Majoritarianism

Let’s first dispose of some necessary trivialities. Majoritarianism comprises the normative thesis that, ceteris paribus, the best procedure for making collectively binding decisions is via majority rule of all those subject to such decisions and that, consequently, political decisions ought to be made by majority rule of all subjects. (Restricted majority rule is insufficient: aristocrats committed to majority rule amongst themselves to the exclusion of other subjects do not qualify.) A more interesting, less trivial point: the foregoing Narrow Majoritarian Thesis (NMT) about the best procedure is most reasonably construed as relying on a normative standard admitting of degrees and hence derived from a more foundational, General Majoritarian Thesis (GMT), according to which, ceteris paribus, the closer a collectively binding decision-making procedure to majority rule, the better, and, consequently, political decisions ought to be made by the decision rule as close to majority rule as possible. Majority rule is therefore the best procedure, according to GMT, because it is—trivially—the procedure closest to majority rule.

In principle, one could envision “narrow majoritarians” committed to NMT but who reject GMT. I set aside narrow majoritarianism for three reasons. First, although narrow majoritarians champion majority rule, they are indifferent between other decision rules once strict majority rule is compromised: as far as they are concerned, dictator rule is just as good as a supermajority rule of 50% + 2. This is implausible: it is hard to see how an adequate normative justification for NMT would not also justify a preference for decision rules like the latter over those like the former. Second, because it is an incomplete normative position and because strict majority rule is often impractical for large-scale decision making (or, for reasons extrinsic to majority rule’s justification, not desirable, all things considered), narrow majoritarianism is typically irrelevant for deciding between the remaining alternatives. Third, presumably because of narrow majoritarianism’s implausibility and irrelevance, contemporary majoritarians typically adhere to the general version, plumping for “a roughly majoritarian method” since strict majority rule is not on offer (Christiano Reference Christiano2008, 103). I therefore reserve the term for the degree-admitting form and define majoritarianism as the commitment to GMT (and to NMT by implication).

Majoritarian democracy, in turn, combines a majoritarian conception of democracy with a democratic justification of majoritarianism. On one hand, it comprises a conception of democracy that equates democracy with majoritarianism—that is, it holds that the greater the departure from majority rule, the greater the derogation from democracy. On the other hand, it holds that the justification for majoritarianism, or GMT, is grounded in democracy’s constitutive commitments—namely, democratic political agency and/or equality. Majoritarian democrats hold that the closer one approximates majority rule, the greater the instantiation of democracy’s core commitments—or, at least, that approximating majority rule is no worse than approximating any other decision rule in this regard and, from amongst the set of equally democratic decision rules, is preferable for reasons extrinsic to, but compatible with, democracy.Footnote ⁷

Are there any majoritarian democrats in the specified sense? To be sure, absolutists—who dispense with the ceteris paribus clause, and hold that departures from strict majority rule are never all-things-considered justified—are rare. But moderate majoritarian democrats, who hold that considerations extrinsic to democracy may sometimes justify departures from majority rule—for example, as a concession to the impracticality of strict majority rule in large polities, to avoid catastrophic outcomes, or to protect individual or minority rights, perhaps via judicial review—abound. What makes them majoritarian democrats, despite moderation, is the view that departures from majority rule, however much justified on extrinsic grounds, derogate from (or place constraints on) democracy and that, within those nondemocratic but justified constraints, the closer we approximate strict majority rule, the better. Majoritarian democrats equate democracy with majoritarianism primarily on political-equality grounds, but some also do so on political-agency grounds.Footnote ⁸

We can formalize majoritarianism’s defining feature via the notion of an expected majority deficit. Let the majority deficit for any possible “division” or vote configuration of all voters be equal to the size of the majority in that division minus the number of voters whose votes agree with the division outcome under the given voting rule. For example, when four out of five voters vote yes and one votes no, the outcome under unanimity rule is no; the majority deficit in such a division is therefore three—the size of the majority (four) minus successful voters (one). By contrast, if a voting rule yields an outcome with which a majority agrees, the majority deficit for that division is zero. Let the expected majority deficit of a voting rule be the expected value of the majority deficit under it.Footnote ⁹ Under majority rule, this will obviously equal zero; any true departure from majority rule will incur a positive value. We can therefore formalize GMT as the thesis that, ceteris paribus, the lower a decision-making procedure’s expected majority deficit, the better and that, consequently, the expected majority deficit ought to be minimized. Majority rule is therefore in principle the best decision-making procedure (NMT) because it uniquely eliminates the expected majority deficit. But the more general, defining feature of majoritarianism (GMT) is the commitment to minimizing it to the extent possible.

