mn>23</mn>\n </mrow>\n </semantics></math>, each of voter 1's actions is more efficacious in expectation than the corresponding action of voter 5, so voter 1 must be deemed more powerful than voter 5. But voter 1 belongs to a structural minority and voter 5 to a structural majority. Voter 1 is in the minority, and voter 5 is in the majority, a fraction <math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>-</mo>\n <mi>p</mi>\n <mo>&gt;</mo>\n <mn>1</mn>\n <mo>/</mo>\n <mn>2</mn>\n </mrow>\n </semantics></math> of the time, due to their positions in the social structure and its influence on voting behavior.</p><p>We assumed without loss of generality that we were dealing with a plebiscite in which voter 1 supports the measure. But we would have reached the same conclusion if we had instead assumed that voter 1 opposed it. Thus, our conclusion does not depend on the actual history leading up to the plebiscite (in particular on the actual realization of the social-structural variable <math>\n <semantics>\n <mrow>\n <msub>\n <mi>S</mi>\n <mi>t</mi>\n </msub>\n </mrow>\n </semantics></math>). It relied only on the assumption that plebiscites are decided by simple majority rule and that the social structural variable <math>\n <semantics>\n <mrow>\n <msub>\n <mi>S</mi>\n <mi>t</mi>\n </msub>\n </mrow>\n </semantics></math> induces probabilistic dependence in voting behavior in the manner described.</p><p>The intuition for the result is simple. Voter 1 can have a higher probability of being in the minority than voter 5, even though she also has a higher probability of casting a decisive vote. If the latter probability is sufficiently high (in our example, if <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>&gt;</mo>\n <mn>3</mn>\n <mo>/</mo>\n <mn>23</mn>\n </mrow>\n </semantics></math>), the conclusion will be that voter 1 is more powerful. The critical assumption is that we assess counterfactual suppositions about voter 1's actions without counterfactualizing the features of her history (the realized value of <math>\n <semantics>\n <mrow>\n <msub>\n <mi>S</mi>\n <mi>t</mi>\n </msub>\n </mrow>\n </semantics></math>) that fix the probability distribution over other voters' behavior and thereby fix the probability that her vote will be decisive.</p><p>This claim about how persistent minorities must be defined is important, because it is all that Abizadeh invokes to avoid the result that the power of numbers thesis “absurdly requires compensating political eccentrics on democratic-equality grounds.” It is his explanation for why “If libertarians or Bolsheviks are persistently outnumbered, democratic equality does not call for formal-procedural inequalities to compensate them.”13</p><p>What exactly is the problem with defining “persistent minority” in terms of actual preferences? The worry seems to be that, so defined, there is something incoherent about asking “what if the persistent minority had different preferences?”, just as there might seem to be something incoherent about asking, “what if every student in the class got a grade better than the median grade?”. But neither question is incoherent provided we take “the persistent minority” and “the median student” to be “rigid designators,” rather than terms whose referents vary across possible worlds. If, in the actual world, the median grade is a B, then the second question is asking what if every student in the class got better than a B. And if you are the one person who, in actual fact, belongs to the minority in every election, then the first question is asking what would happen if <i>you</i> had different preferences and voted differently.</p><p>The justification for defining “persistent minority” in terms of structural position cannot be that this is the only definition that permits a coherent formulation of counterfactual statements about the preferences and actions of the persistent minority. We can intelligibly ask what would happen if a libertarian or Bolshevik, who in <i>actual</i> fact persistently finds himself in the minority, were counterfactually to vote differently from how he will actually vote, and compare that with what would happen if someone who in actual fact is in the mainstream were to vote differently from how she will actually vote. If the answer—in virtue of backtracking counterfactuals—is that the Bolshevik would still be in the minority, and the mainstream voter would still be in the majority, then the power-of-numbers thesis implies there is an inequality in power.</p><p>If one rejects the use of backtracking counterfactuals to answer this question, then the libertarian and the Bolshevik will have no less power than anyone else, at least not in virtue of being persistently outnumbered. But the argument for the power-of-numbers thesis needs backtracking counterfactuals, so the question is whether there is any justification for excluding them when the question concerns libertarians and Bolsheviks, but including them when it concerns structural minorities.</p><p>Backtracking counterfactuals need not be any less plausible in these other “non-structural” cases. Suppose there are 1,000,000 voters who consistently vote in whatever way the Bolshevik party leaders advocate, and 2,000,001 voters who consistently vote against whatever the Bolshevik party leaders advocate. Take any election, and assume (without loss of generality) that the pro-Bolshevik voters are all going to vote <i>yes</i>, the anti-Bolshevik 2,000,001 all <i>no</i>. If a pro-Bolshevik were to vote <i>no</i>, it could only be because the party vanguard had instructed them to vote <i>no</i>, in which case all the anti-Bolshevik voters would vote <i>yes</i>, and our voter would still be in the minority. That judgment is no less reasonable than the judgment that if a black voter were to vote <i>no</i>, in a society polarized along racial lines, it could only be because of social-structural conditions that induce all white voters to vote <i>yes</i>, such that the black voter would still be in the minority. We have argued that backtracking counterfactuals should be excluded from analyses of power in all cases, but if there were any justification for including them in the case of structural minorities, it is unclear why the justification would not carry over to non-structural minorities. And then one gets the absurd conclusion that the power-of-numbers thesis, if it justifies compensating the power deficits of racial or religious minorities, also justifies compensating libertarians and Bolsheviks.</p><p>It is also perhaps worth noting here that once backtracking counterfactuals are admitted, the majority or minority need not be <i>persistent</i> in order for there to be power-of-numbers inequalities. Imagine a one-hit-wonder at the momentary apex of their popularity. Wonder's fans, for a day, form a majority. A vote is taken on that day whether to declare the day Wonder-appreciation-day. Wonder's fans will vote for whatever Wonder decides. If Wonder says <i>yes</i>, then the fans will vote <i>yes</i>; if Wonder, in modesty, says <i>no</i>, then the fans will vote <i>no</i>. Each of the fans in the majority enjoys power of numbers that each nonfan does not. But none of this will persist.</p><p>Even if an agent's power should not <i>in general</i> be assessed on the basis of backtracking counterfactuals, could there nonetheless be a justification for incorporating them into a specific kind of voting power measure? We anticipate an erroneous argument for doing so that proceeds from the reasonable assumption that the desired measure of voting power ought to <i>abstract</i> from features of a voter's actual situation, such as the actual history preceding the vote, and draws the errant conclusion that voting power ought to be assessed by counterfactualizing these features of the voter's situation.</p><p>The assumption that a measure of voting power ought to abstract from certain features of a voter's situation is reasonable. The Penrose–Banzhaf measure, for example, only purports to register the power a voter enjoys in virtue of the voting rule.14 That is why it abstracts from all other features of a voter's situation (such as how the voter can expect other voters to behave) that are plausibly relevant to a <i>comprehensive</i> assessment of agential power. Analogously, if a measure only aims to capture the power a voter enjoys in virtue of (1) the voting rule and (2) their position in the social structure, then it ought to abstract from the voter's actual circumstances.</p><p>The second proposition does not follow from the first, however. There is a logical gap between the reasonable supposition that a measure of voting power ought to abstract from a voter's actual circumstances and the conclusion that it ought to do so by asking backtracking counterfactual questions. To see that the conclusion does not follow, it suffices to establish the existence of a measure that abstracts from the voter's actual circumstances, but eschews backtracking counterfactuals. Consider the following approach.</p><p>For each of the possible circumstances the voter might find herself in, compute the power she would have in those circumstances with normal, non-backtracking counterfactuals in just the way we did for the paradigm case and the example in Section IV. For example, there is the scenario in which a worker is voting on whether to unionize, and her boss has issued a threat to her and all the other workers; and the scenario in which they are voting, but the boss has not issued a threat. In the example from Section IV, there is the scenario in which <math>\n <semantics>\n <mrow>\n <msub>\n <mi>S</mi>\n <mi>t</mi>\n </msub>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n </semantics></math> and another scenario in which <math>\n <semantics>\n <mrow>\n <msub>\n <mi>S</mi>\n <mi>t</mi>\n </msub>\n <mo>=</mo>\n <mn>0</mn>\n </mrow>\n </semantics></math>. Compute the voter's power in each of these scenarios (just as we have already done for the first of these two scenarios in Section IV).15 Now average over the measurements of the voter's power taken in all these possible circumstances, weighting according to the probability of each circumstance. The result does not depend on which of the possible scenarios is the actual one. The power-of-numbers thesis will still fail to hold if one considers it a thesis about “aggregate” voting power in this sense, rather than a thesis about voting power in a particular circumstance. Since no voter derives special power from having numbers on her side in any particular circumstance, no special aggregate power will emerge when one computes a weighted average of her power across the possible circumstances.</p><p>Abstracting is not the same as counterfactualizing. One can measure power in a way that abstracts from a voter's actual circumstances and their history—in a way that makes measurements of power independent of a voter's actual circumstances and history—without counterfactualizing them. Because the use of backtracking counterfactuals in the measurement of voting power does not follow from the (reasonable) assumption that a voting power measure should abstract from the voter's circumstances, a proponent of the power-of-numbers thesis must find some other argument for it. We are not sure what it could be.</p><p>Provided that a posteriori power is measured using standard, non-backtracking counterfactuals, facts about who has more or less of it will not closely track membership in structural minorities and majorities. In our paradigm case, members of structural minorities and majorities have all but equal a posteriori power. In the case from Section IV, members of structural minorities have greater a posteriori power than members of majorities.</p><p>In so far as this article's argument underscores the vagaries of a posteriori power distributions, it may make one skeptical that democracy could require equality in this dimension, or even an approximation to equality. Once we move from a priori to a posteriori measures of voting power, there is no reason to expect an equal scheme of basic political rights to produce an equal distribution of (a posteriori) voting power. Just the opposite. Almost all political institutions, except those that have been calibrated to the actual social-structural conditions just so, will produce asymmetries in a posteriori voting power. And when the social-structural conditions underlying the distribution of preferences change, political institutions will also need to change if they are to preserve equality.</p><p>Moreover, the kind of changes in social-structural conditions that could undermine equal a posteriori voting power are not normally considered threats to democratic equality. So-called swing voters will enjoy more a posteriori voting power than voters who are reliably on one side of a partisan divide. But even when this voting behavior reflects structural causes—the ways in which race, class, and other social-structural identity categories pull some citizens towards one party, some towards the other, and leave some conflicted and open to appeals from both parties—the fact that the rigidly partisan voter has less a posteriori voting power than the swing voter is not plausibly considered a breach of democratic principles.</p><p>One weird implication of the view is that some citizens could violate political equality, as though they were denying someone an equally weighted vote, merely by “freeing” themselves of social-structural influences and thereby altering the probability distribution of preferences in a way that reduces their efficacy. As a simple illustration, suppose an electorate of three voters, 1, 2, and 3, faces a series of binary votes, and due to social-structural causes they always vote as a block and with equal probability for either alternative. Whether one uses backtracking counterfactuals or not, everyone has equal a posteriori power.16</p><p>But now imagine that voter 1 “frees” himself of the causal influence of the background social structure and comes to vote independently of voters 2 and 3, who still vote as a block, and everyone still votes with equal probability for either alternative. Now voter 1 has less power than either of the other two. Each of his actions has an expected efficacy of 1/6: if he were to vote <i>yes</i>, then with probability 1/2, the block would vote <i>yes</i> and his efficacy would be 1/3; and with probability 1/2, the block would vote <i>no</i> and his efficacy would be zero. The expected efficacy of voting <i>no</i> is calculated analogously and also comes out to 1/6.17 But each of voter 2's actions has an expected efficacy of 2/3, assuming we use backtracking counterfactuals in the calculation:18 if she were to vote <i>yes</i>, then with probability 1/2, voters 1 and 3 would each vote <i>yes</i>, in which case her efficacy is 1/3; and with probability 1/2, voter 1 would vote <i>no</i>, but 3 would vote <i>yes</i>, in which case her efficacy is 1. Thus her expected efficacy of voting <i>yes</i> is 1/6 + 1/2 = 2/3. The expected efficacy of voting <i>no</i> is calculated analogously and also comes out to 2/3.