One way to justify the claim that majoritarianism instantiates democratic equality is to show that majority rule gives each person equal agential power in political decision making (justifying NMT) and that the closer one approximates majority rule, the more (or at least no less) equal is such power (justifying GMT). Political equality concerns being treated as an equal with respect to political decisions—that is, the binding laws and policies governing the exercise of political power. If political equality in general concerns both the decision-making process and decision outcomes, the specifically democratic notion of political equality concerns the process: democratic equality consists in equal political agency—that is, equal participation and agential power—in decision making.Footnote ¹⁰ By agential power, I mean agents’ capacity to effect outcomes by way of their intentional actions. (Agential power in my sense contrasts with the more controversial concept of structural power, by which I mean the “passive” power agents have in virtue of which the outcomes they favor or are in their interests obtain thanks to their social-structural position, but independently of their intentional actions.Footnote ¹¹ ) Because democratic equality is a matter of equal agency, it does not require that everyone subject to political decisions actually participate in and exercise power in decision making equally. Actually equal participation and the equal exercise of power could not be a democratic commitment because it is intrinsic to the exercise of agency that it be up to agents themselves to some degree. Democratic equality requires that subjects have equal agential power in decision making—an equal capacity to effect outcomes—but to reflect their own agency they must have some choice over the degree to which they actually participate and exercise it. Equalizing actual participation and the exercise of agential power cannot be a direct target of democratic equality because hitting that target would require eliminating the element of choice constitutive of political agency—for example, by stripping the highly motivated of opportunities to exercise it (cf. Dworkin Reference Dworkin2000, 194–98). Democratic equality instead requires equal agential power in decision-making understood in dispositional terms as a capacity, and hence equal substantive ongoing opportunity for subjects to participate and exercise power (Brighouse Reference Brighouse1996; Kolodny Reference Kolodny2014).

In what sense might majority rule equalize power? For one thing, it guarantees that all voters’ so-called a priori voting power is equal. A priori voting power refers to the voting power one has, once an agenda-set of alternatives is given, solely in virtue of the voting structure constituted by the formal-procedural decision-making rules themselves—that is, the sets of alternative outcomes, actors, and their action profiles and the collective-decision function that maps combinations of actions onto outcomes (Felsenthal and Machover Reference Felsenthal and Machover2002, 12). (It therefore abstracts from the actual distribution of voter preferences.) The concept thus formalizes the conception of power presupposed by thin majoritarian democrats.Footnote ¹²

The classic measure for a priori voting power—the Penrose–Banzhaf measure (PB)—equates it with the proportion of all logically possible divisions in which one’s vote is decisive—that is, in which one could have changed the outcome had one voted differently (Felsenthal and Machover Reference Felsenthal and Machover1998; Reference Felsenthal and Machover2004; cf. Banzhaf Reference Banzhaf1965; Reference Banzhaf1966; Penrose Reference Penrose1946). More formally, let v_i be one of |V| members of the set V = {v ₁, v ₂…v_|V| } of actors/voters, D = {d ₁, d ₂…d_|D| } be the set of all |D| logically possible divisions of V, and e_i (d_j ) be v_i ’s division efficacy score in d_j. Then,

$$ {PB}_i=\mathrm{def}\;\frac{\;{\sum}_{d_j\in D}{e}_i\left({d}_j\right)}{\mid D\mid } $$

where:

$$ {e}_i\left({d}_j\right)=\left\{\begin{array}{c}1\\ {}0\end{array}\right.\hskip1.08em \genfrac{}{}{0pt}{}{\;\mathrm{if}\;{v}_i\;\mathrm{is}\ \mathrm{decisive}\ \mathrm{in}\ \mathrm{division}\;{d}_j}{\mathrm{otherwise}.\hskip4.679997em } $$

For example, under dictator rule, PB = 1 for the dictator, who is decisive in every division; for all others, who are “dummies” not decisive in any division, PB = 0. Under majority rule with five voters, by contrast, each voter is decisive in 12 of 32 possible divisions; therefore, PB = ⅜ for each. PB is frequently interpreted as a voter’s probability of being decisive, on the assumptions of (a) equiprobable voting (the probability a voter votes for any option is equal to the probability she votes for any other) and (b) voting independence (voting inclinations are not correlated), which together imply (c) equiprobable divisions (any division’s probability is equal to that of any other). A posteriori voting power, by contrast, is the voting power one has in light of the actual distribution of preferences and consequent strategies and hence the actual probabilities of divisions.Footnote ¹³

Majority rule is evidently not the only decision rule that satisfies equal a priori voting power. But majority rule respects two further conditions, which some majoritarians argue are also necessary for political equality (Christiano Reference Christiano, John and Wertheimer1990; McGann Reference McGann2006): anonymity (or symmetry), which requires that swapping any sets of voters’ votes make no difference to the outcome (i.e., the identity of the persons whose votes they are be irrelevant), and neutrality, which requires that no alternative be favored by the decision-making rule itself (i.e., if a decision rule would select an outcome on the basis of a given set of votes in favor, any other alternative would also be selected were precisely the same votes to favor it) (May Reference May1952).

Yet even these further conditions are also satisfied by other decision-making rules—in particular, rules deploying lotteries. Consider a decision rule that selects alternatives at random: each person’s a priori voting power is zero (no one is ever decisive) and hence equal, and the decision rule respects anonymity and neutrality.Footnote ¹⁴ More plausibly, consider a voting structure with two alternatives and equiprobable lottery voting (in which the outcome is determined by a single, randomly selected vote) (Amar Reference Amar1984; López-Guerra Reference López-Guerra2011; Saunders Reference Saunders2010a). In each possible division, the probability a voter will be decisive is $ \frac{1}{\mid V\mid } $ . Hence for each voter, PB = $ \frac{1}{\mid V\mid } $ . Lottery voting, like random selection, satisfies not only equal a priori voting power but also anonymity and neutrality.

Therefore, this formal political-equality argument in favor of majoritarian democracy contains two lacunae. First, as even the staunchest majoritarians have recognized, it is insufficient for showing that majority rule instantiates political equality better than all alternatives.Footnote ¹⁵ This is not necessarily fatal: even if random selection and lottery voting satisfy criteria formalizing equality just as well as majority rule, one could still fill in the lacuna and uniquely pick out majoritarian democracy by appeal either to political agency or to values extrinsic to, but still compatible with and hence no compromise of, democracy. Second, and more importantly, the argument merely justifies NMT, not GMT.

A potential way to fill in both lacunae appeals to the fact that overall power is not constant sum: although PB_i ranges from 0 to 1 (from dummy to dictator), the overall amount of a priori voting power $ \sum {PB}_i $ in a voting structure not only varies from structure to structure, it can be greater than 1. For example, under majority-rule voting with five voters, PB_i = ⅜ for each voter; therefore $ \sum {PB}_i $ = $ \frac{15}{8} $ . By contrast, dictator rule (without abstentions) reduces overall voting power to 1. Lottery voting, despite differing from dictator rule in giving each voter equal a priori voting power, mirrors dictator rule in reducing the overall sum to 1. (Random rule reduces it to 0.)

In fact, majority rule uniquely yields the maximum possible overall a priori voting power. This is because any departure from majority rule incurs an expected majority deficit, and the mathematical relation between this deficit and overall a priori voting power is straightforward: minimizing the former turns out to be equivalent to maximizing the latter.Footnote ¹⁶ In other words, the defining feature of majoritarianism—GMT—is equivalent to maximizing overall a priori voting power. The upshot is that, as Saunders (Reference Saunders2010b) and Tuck (Reference Tuck2015, 261) have correctly recognized, majoritarianism’s defining feature concerns not political equality, but political agency (cf. Morriss Reference Morriss2002, 186).

We now seem to have a complete justification of the majoritarian conception of democracy. On the one hand, the appeal to political equality (formalized as equal a priori voting power, anonymity, and neutrality) justifies NMT by picking out strict majority rule as at least as good as any other alternative; on the other hand, the appeal to political agency (formalized as maximizing a priori voting power) justifies GMT by showing that approximating majority rule as much as possible is equivalent to maximizing overall political agency.