</p><p>Some might find this result counterintuitive, because they do not think one can become less powerful in virtue of increasing one's ability to form political opinions independently of social-structural influences. But that is not our point. Our point is rather that if one has reasons to try to preserve political equality, but political equality is understood in terms of equal a posteriori voting power, then the value of political equality gives voter 1 reasons not to cultivate an ability to form political opinions independently of social-structural influences. <i>That</i> implication seems strange.</p><p>These considerations make us doubt that a posteriori voting power is the currency of democratic equality.</p><p>The power-of-numbers thesis is supposed to vindicate the intuition that a member of a persistent minority has a complaint against majoritarian democracy. If the thesis is true, then majoritarian democracy does not in general respect a principle of democratic equality, and the member of the persistent minority can ground their objection by appeal to this principle. We have noted several grounds for doubts about the power-of-numbers thesis, most importantly its reliance on backtracking counterfactuals.</p><p>Supposing one accepts our critique of the power-of-numbers thesis, how else might one try to ground the objection? We noted several alternatives at the outset. One is that the real objection is not to an inequality in voting power, but instead to an inequality in the satisfaction of preferences: namely, that black voters' preferences go unsatisfied vote after vote, whereas white voters' preferences are reliably satisfied. Another alternative is that, again, the real objection is not to an inequality in voting power, but instead to the substantive inequality that results from the policies voted into effect: that black voters are subordinated by a regime of white supremacy. A final alternative recognizes an inequality in voting power, but at the level of groups rather than individuals. Even if no individual member of the structural minority has appreciably less power than any individual member of the structural majority, the latter <i>group</i> has more power than the former. Perhaps the distribution of power across social groups takes on normative significance—above and beyond the distribution of power across individuals—in societies with histories of group-based injustice.19 We do not have space in this article, however, to develop these suggestions and evaluate their merits. Those are tasks for other work.</p><p>None relevant.</p><p>There are no potential conflicts of interest relevant to this article.</p><p>The authors declare human ethics approval was not needed for this study.</p>","PeriodicalId":47624,"journal":{"name":"Journal of Political Philosophy","volume":"31 4","pages":"452-469"},"PeriodicalIF":2.9000,"publicationDate":"2023-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1111/jopp.12307","citationCount":"0","resultStr":"{\"title\":\"Is One More Powerful with Numbers on One's Side?\",\"authors\":\"Sean Ingham,&nbsp;Niko Kolodny\",\"doi\":\"10.1111/jopp.12307\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Suppose that in a jurisdiction there are 2,000,001 white voters and 1,000,000 black voters, all of whom enjoy equally weighted votes. The question of white supremacy is routinely put to a majority-rule plebiscite. In each such plebiscite, all white voters vote <i>yes</i> for white supremacy and all black voters vote <i>no</i>. This has been going on as long as anyone can remember, and it will continue for as long as anyone can foresee. This is a paradigm of a persistent minority, to which, intuitively, each black voter has an objection.</p><p>What is their objection? One answer is that black voters don't get their preferences satisfied. Another answer is that black voters are oppressed by the eventuating policies of white supremacy. Yet another answer considers the white majority <i>as a group</i>. As a group, they have greater power to determine the outcome than have the blacks as a group. Indeed, the white majority as a group is always decisive. In the last plebiscite, all the whites voted for white supremacy and it passed; and if all the whites had voted against, it would have failed. By contrast, the black minority as a group is never decisive. They voted against and it passed; and if any assemblage of them had voted <i>yes</i>, it would still have passed.</p><p>In this article, we raise a number of doubts about Abizadeh's suggestion that the power-of-numbers thesis can vindicate the thought that members of the minority as individuals have less voting power and thereby account for their objection to belonging to a persistent minority. Perhaps the most serious doubt is that while Abizadeh correctly holds that voting power must be assessed in part by counterfactualizing on votes—by asking what would have happened if a voter had voted otherwise than he in fact did—he does not counterfactualize in the right way.</p><p>We cannot vindicate the thought that members of the minority have less voting power if we assume an a priori measure of voting power that abstracts from information about the distribution of political preferences and its causes. An example is the Banzhaf (or Penrose–Banzhaf) measure of voting power, according to which a voter's power is her probability of casting a decisive vote if all other voters vote independently and with equal probability for either alternative.4 (A voter's vote in favor of (or against) a measure is <i>decisive</i> if the measure passes (fails), but would have failed (passed) if the voter had instead voted against (in favor).) An a priori measure of voting power will not register any differences between members of the persistent minority and anyone else, because it ignores, by construction, the facts about the distribution of preferences and its causes, in virtue of which some voters qualify as members of a persistent minority.</p><p>Abizadeh therefore seeks to vindicate the power-of-numbers thesis with an a posteriori measure: power is calculated on the basis of, rather than in abstraction from, information about how social structure influences the distribution of political preferences. This measure of power also differs from the Penrose–Banzhaf measure in making voting power a function of the degree to which the voter can expect her actions to be “efficacious,” where even non-decisive, redundant votes for the winning alternative count as partially efficacious.</p><p>So let us consider voting power in the a posteriori context, in which we take into account information about how voters are likely to vote. The information on the basis of which power is measured may be more or less predictive of voting behavior. At the limit, knowledge of a person's structural position, together with knowledge of how they will vote, fully predicts how everyone else will vote. For the sake of simplicity, we start by assuming that such fully predictive information is available and is the appropriate basis for measuring voting power. This assumption is obviously unrealistic, but it simplifies the discussion by relieving us of the need to calculate probabilities, and it is an innocent simplification because it does not prejudice the assessment of the power-of-numbers thesis. It would be bizarre to argue that a member of a persistent minority has an objection only when there is genuine uncertainty about the future distribution of votes, but not when the future distribution, and her status as a political minority, is certain. We will, in any case, relax this assumption in due course.</p><p>We assume, in particular, that the relevant information perfectly predicts the scenario described above: in every plebiscite, the 2,000,001 white voters are certain to vote in favor of white supremacy and the 1,000,000 black voters are certain to vote against. Our question is whether a given white voter can be said to have significantly more power than a given black voter when voting power is measured on the basis of this information.</p><p>In this situation, everyone's probability of casting a decisive vote is zero. Thus, if a voter's power corresponds to the probability of casting a decisive vote (an assumption behind the Penrose–Banzhaf measure), then every voter, white or black, has the same amount of power.</p><p>What if, following Abizadeh, we grant that a member of the winning side can enjoy partial efficacy, even if they are not decisive? When everyone casts equally weighted votes, as will be true in all our examples, we can assume the degree of partial efficacy is a function of the size of the winning coalition (relative to the size of the electorate). At one end of the spectrum is the case of someone who votes for the winning side along with every other member of the electorate; their vote has some partial efficacy, but not as much as it would have if the relative size of the winning coalition were smaller. At the other end is the case of the fully decisive voter who votes for the winning side and is part of a minimal decisive coalition (for example, a bare majority under simple majority rule).</p><p>In our example, when a white voter votes with the winning coalition, in favor of white supremacy, then his vote has some partial efficacy, namely the degree of partial efficacy that corresponds to voting with 2,000,000 other voters, in an electorate of 3,000,001 voters, for the winning alternative. If the white voter were to vote against white supremacy, the efficacy of his vote would be zero. If a black voter were to vote with the winning coalition, in favor of white supremacy, then her vote would have some partial efficacy, namely the degree of partial efficacy that corresponds to voting with 2,000,001 other voters, in an electorate of 3,000,001 voters, for the winning alternative. When the black voter votes against white supremacy, the efficacy of her vote would be zero. Their partial efficacy scores are virtually identical.</p><p>These claims rest on the tacit assumption that if a voter were to vote differently from how they will actually vote, everyone else would still vote the same as they are going to vote in actual fact. One must reject this assumption if one wishes to argue for the power-of-numbers thesis.</p><p>Some might object to the tacit assumption on the following grounds. Given our stipulations about the case, all white voters are <i>certain</i> to vote in the same way. So, it follows that if a given white voter were to vote <i>against</i> white supremacy rather than <i>for</i>, then all other white voters would also vote against, too. But that does <i>not</i> follow. Suppose newspapers A and B always report the same events; when the scandal breaks, they are each certain to report it. It does not follow that if A were to refrain from reporting the scandal, then B would refrain as well. Probabilistic dependence does not imply counterfactual dependence.5</p><p>One would reject the tacit assumption with good reason if one voter's decision about how to vote causally influenced other voters' decisions. But let us assume that no one's vote has any causal influence over anyone else's vote. If you like, assume everyone votes secretly and simultaneously. The power-of-numbers thesis is not supposed to rest on voters' abilities to influence each other.</p><p>A different reason one might reject the assumption is if one thought that measurements of voting power ought to reflect counterfactualizing not only the voter's action, but also the past events that causally influence the voter's action as well as other voters' behavior. One might reason as follows. The example assumes that all white voters always vote as a block and all black voters always vote as a block. This pattern of correlation can only hold in virtue of underlying social-structural causes that have one (fully determining) impact on white voters' preferences and an opposite (fully determining) effect on black voters' preferences. If a given white voter were to vote against the proposed ballot measure, it would have to be because the underlying social-structural influences caused him and all other white voters to be opposed. Thus if a given white voter were to vote against, they would find themselves on the winning side and would enjoy some partial efficacy, just as they enjoy some partial efficacy when they vote in favor; by contrast, if a given black voter were to vote in favor, she would still be on the losing side and her vote would still be inefficacious, just as her actual vote against is inefficacious. Thus the white voter enjoys greater power than the black voter.</p><p>This reasoning involves “backtracking” counterfactuals: the assumption is that when one counterfactualizes over a white voter's action and looks for the closest possible world in which he voted no, one should include worlds with <i>different histories up until the time of that vote</i>.6 This allows one to say that the white voter would be in the majority even if he were to vote against the proposal, provided that the closest possible world in which he votes <i>no</i> is a world with a different history, one in which some common cause led all the other whites to prefer <i>no</i> and so to vote <i>no</i> as well.</p><p>In the next section, we will argue that the counterfactuals that enter into the measurement of power should not be interpreted in a way that permits backtracking.7 But even if we allow backtracking counterfactuals, it may not help the power-of-numbers thesis. Suppose that the closest world in which a given white voter votes no is a world in which the normal link between social-structural causes and political preferences is broken for him alone, but not for other white voters (he regularly converses about racial justice with a colleague, and the closest possible world in which he votes against white supremacy is one in which these conversations induce a moral epiphany, severing the causal link that continues to make other white voters' political preferences a deterministic function of their position in the racial hierarchy). Then the crucial counterfactual would be false: if he were to vote against white supremacy, the other whites would still vote in favor, and his situation is not appreciably different from the situation of any black voter. Nor would it help much if the closest world is one in which he and, say, a thousand others had racial justice epiphanies.</p><p>One worry about the power-of-numbers thesis is that, offhand, it's not clear why the closest possible world in which a given white voter votes no (even among those worlds with different histories) is one in which the social-structural causes that influence all other white voters' preferences are different. So even if we permit backtracking, it is not clear why, in our example, it would be true that if a given white voter were to vote against white supremacy, then all other white voters (or at least a subset that constitutes a majority) would also vote <i>no</i>. Another worry about the power-of-numbers thesis, as a diagnosis of our intuitive objection to a paradigm case of a persistent minority, is that the intuitive objection does not seem to depend on which of these backtracking counterfactuals is correct, but the power-of-numbers thesis does.</p><p>In any event, this backtracking counterfactual isn't the relevant conditional for assessing agential power generally or voting power in particular. If we are measuring the power of a given white voter on the day of the election, we should ask what would happen if he were to vote <i>no</i>, <i>holding fixed the actual history of the world up until the time he votes</i>. The following cases illustrate the importance of excluding backtracking from the analysis of agential power.</p><p>Jones sometimes prays for rain after eating breakfast. To be precise, he prays on all and only those mornings when, while eating breakfast and reading the newspaper, he notices that the weather forecast predicts rain. Thus, conditional on praying for rain in the morning, the probability that it will rain in the afternoon is high. And conditional on not praying, the probability that it will rain is low (because, conditional on not praying, there was no rain in the forecast, and forecasting the weather in his area of southern California is not hard). Jones does not have the power to make it rain, of course. We can explain this fact with normal, non-backtracking counterfactuals, but not with backtracking counterfactuals. On any given morning when Jones prays, the probability of rain is high, but would remain high even if he were not to pray for rain; and on any given morning when he does not pray, the probability of rain is low, but would remain low even if he were to pray. If agential power is measured with backtracking counterfactuals, however, Jones does have the power to make it rain. With backtracking, we have: on any given morning, if Jones were to pray, then (it would be because rain was in the forecast and so) it would probably rain, and if he were not to pray, then (it would be because rain was not in the forecast and so) it would probably not rain.</p><p>Backtracking counterfactuals are no more plausible a basis for ascriptions of power when we move from the single-agent example to a voting situation. A boss threatens to fire anyone who votes for a proposal to unionize. As a result, each worker knows that if she votes for the proposal to unionize and it fails, she will lose her job; otherwise she will keep it. Needless to say, no one sticks her neck out for the union. It seems false to say, and a cruel joke in the mouth of the boss, that each worker has the power to effect, with the others, the passage of the unionization proposal.8 And yet it is plausibly true if we assess a worker's power using backtracking counterfactuals. The closest world in which she votes for the union might be one in which the boss never made the threat and no one faces the prospect of losing their job for voting for it, in which case everyone else would also vote for the union and the unionization proposal would pass. Thus if she were to vote in favor of the unionization proposal, then—backtracking and counterfactualizing the boss's past behavior—it would pass, in which case she would have effected, with others, the passage of the proposal.</p><p>Or suppose an American football coach is deciding whether to go for a touchdown on fourth down or instead have his kicker kick a field goal. One consideration when he makes such decisions is the wind: if the wind is too strong, the kicker will not have the power to make the field goal, and so in such circumstances the coach decides to go for the touchdown. Suppose in actual fact the winds are strong, so he decides to go for the touchdown. After they fail, his critics say: “Coach made the wrong call. The kicker <i>did</i> have the power to make a field goal. In the closest possible world in which he kicked, he kicked because coach told him to, and coach told him to because the wind conditions were favorable, and since they were favorable, kicker made the field goal.” We have taken Monday-morning quarterbacking to a whole new level.</p><p>We have so far considered an example in which the measurement of voting power is made on the basis of information that is fully predictive of everyone's voting behavior. Under this assumption, the differences in voting power between an individual member of the persistent majority and an individual member of the persistent minority were negligible. If one relaxes this assumption and measures power instead on the basis of information that is only partially predictive of the distribution of votes, there are cases in which members of a structural minority have <i>more</i> power in expectation than members of a structural majority—provided, again, that one does not measure power with backtracking counterfactuals.</p><p>Voter 1 belongs to a structural minority, voters 2, 3, 4, and 5 to a structural majority. For each plebiscite <i>t</i>, there is a social-structural variable <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>S</mi>\\n <mi>t</mi>\\n </msub>\\n </mrow>\\n </semantics></math>  that influences the voting behavior of all five voters. Either <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>S</mi>\\n <mi>t</mi>\\n </msub>\\n <mo>=</mo>\\n <mn>0</mn>\\n </mrow>\\n </semantics></math>   or <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>S</mi>\\n <mi>t</mi>\\n </msub>\\n <mo>=</mo>\\n <mn>1</mn>\\n </mrow>\\n </semantics></math>. If  <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>S</mi>\\n <mi>t</mi>\\n </msub>\\n <mo>=</mo>\\n <mn>1</mn>\\n </mrow>\\n </semantics></math>, then the probability that voters 1, 2, and 3 will support the measure and voters 4 and 5 will oppose it is <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n </mrow>\\n </semantics></math>; and the probability that voter 1 supports it, but voters 2, 3, 4, and 5 oppose it is <math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>−</mo>\\n <mi>p</mi>\\n </mrow>\\n </semantics></math>. If  <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>S</mi>\\n <mi>t</mi>\\n </msub>\\n <mo>=</mo>\\n <mn>0</mn>\\n </mrow>\\n </semantics></math>, then the probability that voters 1, 2, and 3 will oppose the measure and voters 4 and 5 will support it is <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n </mrow>\\n </semantics></math>; and the probability that voter 1 opposes it, but voters 2, 3, 4, and 5 support it is <math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>−</mo>\\n <mi>p</mi>\\n </mrow>\\n </semantics></math>.9 Assume <math>\\n <semantics>\\n <mrow>\\n <mn>0</mn>\\n <mo>&lt;</mo>\\n <mi>p</mi>\\n <mo>&lt;</mo>\\n <mn>1</mn>\\n <mo>/</mo>\\n <mn>2</mn>\\n </mrow>\\n </semantics></math>. Most of the time (a fraction <math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>−</mo>\\n <mi>p</mi>\\n <mo>&gt;</mo>\\n <mn>1</mn>\\n <mo>/</mo>\\n <mn>2</mn>\\n </mrow>\\n </semantics></math> of the time), voter 1 finds herself in the minority and voters 2, 3, 4, and 5 find themselves in the majority.</p><p>We now show that if <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n <mo>&gt;</mo>\\n <mn>3</mn>\\n <mo>/</mo>\\n <mn>23</mn>\\n </mrow>\\n </semantics></math>, then in any given plebiscite, voter 1 has more power than voter 5, despite the fact that voter 5 has numbers on his side. This conclusion holds for every plebiscite <i>t</i>, whatever the history preceding the plebiscite and, in particular, whatever the realized value of <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>S</mi>\\n <mi>t</mi>\\n </msub>\\n </mrow>\\n </semantics></math> . The argument for the conclusion abstracts from all facts about the voters' situation other than (1) the voting rule and (2) the social structure and probabilistic dependence that it induces in voting behavior.</p><p>Our argument uses the measure of partial efficacy proposed by Abizadeh.10 According to this measure, in an electorate of five voters, where everyone casts equally weighted votes, a vote for the winning alternative has an efficacy score of 1 if it is one of three votes cast for the winning alternative, a score of 3/4 if it is one of four votes cast for the winning alternative, and an efficacy score of 3/5 if it is one of five votes cast for the winning alternative. A vote cast for the losing alternative has no efficacy. Perhaps a different measure of efficacy could be proposed. But so long as the measure holds that a successful voter's efficacy is a function <math>\\n <semantics>\\n <mrow>\\n <mi>f</mi>\\n <mo>(</mo>\\n <mi>x</mi>\\n <mo>)</mo>\\n </mrow>\\n </semantics></math> of the fraction <math>\\n <semantics>\\n <mrow>\\n <mi>x</mi>\\n </mrow>\\n </semantics></math> of the electorate voting for the winning alternative, and (in the 5-voter case) the function satisfies <math>\\n <semantics>\\n <mrow>\\n <mn>0</mn>\\n <mo>&lt;</mo>\\n <mi>f</mi>\\n <mo>(</mo>\\n <mn>5</mn>\\n <mo>/</mo>\\n <mn>5</mn>\\n <mo>)</mo>\\n <mo>&lt;</mo>\\n <mi>f</mi>\\n <mo>(</mo>\\n <mn>4</mn>\\n <mo>/</mo>\\n <mn>5</mn>\\n <mo>)</mo>\\n <mo>&lt;</mo>\\n <mi>f</mi>\\n <mo>(</mo>\\n <mn>3</mn>\\n <mo>/</mo>\\n <mn>5</mn>\\n <mo>)</mo>\\n </mrow>\\n </semantics></math>, one can construct an example similar to the one below (see Appendix).</p><p>The critical assumption here is that each counterfactual must be assessed without backtracking. Conditional on its being the case that voter 1 supports the measure, voters 4 and 5 are sure to vote against, and voters 2 and 3 will vote for the measure with probability <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n </mrow>\\n </semantics></math> and vote against it with probability <math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>-</mo>\\n <mi>p</mi>\\n </mrow>\\n </semantics></math>. This describes the way in which the other voters' behavior is probabilistically dependent on voter 1's behavior. But it does not imply counterfactual dependence (unless one allows for backtracking counterfactuals). Their voting behavior is counterfactually independent of what voter 1 chooses to do, so even if voter 1 were to vote against the measure, it would still be the case that voters 4 and 5 are sure to vote against and voters 2 and 3 vote in favor with probability <i>p</i>.11</p><p>Again, the critical assumption is that the behavior of the other voters is counterfactually independent of what voter 5 chooses to do.</p><p>Now notice that for any nonzero value of <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n </mrow>\\n </semantics></math>, voter 1's action of voting <i>yes</i> has a greater expected efficacy than voter 5's action of voting <i>yes</i>. And voter 1's action of voting <i>no</i> has a greater expected efficacy than voter 5's action of voting <i>no</i> so long as <math>\\n <semantics>\\n <mrow>\\n <mn>3</mn>\\n <mo>/</mo>\\n <mn>5</mn>\\n <mo>+</mo>\\n <mn>2</mn>\\n <mo>/</mo>\\n <mn>5</mn>\\n <mi>p</mi>\\n <mo>&gt;</mo>\\n <mn>3</mn>\\n <mo>/</mo>\\n <mn>4</mn>\\n <mo>(</mo>\\n <mn>1</mn>\\n <mo>-</mo>\\n <mi>p</mi>\\n <mo>)</mo>\\n </mrow>\\n </semantics></math>, which is equivalent to  <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n <mo>&gt;</mo>\\n <mn>3</mn>\\n <mo>/</mo>\\n <mn>23</mn>\\n </mrow>\\n </semantics></math>. Thus, provided <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n <mo>&gt;</mo>\\n <mn>3</mn>\\n <mo>/</mo>\\n <mn>23</mn>\\n </mrow>\\n </semantics></math>, each of voter 1's actions is more efficacious in expectation than the corresponding action of voter 5, so voter 1 must be deemed more powerful than voter 5. But voter 1 belongs to a structural minority and voter 5 to a structural majority. Voter 1 is in the minority, and voter 5 is in the majority, a fraction <math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>-</mo>\\n <mi>p</mi>\\n <mo>&gt;</mo>\\n <mn>1</mn>\\n <mo>/</mo>\\n <mn>2</mn>\\n </mrow>\\n </semantics></math> of the time, due to their positions in the social structure and its influence on voting behavior.</p><p>We assumed without loss of generality that we were dealing with a plebiscite in which voter 1 supports the measure. But we would have reached the same conclusion if we had instead assumed that voter 1 opposed it. Thus, our conclusion does not depend on the actual history leading up to the plebiscite (in particular on the actual realization of the social-structural variable <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>S</mi>\\n <mi>t</mi>\\n </msub>\\n </mrow>\\n </semantics></math>). It relied only on the assumption that plebiscites are decided by simple majority rule and that the social structural variable <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>S</mi>\\n <mi>t</mi>\\n </msub>\\n </mrow>\\n </semantics></math> induces probabilistic dependence in voting behavior in the manner described.</p><p>The intuition for the result is simple. Voter 1 can have a higher probability of being in the minority than voter 5, even though she also has a higher probability of casting a decisive vote. If the latter probability is sufficiently high (in our example, if <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n <mo>&gt;</mo>\\n <mn>3</mn>\\n <mo>/</mo>\\n <mn>23</mn>\\n </mrow>\\n </semantics></math>), the conclusion will be that voter 1 is more powerful. The critical assumption is that we assess counterfactual suppositions about voter 1's actions without counterfactualizing the features of her history (the realized value of <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>S</mi>\\n <mi>t</mi>\\n </msub>\\n </mrow>\\n </semantics></math>) that fix the probability distribution over other voters' behavior and thereby fix the probability that her vote will be decisive.</p><p>This claim about how persistent minorities must be defined is important, because it is all that Abizadeh invokes to avoid the result that the power of numbers thesis “absurdly requires compensating political eccentrics on democratic-equality grounds.” It is his explanation for why “If libertarians or Bolsheviks are persistently outnumbered, democratic equality does not call for formal-procedural inequalities to compensate them.”13</p><p>What exactly is the problem with defining “persistent minority” in terms of actual preferences? The worry seems to be that, so defined, there is something incoherent about asking “what if the persistent minority had different preferences?”, just as there might seem to be something incoherent about asking, “what if every student in the class got a grade better than the median grade?”. But neither question is incoherent provided we take “the persistent minority” and “the median student” to be “rigid designators,” rather than terms whose referents vary across possible worlds. If, in the actual world, the median grade is a B, then the second question is asking what if every student in the class got better than a B. And if you are the one person who, in actual fact, belongs to the minority in every election, then the first question is asking what would happen if <i>you</i> had different preferences and voted differently.