Federalism and Representation

With these four formal criteria to hand, we can now state the precise sense in which, according to majoritarian democrats, federalism derogates from democracy.

Vertical federalist arrangements constitutionally entrench a division of powers between federal and provincial jurisdictions; horizontal federalist arrangements seek to represent provincial interests in federal-level decision making (e.g., a Senate with representation by provinces). The first point is that horizontally to federalize collectively binding decisions of polity-wide scope is effectively to establish a two-tier (or multitier) decision-making procedure. This derogates from democracy, on the majoritarian view, because by departing from strict majority rule, it incurs an expected majority deficit, thereby lowering overall a priori voting power—an affront to political agency. True, some argue vertical federalism may enhance democratic agency, by giving subunits jurisdiction over issues whose scope could be restricted to the subunit (Weinstock Reference Weinstock2001). But this is an argument for subsidiarity, not federalism: vertical federalism, unlike merely discretionary subsidiarity, entails constitutionally entrenching these jurisdictional boundaries, such that a sheer majority at the federal level cannot itself decide the jurisdictional questions (Levy Reference Levy, Fleming and Levy2014)—which incurs an expected majority deficit over polity-wide questions about jurisdictional boundaries.

Second, creating multiple subunits in a two-tier decision-making process violates anonymity because outcomes become sensitive to the distribution of support across subunits. Third, unless subunits are all identical in size, the decision-making process will normally violate voters’ equal a priori voting power—unless the a priori voting power of each subunit in tier 2 is proportional to the square root of the subunit’s number of voters (Felsenthal and Machover Reference Felsenthal and Machover1998, 66–7). Fourth, if subunits were granted, as is typical under federalism, some extra power to block decisions, then—even if each subunit enjoyed the same amount of extra blocking power—neutrality would be violated.Footnote ¹⁷ Finally, the typical federalist arrangement of giving extra voting weight to subunits representing persistent minorities—for example, giving Quebec proportionally greater seats in the House of Commons, or an effective veto over constitutional matters—violates both equal a priori voting power and anonymity.Footnote ¹⁸

An even more far-reaching implication: the majoritarian conception cannot but treat representation as a derogation from democracy. First, any truly two-tier procedure incurs an expected majority deficit, which, again, is mathematically equivalent to reducing overall a priori voting power.Footnote ¹⁹ Second, any truly two-tier procedure with more than one electoral subunit in tier 1 violates anonymity.

None of this implies that (moderate) majoritarian democrats cannot accommodate representative democracy. They can. But the fallout is twofold. First, either representative democracy is an accommodation with values extrinsic to democracy that are in tension with, or traded off against, the majoritarian interpretation of democracy’s two core commitments, or it is a compromise of democratic values in light of practical constraints.Footnote ²⁰ For example, some may prefer exercising nonpolitical agency in private to political participation; and directly participating in all political decisions may in practice be too time consuming to be feasible for everyone. Representation may be a way to institutionalize some level of political agency and equality in light of such constraints.Footnote ²¹ Second, and more importantly, to whatever degree these values and practical constraints are accommodated, GMT implies that the expected majority deficit, even if it cannot be eliminated, still ought to be minimized within these constraints.

This last point confronts the majoritarian accommodation with a fatal problem. Imagine the majoritarian democrat acquiesces to a two-tier representative democracy as the structure closest to majority rule feasible under the circumstances. One might expect approximating majority rule always to tend towards equalizing power (Brighouse Reference Brighouse1996, 132–33). It does not. The problem is this: once a two-tier procedure is introduced and hence strict majority rule compromised, then minimizing the expected majority deficit (which, despite the compromise, remains majoritarianism’s defining feature) and ipso facto maximizing overall a priori voting power will often sharply clash with equalizing a priori voting power. Indeed, it turns out that, in many circumstances, minimizing the expected majority deficit would require giving some voters zero a priori voting power over second-tier decisions—effectively disenfranchising them.Footnote ²² GMT, majoritarianism’s defining commitment, cuts against the majoritarian conception’s own interpretation of political equality.