</p><p>The justification for defining “persistent minority” in terms of structural position cannot be that this is the only definition that permits a coherent formulation of counterfactual statements about the preferences and actions of the persistent minority. We can intelligibly ask what would happen if a libertarian or Bolshevik, who in <i>actual</i> fact persistently finds himself in the minority, were counterfactually to vote differently from how he will actually vote, and compare that with what would happen if someone who in actual fact is in the mainstream were to vote differently from how she will actually vote. If the answer—in virtue of backtracking counterfactuals—is that the Bolshevik would still be in the minority, and the mainstream voter would still be in the majority, then the power-of-numbers thesis implies there is an inequality in power.</p><p>If one rejects the use of backtracking counterfactuals to answer this question, then the libertarian and the Bolshevik will have no less power than anyone else, at least not in virtue of being persistently outnumbered. But the argument for the power-of-numbers thesis needs backtracking counterfactuals, so the question is whether there is any justification for excluding them when the question concerns libertarians and Bolsheviks, but including them when it concerns structural minorities.</p><p>Backtracking counterfactuals need not be any less plausible in these other “non-structural” cases. Suppose there are 1,000,000 voters who consistently vote in whatever way the Bolshevik party leaders advocate, and 2,000,001 voters who consistently vote against whatever the Bolshevik party leaders advocate. Take any election, and assume (without loss of generality) that the pro-Bolshevik voters are all going to vote <i>yes</i>, the anti-Bolshevik 2,000,001 all <i>no</i>. If a pro-Bolshevik were to vote <i>no</i>, it could only be because the party vanguard had instructed them to vote <i>no</i>, in which case all the anti-Bolshevik voters would vote <i>yes</i>, and our voter would still be in the minority. That judgment is no less reasonable than the judgment that if a black voter were to vote <i>no</i>, in a society polarized along racial lines, it could only be because of social-structural conditions that induce all white voters to vote <i>yes</i>, such that the black voter would still be in the minority. We have argued that backtracking counterfactuals should be excluded from analyses of power in all cases, but if there were any justification for including them in the case of structural minorities, it is unclear why the justification would not carry over to non-structural minorities. And then one gets the absurd conclusion that the power-of-numbers thesis, if it justifies compensating the power deficits of racial or religious minorities, also justifies compensating libertarians and Bolsheviks.</p><p>It is also perhaps worth noting here that once backtracking counterfactuals are admitted, the majority or minority need not be <i>persistent</i> in order for there to be power-of-numbers inequalities. Imagine a one-hit-wonder at the momentary apex of their popularity. Wonder's fans, for a day, form a majority. A vote is taken on that day whether to declare the day Wonder-appreciation-day. Wonder's fans will vote for whatever Wonder decides. If Wonder says <i>yes</i>, then the fans will vote <i>yes</i>; if Wonder, in modesty, says <i>no</i>, then the fans will vote <i>no</i>. Each of the fans in the majority enjoys power of numbers that each nonfan does not. But none of this will persist.</p><p>Even if an agent's power should not <i>in general</i> be assessed on the basis of backtracking counterfactuals, could there nonetheless be a justification for incorporating them into a specific kind of voting power measure? We anticipate an erroneous argument for doing so that proceeds from the reasonable assumption that the desired measure of voting power ought to <i>abstract</i> from features of a voter's actual situation, such as the actual history preceding the vote, and draws the errant conclusion that voting power ought to be assessed by counterfactualizing these features of the voter's situation.</p><p>The assumption that a measure of voting power ought to abstract from certain features of a voter's situation is reasonable. The Penrose–Banzhaf measure, for example, only purports to register the power a voter enjoys in virtue of the voting rule.14 That is why it abstracts from all other features of a voter's situation (such as how the voter can expect other voters to behave) that are plausibly relevant to a <i>comprehensive</i> assessment of agential power. Analogously, if a measure only aims to capture the power a voter enjoys in virtue of (1) the voting rule and (2) their position in the social structure, then it ought to abstract from the voter's actual circumstances.</p><p>The second proposition does not follow from the first, however. There is a logical gap between the reasonable supposition that a measure of voting power ought to abstract from a voter's actual circumstances and the conclusion that it ought to do so by asking backtracking counterfactual questions. To see that the conclusion does not follow, it suffices to establish the existence of a measure that abstracts from the voter's actual circumstances, but eschews backtracking counterfactuals. Consider the following approach.</p><p>For each of the possible circumstances the voter might find herself in, compute the power she would have in those circumstances with normal, non-backtracking counterfactuals in just the way we did for the paradigm case and the example in Section IV. For example, there is the scenario in which a worker is voting on whether to unionize, and her boss has issued a threat to her and all the other workers; and the scenario in which they are voting, but the boss has not issued a threat. In the example from Section IV, there is the scenario in which <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>S</mi>\\n <mi>t</mi>\\n </msub>\\n <mo>=</mo>\\n <mn>1</mn>\\n </mrow>\\n </semantics></math> and another scenario in which <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>S</mi>\\n <mi>t</mi>\\n </msub>\\n <mo>=</mo>\\n <mn>0</mn>\\n </mrow>\\n </semantics></math>. Compute the voter's power in each of these scenarios (just as we have already done for the first of these two scenarios in Section IV).15 Now average over the measurements of the voter's power taken in all these possible circumstances, weighting according to the probability of each circumstance. The result does not depend on which of the possible scenarios is the actual one. The power-of-numbers thesis will still fail to hold if one considers it a thesis about “aggregate” voting power in this sense, rather than a thesis about voting power in a particular circumstance. Since no voter derives special power from having numbers on her side in any particular circumstance, no special aggregate power will emerge when one computes a weighted average of her power across the possible circumstances.</p><p>Abstracting is not the same as counterfactualizing. One can measure power in a way that abstracts from a voter's actual circumstances and their history—in a way that makes measurements of power independent of a voter's actual circumstances and history—without counterfactualizing them. Because the use of backtracking counterfactuals in the measurement of voting power does not follow from the (reasonable) assumption that a voting power measure should abstract from the voter's circumstances, a proponent of the power-of-numbers thesis must find some other argument for it. We are not sure what it could be.</p><p>Provided that a posteriori power is measured using standard, non-backtracking counterfactuals, facts about who has more or less of it will not closely track membership in structural minorities and majorities. In our paradigm case, members of structural minorities and majorities have all but equal a posteriori power. In the case from Section IV, members of structural minorities have greater a posteriori power than members of majorities.</p><p>In so far as this article's argument underscores the vagaries of a posteriori power distributions, it may make one skeptical that democracy could require equality in this dimension, or even an approximation to equality. Once we move from a priori to a posteriori measures of voting power, there is no reason to expect an equal scheme of basic political rights to produce an equal distribution of (a posteriori) voting power. Just the opposite. Almost all political institutions, except those that have been calibrated to the actual social-structural conditions just so, will produce asymmetries in a posteriori voting power. And when the social-structural conditions underlying the distribution of preferences change, political institutions will also need to change if they are to preserve equality.</p><p>Moreover, the kind of changes in social-structural conditions that could undermine equal a posteriori voting power are not normally considered threats to democratic equality. So-called swing voters will enjoy more a posteriori voting power than voters who are reliably on one side of a partisan divide. But even when this voting behavior reflects structural causes—the ways in which race, class, and other social-structural identity categories pull some citizens towards one party, some towards the other, and leave some conflicted and open to appeals from both parties—the fact that the rigidly partisan voter has less a posteriori voting power than the swing voter is not plausibly considered a breach of democratic principles.</p><p>One weird implication of the view is that some citizens could violate political equality, as though they were denying someone an equally weighted vote, merely by “freeing” themselves of social-structural influences and thereby altering the probability distribution of preferences in a way that reduces their efficacy. As a simple illustration, suppose an electorate of three voters, 1, 2, and 3, faces a series of binary votes, and due to social-structural causes they always vote as a block and with equal probability for either alternative. Whether one uses backtracking counterfactuals or not, everyone has equal a posteriori power.16</p><p>But now imagine that voter 1 “frees” himself of the causal influence of the background social structure and comes to vote independently of voters 2 and 3, who still vote as a block, and everyone still votes with equal probability for either alternative. Now voter 1 has less power than either of the other two. Each of his actions has an expected efficacy of 1/6: if he were to vote <i>yes</i>, then with probability 1/2, the block would vote <i>yes</i> and his efficacy would be 1/3; and with probability 1/2, the block would vote <i>no</i> and his efficacy would be zero. The expected efficacy of voting <i>no</i> is calculated analogously and also comes out to 1/6.17 But each of voter 2's actions has an expected efficacy of 2/3, assuming we use backtracking counterfactuals in the calculation:18 if she were to vote <i>yes</i>, then with probability 1/2, voters 1 and 3 would each vote <i>yes</i>, in which case her efficacy is 1/3; and with probability 1/2, voter 1 would vote <i>no</i>, but 3 would vote <i>yes</i>, in which case her efficacy is 1. Thus her expected efficacy of voting <i>yes</i> is 1/6 + 1/2 = 2/3. The expected efficacy of voting <i>no</i> is calculated analogously and also comes out to 2/3.</p><p>Some might find this result counterintuitive, because they do not think one can become less powerful in virtue of increasing one's ability to form political opinions independently of social-structural influences. But that is not our point. Our point is rather that if one has reasons to try to preserve political equality, but political equality is understood in terms of equal a posteriori voting power, then the value of political equality gives voter 1 reasons not to cultivate an ability to form political opinions independently of social-structural influences. <i>That</i> implication seems strange.</p><p>These considerations make us doubt that a posteriori voting power is the currency of democratic equality.</p><p>The power-of-numbers thesis is supposed to vindicate the intuition that a member of a persistent minority has a complaint against majoritarian democracy. If the thesis is true, then majoritarian democracy does not in general respect a principle of democratic equality, and the member of the persistent minority can ground their objection by appeal to this principle. We have noted several grounds for doubts about the power-of-numbers thesis, most importantly its reliance on backtracking counterfactuals.</p><p>Supposing one accepts our critique of the power-of-numbers thesis, how else might one try to ground the objection? We noted several alternatives at the outset. One is that the real objection is not to an inequality in voting power, but instead to an inequality in the satisfaction of preferences: namely, that black voters' preferences go unsatisfied vote after vote, whereas white voters' preferences are reliably satisfied. Another alternative is that, again, the real objection is not to an inequality in voting power, but instead to the substantive inequality that results from the policies voted into effect: that black voters are subordinated by a regime of white supremacy. A final alternative recognizes an inequality in voting power, but at the level of groups rather than individuals. Even if no individual member of the structural minority has appreciably less power than any individual member of the structural majority, the latter <i>group</i> has more power than the former. Perhaps the distribution of power across social groups takes on normative significance—above and beyond the distribution of power across individuals—in societies with histories of group-based injustice.19 We do not have space in this article, however, to develop these suggestions and evaluate their merits. Those are tasks for other work.</p><p>None relevant.</p><p>There are no potential conflicts of interest relevant to this article.</p><p>The authors declare human ethics approval was not needed for this study.</p>\",\"PeriodicalId\":47624,\"journal\":{\"name\":\"Journal of Political Philosophy\",\"volume\":\"31 4\",\"pages\":\"452-469\"},\"PeriodicalIF\":2.9000,\"publicationDate\":\"2023-08-02\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1111/jopp.12307\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Political Philosophy\",\"FirstCategoryId\":\"98\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1111/jopp.12307\",\"RegionNum\":1,\"RegionCategory\":\"哲学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ETHICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Political Philosophy","FirstCategoryId":"98","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1111/jopp.12307","RegionNum":1,"RegionCategory":"哲学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ETHICS","Score":null,"Total":0}
引用次数: 0

摘要

因此,如果选民的权力与投出决定性选票的概率相对应(Penrose-Banzhaf衡量标准背后的假设),那么每个选民,无论白人还是黑人,都拥有相同的权力。如果在阿比扎德之后,我们承认获胜一方的一名成员可以享有部分效力,即使他们不是决定性的,该怎么办?当每个人都投下同等权重的选票时,正如我们所有的例子一样,我们可以假设部分效力的程度是获胜联盟规模(相对于选民规模)的函数。在光谱的一端,有人和其他选民一起投票给获胜的一方;他们的投票有一定的部分效果,但不如获胜联盟的相对规模较小时那么有效。另一端是完全决定性的选民,他投票给获胜的一方,是最小决定性联盟的一部分(例如,简单多数规则下的绝对多数)。在我们的例子中,当白人选民投票给获胜联盟,支持白人至上主义时,他的投票具有一定的部分效力,即在3000001名选民中,与2000000名其他选民一起投票选出获胜的备选方案相对应的部分效力程度。如果白人选民投票反对白人至上主义,他的投票效果将为零。如果一名黑人选民投票支持获胜的联盟,支持白人至上主义,那么她的选票将具有一定的部分效力,即在3000001名选民中,与2000001名其他选民一起投票支持获胜候选人的部分效力程度。当黑人选民投票反对白人至上主义时,她的投票效果将为零。他们的部分疗效评分几乎相同。这些说法建立在一种默认的假设之上,即如果选民的投票方式与他们实际的投票方式不同,那么其他人的投票方式仍然与他们实际投票的方式相同。如果一个人想为数字的幂理论辩护,就必须拒绝这种假设。有些人可能会基于以下理由反对这种默认假设。鉴于我们对此案的规定,所有白人选民肯定会以同样的方式投票。因此,如果一个特定的白人选民投票反对白人至上主义,而不是支持,那么所有其他白人选民也会投票反对。但事实并非如此。假设A报和B报总是报道同样的事件;当丑闻爆发时,他们每个人都一定会举报。这并不是说如果A不举报丑闻,那么B也会不举报。概率依赖并不意味着反事实依赖。5如果一名选民关于如何投票的决定对其他选民的决定产生了因果影响,人们会有充分的理由拒绝默认假设。但让我们假设,没有人的投票对其他人的投票有任何因果影响。如果你愿意,假设每个人都同时秘密投票。数字理论的力量不应该取决于选民相互影响的能力。人们可能会拒绝这一假设的另一个原因是,如果人们认为投票权的衡量不仅应该反映选民的行为,还应该反映过去对选民行为以及其他选民行为产生因果影响的事件。原因可能如下。这个例子假设所有白人选民总是作为一个群体投票,所有黑人选民总是作为群体投票。这种相关性模式只有在潜在的社会结构原因的作用下才能成立,这些原因对白人选民的偏好有一个(完全决定的)影响,而对黑人选民的偏好则有相反的(完全决定)影响。如果某个白人选民投票反对拟议的投票措施,那一定是因为潜在的社会结构影响导致他和所有其他白人选民反对。因此,如果一个特定的白人选民投反对票,他们会发现自己站在获胜的一边,并会享有一些部分效力,就像他们投赞成票时享有一些部分功效一样;相比之下,如果一位黑人选民投了赞成票,她仍然会输,她的投票仍然无效,就像她实际投的反对票无效一样。因此,白人选民比黑人选民享有更大的权力。这种推理涉及“回溯”反事实:假设当一个人对白人选民的行为进行反事实化,并寻找他投反对票的最接近的世界时,应该包括在投票前具有不同历史的世界。 她投票支持工会的最接近的世界可能是老板从未发出威胁,也没有人因为投票支持工会而面临失业的前景,在这种情况下,其他人也会投票支持工会,工会提案就会通过。因此,如果她投赞成工会提案的票,然后——回溯和反实现老板过去的行为——它就会通过,在这种情况下,她会和其他人一起通过提案。或者假设一位美国足球教练正在决定是在第四次落后时触地得分,还是让他的踢球者踢出一个界外球。当他做出这样的决定时,一个考虑因素是风:如果风太大,踢球者将没有力量射门,因此在这种情况下,教练决定触地得分。假设事实上风很大,所以他决定触地得分。在他们失败后,他的批评者说:“教练做出了错误的决定。踢球者确实有能力在球场上进球。在他踢球的最接近的世界里,他踢球是因为教练告诉他,教练告诉他这样做是因为风的条件很好,既然风的条件好,踢球者就进球了。”我们把周一早上的四分卫提升到了一个全新的水平。到目前为止,我们已经考虑了一个例子,在这个例子中,投票权的衡量是基于完全可以预测每个人投票行为的信息。在这种假设下,持续多数派的个人成员和持续少数派的个人会员之间的投票权差异可以忽略不计。如果人们放松这一假设,转而根据仅部分预测选票分布的信息来衡量权力,那么在某些情况下,结构性少数群体的成员比结构性多数群体的成员在预期中拥有更大的权力——同样,前提是人们不用回溯的反事实来衡量权力。选民1属于结构性少数,选民2、3、4和5属于结构性多数。对于每一次公民投票,都有一个社会结构变量S t影响所有五名选民的投票行为。S t=0  或S t=1。如果  S t=1,则选民1、2和3将支持该措施,而选民4和5将反对该措施的概率为p;选民1支持,但选民2、3、4和5反对的概率为1−p。如果  S t=0,则选民1、2和3将反对该措施,而选民4和5将支持该措施的概率为p;选民1反对,但选民2、3、4和5支持的概率为1−p。9假设0&lt;p&lt;1/2。大多数情况下(一小部分1−p&gt;1/2的时间),选民1发现自己处于少数,而选民2、3、4和5发现自己处于多数。 但选民1属于结构性少数,选民5属于结构性多数。选民1占少数,而选民5占多数,分数为1-p&gt;1/2的时间,由于他们在社会结构中的地位及其对投票行为的影响。在不失普遍性的情况下,我们假设我们正在进行公民投票,1号选民支持这项措施。但如果我们假设选民1反对,我们也会得出同样的结论。因此,我们的结论并不取决于公民投票前的实际历史(特别是社会结构变量ST的实际实现)。它只依赖于这样一个假设,即公民投票是由简单多数规则决定的,并且社会结构变量S t以所描述的方式在投票行为中引起概率依赖。结果的直觉很简单。选民1比选民5更有可能成为少数派,尽管她也有更高的概率投出决定性的一票。如果后一种概率足够高(在我们的例子中,如果p>3/23),则结论将是投票者1更强大。关键的假设是,我们评估了关于选民1行为的反事实假设,而没有反事实化她的历史特征(s t的已实现值),这些特征固定了其他选民行为的概率分布从而确定她的投票具有决定性的可能性。关于必须如何定义顽固的少数群体的说法很重要,因为Abizadeh所援引的一切都是为了避免数字力量论“荒谬地要求以民主平等为由补偿政治怪癖”的结果。这是他对为什么“如果自由意志主义者或布尔什维克的人数持续超过他们,民主平等就不需要正式的程序不平等来补偿他们。”13根据实际偏好来定义“持续的少数群体”到底有什么问题?令人担忧的是,根据定义,问“如果顽固的少数人有不同的偏好怎么办?”似乎有些语无伦次,就像问“如果班上每个学生的成绩都比平均成绩好怎么办。但这两个问题都不连贯,前提是我们将“持续的少数群体”和“中等学生”视为“刚性指标”,而不是指代在可能的世界中不同的术语。如果在实际世界中,平均成绩是B,那么第二个问题是问,如果班上的每个学生都比B好怎么办。如果你实际上在每次选举中都属于少数,那么第一个问题是,如果你有不同的偏好,投不同的票,会发生什么。用结构立场来定义“持久少数群体”的理由不可能是,这是唯一一个允许对持久少数群体的偏好和行动进行连贯的反事实陈述的定义。我们可以清楚地问,如果一个事实上一直处于少数派的自由意志主义者或布尔什维克反事实地以不同于他实际投票方式的方式投票,会发生什么,并将其与如果一个实际上处于主流的人以不同于她实际投票方式投票会发生什么进行比较。如果答案——根据回溯的反事实——是布尔什维克仍然是少数,而主流选民仍然是多数,那么数字的力量论意味着权力不平等。如果有人拒绝使用回溯的反事实来回答这个问题,那么自由意志主义者和布尔什维克的权力将不亚于任何人,至少不会因为人数持续不足。 他的每一项行动都有1/6的预期效果:如果他投赞成票,那么在概率为1/2的情况下,该区块将投赞成票。他的效果将为1/3;在概率为1/2的情况下,该区块将投反对票,他的效力将为零。投反对票的预期效果是以类似的方式计算的,也得出1/6.17。但是,假设我们在计算中使用回溯反事实,选民2的每一个行为的预期效果都是2/3:18如果她投赞成票,那么在概率为1/2的情况下,选民1和选民3都会投赞成票。在这种情况下,她的效果是1/3;在概率为1/2的情况下,选民1会投反对票,但3会投赞成票,在这种情况下,她的有效性为1。因此,她投赞成票的预期效果是1/6 + 1/2 = 2/3.投反对票的预期效果是类似的计算,也得出了2/3。有些人可能会觉得这个结果违反直觉,因为他们不认为一个人可以因为独立于社会结构影响形成政治观点的能力的提高而变得不那么强大。但这不是我们的重点。我们的观点是,如果一个人有理由试图维护政治平等,但政治平等是从平等的后验投票权的角度来理解的,那么政治平等的价值就给了选民1不培养独立于社会结构影响形成政治观点的能力的理由。这种暗示似乎很奇怪。这些考虑使我们怀疑后验投票权是民主平等的货币。数字的力量理论应该证明一种直觉,即一个顽固的少数群体的成员对多数民主有抱怨。如果这一论点是正确的,那么多数民主一般不尊重民主平等原则,而顽固的少数群体成员可以通过诉诸这一原则来提出反对意见。我们已经注意到了对数字命题的力量的怀疑的几个理由,最重要的是它对回溯反事实的依赖。假设一个人接受了我们对数字力量论的批评,否则怎么可能试图提出反对意见呢?我们一开始就注意到了几个备选方案。一个是,真正的反对意见不是投票权的不平等,而是偏好满意度的不平等:即黑人选民的偏好一次又一次地得不到满足,而白人选民的偏好得到了可靠的满足。另一种选择是,同样,真正的反对意见不是投票权的不平等,而是投票生效的政策造成的实质性不平等:黑人选民服从白人至上的政权。最后一种选择是承认投票权的不平等,但在群体而非个人层面。即使结构性少数群体中没有一个成员的权力明显低于结构性多数群体中的任何一个成员,后者的权力也高于前者。也许在有基于群体的不公正历史的社会中,权力在社会群体之间的分配具有规范意义——超越了权力在个人之间的分配。19然而,我们在这篇文章中没有空间来发展这些建议并评估其优点。这些都是其他工作的任务。无相关。本条不存在潜在的利益冲突。作者宣称这项研究不需要人类伦理的批准。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
查看原文
分享 分享
微信好友 朋友圈 QQ好友 复制链接
本刊更多论文
Is One More Powerful with Numbers on One's Side?

Suppose that in a jurisdiction there are 2,000,001 white voters and 1,000,000 black voters, all of whom enjoy equally weighted votes. The question of white supremacy is routinely put to a majority-rule plebiscite. In each such plebiscite, all white voters vote yes for white supremacy and all black voters vote no. This has been going on as long as anyone can remember, and it will continue for as long as anyone can foresee. This is a paradigm of a persistent minority, to which, intuitively, each black voter has an objection.

What is their objection? One answer is that black voters don't get their preferences satisfied. Another answer is that black voters are oppressed by the eventuating policies of white supremacy. Yet another answer considers the white majority as a group. As a group, they have greater power to determine the outcome than have the blacks as a group. Indeed, the white majority as a group is always decisive. In the last plebiscite, all the whites voted for white supremacy and it passed; and if all the whites had voted against, it would have failed. By contrast, the black minority as a group is never decisive. They voted against and it passed; and if any assemblage of them had voted yes, it would still have passed.

In this article, we raise a number of doubts about Abizadeh's suggestion that the power-of-numbers thesis can vindicate the thought that members of the minority as individuals have less voting power and thereby account for their objection to belonging to a persistent minority. Perhaps the most serious doubt is that while Abizadeh correctly holds that voting power must be assessed in part by counterfactualizing on votes—by asking what would have happened if a voter had voted otherwise than he in fact did—he does not counterfactualize in the right way.

We cannot vindicate the thought that members of the minority have less voting power if we assume an a priori measure of voting power that abstracts from information about the distribution of political preferences and its causes. An example is the Banzhaf (or Penrose–Banzhaf) measure of voting power, according to which a voter's power is her probability of casting a decisive vote if all other voters vote independently and with equal probability for either alternative.4 (A voter's vote in favor of (or against) a measure is decisive if the measure passes (fails), but would have failed (passed) if the voter had instead voted against (in favor).) An a priori measure of voting power will not register any differences between members of the persistent minority and anyone else, because it ignores, by construction, the facts about the distribution of preferences and its causes, in virtue of which some voters qualify as members of a persistent minority.

Abizadeh therefore seeks to vindicate the power-of-numbers thesis with an a posteriori measure: power is calculated on the basis of, rather than in abstraction from, information about how social structure influences the distribution of political preferences. This measure of power also differs from the Penrose–Banzhaf measure in making voting power a function of the degree to which the voter can expect her actions to be “efficacious,” where even non-decisive, redundant votes for the winning alternative count as partially efficacious.

So let us consider voting power in the a posteriori context, in which we take into account information about how voters are likely to vote. The information on the basis of which power is measured may be more or less predictive of voting behavior. At the limit, knowledge of a person's structural position, together with knowledge of how they will vote, fully predicts how everyone else will vote. For the sake of simplicity, we start by assuming that such fully predictive information is available and is the appropriate basis for measuring voting power. This assumption is obviously unrealistic, but it simplifies the discussion by relieving us of the need to calculate probabilities, and it is an innocent simplification because it does not prejudice the assessment of the power-of-numbers thesis. It would be bizarre to argue that a member of a persistent minority has an objection only when there is genuine uncertainty about the future distribution of votes, but not when the future distribution, and her status as a political minority, is certain. We will, in any case, relax this assumption in due course.

We assume, in particular, that the relevant information perfectly predicts the scenario described above: in every plebiscite, the 2,000,001 white voters are certain to vote in favor of white supremacy and the 1,000,000 black voters are certain to vote against. Our question is whether a given white voter can be said to have significantly more power than a given black voter when voting power is measured on the basis of this information.

In this situation, everyone's probability of casting a decisive vote is zero. Thus, if a voter's power corresponds to the probability of casting a decisive vote (an assumption behind the Penrose–Banzhaf measure), then every voter, white or black, has the same amount of power.

What if, following Abizadeh, we grant that a member of the winning side can enjoy partial efficacy, even if they are not decisive? When everyone casts equally weighted votes, as will be true in all our examples, we can assume the degree of partial efficacy is a function of the size of the winning coalition (relative to the size of the electorate). At one end of the spectrum is the case of someone who votes for the winning side along with every other member of the electorate; their vote has some partial efficacy, but not as much as it would have if the relative size of the winning coalition were smaller. At the other end is the case of the fully decisive voter who votes for the winning side and is part of a minimal decisive coalition (for example, a bare majority under simple majority rule).

In our example, when a white voter votes with the winning coalition, in favor of white supremacy, then his vote has some partial efficacy, namely the degree of partial efficacy that corresponds to voting with 2,000,000 other voters, in an electorate of 3,000,001 voters, for the winning alternative. If the white voter were to vote against white supremacy, the efficacy of his vote would be zero. If a black voter were to vote with the winning coalition, in favor of white supremacy, then her vote would have some partial efficacy, namely the degree of partial efficacy that corresponds to voting with 2,000,001 other voters, in an electorate of 3,000,001 voters, for the winning alternative. When the black voter votes against white supremacy, the efficacy of her vote would be zero. Their partial efficacy scores are virtually identical.

These claims rest on the tacit assumption that if a voter were to vote differently from how they will actually vote, everyone else would still vote the same as they are going to vote in actual fact. One must reject this assumption if one wishes to argue for the power-of-numbers thesis.

Some might object to the tacit assumption on the following grounds. Given our stipulations about the case, all white voters are certain to vote in the same way. So, it follows that if a given white voter were to vote against white supremacy rather than for, then all other white voters would also vote against, too. But that does not follow. Suppose newspapers A and B always report the same events; when the scandal breaks, they are each certain to report it. It does not follow that if A were to refrain from reporting the scandal, then B would refrain as well. Probabilistic dependence does not imply counterfactual dependence.5

One would reject the tacit assumption with good reason if one voter's decision about how to vote causally influenced other voters' decisions. But let us assume that no one's vote has any causal influence over anyone else's vote. If you like, assume everyone votes secretly and simultaneously. The power-of-numbers thesis is not supposed to rest on voters' abilities to influence each other.

A different reason one might reject the assumption is if one thought that measurements of voting power ought to reflect counterfactualizing not only the voter's action, but also the past events that causally influence the voter's action as well as other voters' behavior. One might reason as follows. The example assumes that all white voters always vote as a block and all black voters always vote as a block. This pattern of correlation can only hold in virtue of underlying social-structural causes that have one (fully determining) impact on white voters' preferences and an opposite (fully determining) effect on black voters' preferences. If a given white voter were to vote against the proposed ballot measure, it would have to be because the underlying social-structural influences caused him and all other white voters to be opposed. Thus if a given white voter were to vote against, they would find themselves on the winning side and would enjoy some partial efficacy, just as they enjoy some partial efficacy when they vote in favor; by contrast, if a given black voter were to vote in favor, she would still be on the losing side and her vote would still be inefficacious, just as her actual vote against is inefficacious. Thus the white voter enjoys greater power than the black voter.

This reasoning involves “backtracking” counterfactuals: the assumption is that when one counterfactualizes over a white voter's action and looks for the closest possible world in which he voted no, one should include worlds with different histories up until the time of that vote.6 This allows one to say that the white voter would be in the majority even if he were to vote against the proposal, provided that the closest possible world in which he votes no is a world with a different history, one in which some common cause led all the other whites to prefer no and so to vote no as well.

In the next section, we will argue that the counterfactuals that enter into the measurement of power should not be interpreted in a way that permits backtracking.7 But even if we allow backtracking counterfactuals, it may not help the power-of-numbers thesis. Suppose that the closest world in which a given white voter votes no is a world in which the normal link between social-structural causes and political preferences is broken for him alone, but not for other white voters (he regularly converses about racial justice with a colleague, and the closest possible world in which he votes against white supremacy is one in which these conversations induce a moral epiphany, severing the causal link that continues to make other white voters' political preferences a deterministic function of their position in the racial hierarchy). Then the crucial counterfactual would be false: if he were to vote against white supremacy, the other whites would still vote in favor, and his situation is not appreciably different from the situation of any black voter. Nor would it help much if the closest world is one in which he and, say, a thousand others had racial justice epiphanies.

One worry about the power-of-numbers thesis is that, offhand, it's not clear why the closest possible world in which a given white voter votes no (even among those worlds with different histories) is one in which the social-structural causes that influence all other white voters' preferences are different. So even if we permit backtracking, it is not clear why, in our example, it would be true that if a given white voter were to vote against white supremacy, then all other white voters (or at least a subset that constitutes a majority) would also vote no. Another worry about the power-of-numbers thesis, as a diagnosis of our intuitive objection to a paradigm case of a persistent minority, is that the intuitive objection does not seem to depend on which of these backtracking counterfactuals is correct, but the power-of-numbers thesis does.

In any event, this backtracking counterfactual isn't the relevant conditional for assessing agential power generally or voting power in particular. If we are measuring the power of a given white voter on the day of the election, we should ask what would happen if he were to vote no, holding fixed the actual history of the world up until the time he votes. The following cases illustrate the importance of excluding backtracking from the analysis of agential power.

Jones sometimes prays for rain after eating breakfast. To be precise, he prays on all and only those mornings when, while eating breakfast and reading the newspaper, he notices that the weather forecast predicts rain. Thus, conditional on praying for rain in the morning, the probability that it will rain in the afternoon is high. And conditional on not praying, the probability that it will rain is low (because, conditional on not praying, there was no rain in the forecast, and forecasting the weather in his area of southern California is not hard). Jones does not have the power to make it rain, of course. We can explain this fact with normal, non-backtracking counterfactuals, but not with backtracking counterfactuals. On any given morning when Jones prays, the probability of rain is high, but would remain high even if he were not to pray for rain; and on any given morning when he does not pray, the probability of rain is low, but would remain low even if he were to pray. If agential power is measured with backtracking counterfactuals, however, Jones does have the power to make it rain. With backtracking, we have: on any given morning, if Jones were to pray, then (it would be because rain was in the forecast and so) it would probably rain, and if he were not to pray, then (it would be because rain was not in the forecast and so) it would probably not rain.

Backtracking counterfactuals are no more plausible a basis for ascriptions of power when we move from the single-agent example to a voting situation. A boss threatens to fire anyone who votes for a proposal to unionize. As a result, each worker knows that if she votes for the proposal to unionize and it fails, she will lose her job; otherwise she will keep it. Needless to say, no one sticks her neck out for the union. It seems false to say, and a cruel joke in the mouth of the boss, that each worker has the power to effect, with the others, the passage of the unionization proposal.8 And yet it is plausibly true if we assess a worker's power using backtracking counterfactuals. The closest world in which she votes for the union might be one in which the boss never made the threat and no one faces the prospect of losing their job for voting for it, in which case everyone else would also vote for the union and the unionization proposal would pass. Thus if she were to vote in favor of the unionization proposal, then—backtracking and counterfactualizing the boss's past behavior—it would pass, in which case she would have effected, with others, the passage of the proposal.

Or suppose an American football coach is deciding whether to go for a touchdown on fourth down or instead have his kicker kick a field goal. One consideration when he makes such decisions is the wind: if the wind is too strong, the kicker will not have the power to make the field goal, and so in such circumstances the coach decides to go for the touchdown. Suppose in actual fact the winds are strong, so he decides to go for the touchdown. After they fail, his critics say: “Coach made the wrong call. The kicker did have the power to make a field goal. In the closest possible world in which he kicked, he kicked because coach told him to, and coach told him to because the wind conditions were favorable, and since they were favorable, kicker made the field goal.” We have taken Monday-morning quarterbacking to a whole new level.

We have so far considered an example in which the measurement of voting power is made on the basis of information that is fully predictive of everyone's voting behavior. Under this assumption, the differences in voting power between an individual member of the persistent majority and an individual member of the persistent minority were negligible. If one relaxes this assumption and measures power instead on the basis of information that is only partially predictive of the distribution of votes, there are cases in which members of a structural minority have more power in expectation than members of a structural majority—provided, again, that one does not measure power with backtracking counterfactuals.

Voter 1 belongs to a structural minority, voters 2, 3, 4, and 5 to a structural majority. For each plebiscite t, there is a social-structural variable S t   that influences the voting behavior of all five voters. Either S t = 0   or S t = 1 . If   S t = 1 , then the probability that voters 1, 2, and 3 will support the measure and voters 4 and 5 will oppose it is p ; and the probability that voter 1 supports it, but voters 2, 3, 4, and 5 oppose it is 1 p . If   S t = 0 , then the probability that voters 1, 2, and 3 will oppose the measure and voters 4 and 5 will support it is p ; and the probability that voter 1 opposes it, but voters 2, 3, 4, and 5 support it is 1 p .9 Assume 0 < p < 1 / 2 . Most of the time (a fraction 1 p > 1 / 2 of the time), voter 1 finds herself in the minority and voters 2, 3, 4, and 5 find themselves in the majority.

We now show that if p > 3 / 23 , then in any given plebiscite, voter 1 has more power than voter 5, despite the fact that voter 5 has numbers on his side. This conclusion holds for every plebiscite t, whatever the history preceding the plebiscite and, in particular, whatever the realized value of S t . The argument for the conclusion abstracts from all facts about the voters' situation other than (1) the voting rule and (2) the social structure and probabilistic dependence that it induces in voting behavior.

Our argument uses the measure of partial efficacy proposed by Abizadeh.10 According to this measure, in an electorate of five voters, where everyone casts equally weighted votes, a vote for the winning alternative has an efficacy score of 1 if it is one of three votes cast for the winning alternative, a score of 3/4 if it is one of four votes cast for the winning alternative, and an efficacy score of 3/5 if it is one of five votes cast for the winning alternative. A vote cast for the losing alternative has no efficacy. Perhaps a different measure of efficacy could be proposed. But so long as the measure holds that a successful voter's efficacy is a function f ( x ) of the fraction x of the electorate voting for the winning alternative, and (in the 5-voter case) the function satisfies 0 < f ( 5 / 5 ) < f ( 4 / 5 ) < f ( 3 / 5 ) , one can construct an example similar to the one below (see Appendix).

The critical assumption here is that each counterfactual must be assessed without backtracking. Conditional on its being the case that voter 1 supports the measure, voters 4 and 5 are sure to vote against, and voters 2 and 3 will vote for the measure with probability p and vote against it with probability 1 - p . This describes the way in which the other voters' behavior is probabilistically dependent on voter 1's behavior. But it does not imply counterfactual dependence (unless one allows for backtracking counterfactuals). Their voting behavior is counterfactually independent of what voter 1 chooses to do, so even if voter 1 were to vote against the measure, it would still be the case that voters 4 and 5 are sure to vote against and voters 2 and 3 vote in favor with probability p.11

Again, the critical assumption is that the behavior of the other voters is counterfactually independent of what voter 5 chooses to do.

Now notice that for any nonzero value of p , voter 1's action of voting yes has a greater expected efficacy than voter 5's action of voting yes. And voter 1's action of voting no has a greater expected efficacy than voter 5's action of voting no so long as 3 / 5 + 2 / 5 p > 3 / 4 ( 1 - p ) , which is equivalent to   p > 3 / 23 . Thus, provided p > 3 / 23 , each of voter 1's actions is more efficacious in expectation than the corresponding action of voter 5, so voter 1 must be deemed more powerful than voter 5. But voter 1 belongs to a structural minority and voter 5 to a structural majority. Voter 1 is in the minority, and voter 5 is in the majority, a fraction 1 - p > 1 / 2 of the time, due to their positions in the social structure and its influence on voting behavior.

We assumed without loss of generality that we were dealing with a plebiscite in which voter 1 supports the measure. But we would have reached the same conclusion if we had instead assumed that voter 1 opposed it. Thus, our conclusion does not depend on the actual history leading up to the plebiscite (in particular on the actual realization of the social-structural variable S t ). It relied only on the assumption that plebiscites are decided by simple majority rule and that the social structural variable S t induces probabilistic dependence in voting behavior in the manner described.

The intuition for the result is simple. Voter 1 can have a higher probability of being in the minority than voter 5, even though she also has a higher probability of casting a decisive vote. If the latter probability is sufficiently high (in our example, if p > 3 / 23 ), the conclusion will be that voter 1 is more powerful. The critical assumption is that we assess counterfactual suppositions about voter 1's actions without counterfactualizing the features of her history (the realized value of S t ) that fix the probability distribution over other voters' behavior and thereby fix the probability that her vote will be decisive.

This claim about how persistent minorities must be defined is important, because it is all that Abizadeh invokes to avoid the result that the power of numbers thesis “absurdly requires compensating political eccentrics on democratic-equality grounds.” It is his explanation for why “If libertarians or Bolsheviks are persistently outnumbered, democratic equality does not call for formal-procedural inequalities to compensate them.”13

What exactly is the problem with defining “persistent minority” in terms of actual preferences? The worry seems to be that, so defined, there is something incoherent about asking “what if the persistent minority had different preferences?”, just as there might seem to be something incoherent about asking, “what if every student in the class got a grade better than the median grade?”. But neither question is incoherent provided we take “the persistent minority” and “the median student” to be “rigid designators,” rather than terms whose referents vary across possible worlds. If, in the actual world, the median grade is a B, then the second question is asking what if every student in the class got better than a B. And if you are the one person who, in actual fact, belongs to the minority in every election, then the first question is asking what would happen if you had different preferences and voted differently.

The justification for defining “persistent minority” in terms of structural position cannot be that this is the only definition that permits a coherent formulation of counterfactual statements about the preferences and actions of the persistent minority. We can intelligibly ask what would happen if a libertarian or Bolshevik, who in actual fact persistently finds himself in the minority, were counterfactually to vote differently from how he will actually vote, and compare that with what would happen if someone who in actual fact is in the mainstream were to vote differently from how she will actually vote. If the answer—in virtue of backtracking counterfactuals—is that the Bolshevik would still be in the minority, and the mainstream voter would still be in the majority, then the power-of-numbers thesis implies there is an inequality in power.

If one rejects the use of backtracking counterfactuals to answer this question, then the libertarian and the Bolshevik will have no less power than anyone else, at least not in virtue of being persistently outnumbered. But the argument for the power-of-numbers thesis needs backtracking counterfactuals, so the question is whether there is any justification for excluding them when the question concerns libertarians and Bolsheviks, but including them when it concerns structural minorities.

Backtracking counterfactuals need not be any less plausible in these other “non-structural” cases. Suppose there are 1,000,000 voters who consistently vote in whatever way the Bolshevik party leaders advocate, and 2,000,001 voters who consistently vote against whatever the Bolshevik party leaders advocate. Take any election, and assume (without loss of generality) that the pro-Bolshevik voters are all going to vote yes, the anti-Bolshevik 2,000,001 all no. If a pro-Bolshevik were to vote no, it could only be because the party vanguard had instructed them to vote no, in which case all the anti-Bolshevik voters would vote yes, and our voter would still be in the minority. That judgment is no less reasonable than the judgment that if a black voter were to vote no, in a society polarized along racial lines, it could only be because of social-structural conditions that induce all white voters to vote yes, such that the black voter would still be in the minority. We have argued that backtracking counterfactuals should be excluded from analyses of power in all cases, but if there were any justification for including them in the case of structural minorities, it is unclear why the justification would not carry over to non-structural minorities. And then one gets the absurd conclusion that the power-of-numbers thesis, if it justifies compensating the power deficits of racial or religious minorities, also justifies compensating libertarians and Bolsheviks.

It is also perhaps worth noting here that once backtracking counterfactuals are admitted, the majority or minority need not be persistent in order for there to be power-of-numbers inequalities. Imagine a one-hit-wonder at the momentary apex of their popularity. Wonder's fans, for a day, form a majority. A vote is taken on that day whether to declare the day Wonder-appreciation-day. Wonder's fans will vote for whatever Wonder decides. If Wonder says yes, then the fans will vote yes; if Wonder, in modesty, says no, then the fans will vote no. Each of the fans in the majority enjoys power of numbers that each nonfan does not. But none of this will persist.

Even if an agent's power should not in general be assessed on the basis of backtracking counterfactuals, could there nonetheless be a justification for incorporating them into a specific kind of voting power measure? We anticipate an erroneous argument for doing so that proceeds from the reasonable assumption that the desired measure of voting power ought to abstract from features of a voter's actual situation, such as the actual history preceding the vote, and draws the errant conclusion that voting power ought to be assessed by counterfactualizing these features of the voter's situation.

The assumption that a measure of voting power ought to abstract from certain features of a voter's situation is reasonable. The Penrose–Banzhaf measure, for example, only purports to register the power a voter enjoys in virtue of the voting rule.14 That is why it abstracts from all other features of a voter's situation (such as how the voter can expect other voters to behave) that are plausibly relevant to a comprehensive assessment of agential power. Analogously, if a measure only aims to capture the power a voter enjoys in virtue of (1) the voting rule and (2) their position in the social structure, then it ought to abstract from the voter's actual circumstances.

The second proposition does not follow from the first, however. There is a logical gap between the reasonable supposition that a measure of voting power ought to abstract from a voter's actual circumstances and the conclusion that it ought to do so by asking backtracking counterfactual questions. To see that the conclusion does not follow, it suffices to establish the existence of a measure that abstracts from the voter's actual circumstances, but eschews backtracking counterfactuals. Consider the following approach.

For each of the possible circumstances the voter might find herself in, compute the power she would have in those circumstances with normal, non-backtracking counterfactuals in just the way we did for the paradigm case and the example in Section IV. For example, there is the scenario in which a worker is voting on whether to unionize, and her boss has issued a threat to her and all the other workers; and the scenario in which they are voting, but the boss has not issued a threat. In the example from Section IV, there is the scenario in which S t = 1 and another scenario in which S t = 0 . Compute the voter's power in each of these scenarios (just as we have already done for the first of these two scenarios in Section IV).15 Now average over the measurements of the voter's power taken in all these possible circumstances, weighting according to the probability of each circumstance. The result does not depend on which of the possible scenarios is the actual one. The power-of-numbers thesis will still fail to hold if one considers it a thesis about “aggregate” voting power in this sense, rather than a thesis about voting power in a particular circumstance. Since no voter derives special power from having numbers on her side in any particular circumstance, no special aggregate power will emerge when one computes a weighted average of her power across the possible circumstances.

Abstracting is not the same as counterfactualizing. One can measure power in a way that abstracts from a voter's actual circumstances and their history—in a way that makes measurements of power independent of a voter's actual circumstances and history—without counterfactualizing them. Because the use of backtracking counterfactuals in the measurement of voting power does not follow from the (reasonable) assumption that a voting power measure should abstract from the voter's circumstances, a proponent of the power-of-numbers thesis must find some other argument for it. We are not sure what it could be.

Provided that a posteriori power is measured using standard, non-backtracking counterfactuals, facts about who has more or less of it will not closely track membership in structural minorities and majorities. In our paradigm case, members of structural minorities and majorities have all but equal a posteriori power. In the case from Section IV, members of structural minorities have greater a posteriori power than members of majorities.

In so far as this article's argument underscores the vagaries of a posteriori power distributions, it may make one skeptical that democracy could require equality in this dimension, or even an approximation to equality. Once we move from a priori to a posteriori measures of voting power, there is no reason to expect an equal scheme of basic political rights to produce an equal distribution of (a posteriori) voting power. Just the opposite. Almost all political institutions, except those that have been calibrated to the actual social-structural conditions just so, will produce asymmetries in a posteriori voting power. And when the social-structural conditions underlying the distribution of preferences change, political institutions will also need to change if they are to preserve equality.

Moreover, the kind of changes in social-structural conditions that could undermine equal a posteriori voting power are not normally considered threats to democratic equality. So-called swing voters will enjoy more a posteriori voting power than voters who are reliably on one side of a partisan divide. But even when this voting behavior reflects structural causes—the ways in which race, class, and other social-structural identity categories pull some citizens towards one party, some towards the other, and leave some conflicted and open to appeals from both parties—the fact that the rigidly partisan voter has less a posteriori voting power than the swing voter is not plausibly considered a breach of democratic principles.

One weird implication of the view is that some citizens could violate political equality, as though they were denying someone an equally weighted vote, merely by “freeing” themselves of social-structural influences and thereby altering the probability distribution of preferences in a way that reduces their efficacy. As a simple illustration, suppose an electorate of three voters, 1, 2, and 3, faces a series of binary votes, and due to social-structural causes they always vote as a block and with equal probability for either alternative. Whether one uses backtracking counterfactuals or not, everyone has equal a posteriori power.16

But now imagine that voter 1 “frees” himself of the causal influence of the background social structure and comes to vote independently of voters 2 and 3, who still vote as a block, and everyone still votes with equal probability for either alternative. Now voter 1 has less power than either of the other two. Each of his actions has an expected efficacy of 1/6: if he were to vote yes, then with probability 1/2, the block would vote yes and his efficacy would be 1/3; and with probability 1/2, the block would vote no and his efficacy would be zero. The expected efficacy of voting no is calculated analogously and also comes out to 1/6.17 But each of voter 2's actions has an expected efficacy of 2/3, assuming we use backtracking counterfactuals in the calculation:18 if she were to vote yes, then with probability 1/2, voters 1 and 3 would each vote yes, in which case her efficacy is 1/3; and with probability 1/2, voter 1 would vote no, but 3 would vote yes, in which case her efficacy is 1. Thus her expected efficacy of voting yes is 1/6 + 1/2 = 2/3. The expected efficacy of voting no is calculated analogously and also comes out to 2/3.

Some might find this result counterintuitive, because they do not think one can become less powerful in virtue of increasing one's ability to form political opinions independently of social-structural influences. But that is not our point. Our point is rather that if one has reasons to try to preserve political equality, but political equality is understood in terms of equal a posteriori voting power, then the value of political equality gives voter 1 reasons not to cultivate an ability to form political opinions independently of social-structural influences. That implication seems strange.

These considerations make us doubt that a posteriori voting power is the currency of democratic equality.

The power-of-numbers thesis is supposed to vindicate the intuition that a member of a persistent minority has a complaint against majoritarian democracy. If the thesis is true, then majoritarian democracy does not in general respect a principle of democratic equality, and the member of the persistent minority can ground their objection by appeal to this principle. We have noted several grounds for doubts about the power-of-numbers thesis, most importantly its reliance on backtracking counterfactuals.

Supposing one accepts our critique of the power-of-numbers thesis, how else might one try to ground the objection? We noted several alternatives at the outset. One is that the real objection is not to an inequality in voting power, but instead to an inequality in the satisfaction of preferences: namely, that black voters' preferences go unsatisfied vote after vote, whereas white voters' preferences are reliably satisfied. Another alternative is that, again, the real objection is not to an inequality in voting power, but instead to the substantive inequality that results from the policies voted into effect: that black voters are subordinated by a regime of white supremacy. A final alternative recognizes an inequality in voting power, but at the level of groups rather than individuals. Even if no individual member of the structural minority has appreciably less power than any individual member of the structural majority, the latter group has more power than the former. Perhaps the distribution of power across social groups takes on normative significance—above and beyond the distribution of power across individuals—in societies with histories of group-based injustice.19 We do not have space in this article, however, to develop these suggestions and evaluate their merits. Those are tasks for other work.

None relevant.

There are no potential conflicts of interest relevant to this article.

The authors declare human ethics approval was not needed for this study.

求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
CiteScore
4.10
自引率
5.60%
发文量
17
期刊介绍: The Journal of Political Philosophy is an international journal devoted to the study of theoretical issues arising out of moral, legal and political life. It welcomes, and hopes to foster, work cutting across a variety of disciplinary concerns, among them philosophy, sociology, history, economics and political science. The journal encourages new approaches, including (but not limited to): feminism; environmentalism; critical theory, post-modernism and analytical Marxism; social and public choice theory; law and economics, critical legal studies and critical race studies; and game theoretic, socio-biological and anthropological approaches to politics. It also welcomes work in the history of political thought which builds to a larger philosophical point and work in the philosophy of the social sciences and applied ethics with broader political implications. Featuring a distinguished editorial board from major centres of thought from around the globe, the journal draws equally upon the work of non-philosophers and philosophers and provides a forum of debate between disparate factions who usually keep to their own separate journals.
期刊最新文献
Evaluating International Agreements: The Voluntarist Reply and Its Limits Issue Information The Journal of Political Philosophy Index, Volume 31 (2023) The challenge of policing minorities in a liberal society Noncompliance and the Demands of Public Reason
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
0
微信
客服QQ
Book学术公众号 扫码关注我们
反馈
×
意见反馈
请填写您的意见或建议
请填写您的手机或邮箱
已复制链接
已复制链接
快去分享给好友吧!
我知道了
×
扫码分享
扫码分享
Book学术官方微信
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术
文献互助 智能选刊 最新文献 互助须知 联系我们:info@booksci.cn
Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。
Copyright © 2023 Book学术 All rights reserved.
ghs 京公网安备 11010802042870号 京ICP备2023020795号-1