The implication is that those who defend a majoritarian conception of democracy by appeal to political equality have no leg to stand on once they move from direct to representative democracy. The issue is not just that according to the majoritarian conception representation in one respect compromises democracy. It is that what majoritarianism demands of representative democracy is incompatible with its own interpretation of political equality. True, the decision-making procedure favored as best by majoritarian democrats under ideal circumstances—strict majority rule—satisfies formal criteria majoritarians associate with political agency and equality. But once majoritarians face nonideal circumstances—as they invariably do—and accommodate other values and practical constraints, thus departing from strict majority rule but seeking to come as close as possible under the circumstances, then their constitutive commitment simply takes no account of—indeed, often cuts against—political equality. It therefore makes no sense to equate democracy with majoritarianism: GMT is in conflict with one of democracy’s core normative commitments as interpreted by majoritarianism itself. Of course, it might very well be true that democracy’s two normative commitments sometimes face trade-offs; this would render the notion of democracy complex, not incoherent. Indeed, I conclude below they are in tension. But despite majoritarian democrats’ pretension to serving both democratic masters, majoritarianism ultimately keeps faith with only one and stands against the other: in real-world circumstances, there is a trade-off between majoritarianism (reducing the expected majority deficit) and political equality as interpreted by majoritarianism. The upshot is that democracy cannot be equated with majoritarianism. The majoritarian conception fails on its own terms.Footnote ²³

So much for internal critique, which reveals the majoritarian conception of democracy to be incoherent. Now for external critique: even if majoritarian democrats did not jettison political equality, they would lack the conceptual resources to explain the political inequality of persistent-minority members.

Persistent Minorities

A recurring question in democratic theory has been why the results of majoritarian decision making should have any normative purchase for those who voted against the outcome. A typical Madisonian response is that in large pluralist societies, cleavages and coalitions would shift from issue to issue, so that under majoritarian decision making losers today could expect to be winners tomorrow (Hamilton, Jay, and Madison Reference Hamilton, Jay, Madison, Carey and McClellan2001, #10; cf. Przeworski Reference Przeworski, Shapiro and Hacker-Cordón1999). Indeed, some argue winners will invariably rotate because Condorcet winners over the feasible alternatives are practically impossible and hence cycling inevitable (McGann Reference McGann2006, 25–6).Footnote ²⁴ Yet this pluralist response, whatever its merits, fails to address the problem of persistent minorities, groups whose members’ preferences are positively correlated—and negatively correlated with those of a persistent majority—over a broad range of issues over time. Persistent-minority members such as French Canadians would repeatedly lose in majoritarian decision making.

One potential explanation for why this is a normative problem refers to the interests of persistent-minority members: their interest in having their preferences satisfied as such, as well as their other, substantive interests (assuming their preferences are a reliable guide to the latter). As Jones (Reference Jones, Miller and Siedentop1983, 168) puts it, “individuals who make up a persistent minority can properly complain that repeated applications of the majority principle are not consistent with their equal entitlement as individuals to want-satisfaction or interest-promotion.” In contrast to lottery voting, majoritarian decision making would give persistent-minority members neither equal prospects for success nor even prospects proportional to their numbers (Jones Reference Jones, Miller and Siedentop1983; Saunders Reference Saunders2010a). Yet this explanation in terms of want- or interest-satisfaction appeals not to features intrinsic to the decision-making process itself—such as equal agential power—but to substantive decision outcomes. Some have accordingly concluded that the persistent-minority problem shows political equality cannot be a solely procedural norm but must also concern the pattern of outcomes (Christiano Reference Christiano1994).Footnote ²⁵ Thus Beitz argues political equality should be thought of as a criterion not directly applied to the formal-procedural rules of decision making, but constraining the validity of justifications for institutional arrangements—justifications that must refer both to features intrinsic to the procedure and to the pattern of outcomes those procedures would likely produce. Political equality may, therefore, require formal-procedural inequalities (Beitz Reference Beitz1989, 17–18, 24).

This outcome-oriented approach has prompted the reply, by Kolodny (Reference Kolodny2014, 206–8, 321–28) for example, that no one is in general entitled to have her preferences satisfied by others and that the interest of concern to political equality does not comprise all interests in general, or preference satisfaction in particular, but consists in the interest in opportunities for “contributory influence” over the outcome—which Kolodny interprets as equal a priori voting power.Footnote ²⁶ Adopting a similar approach, Brighouse (Reference Brighouse1996, 139–40) concludes that a persistent minority who repeatedly loses but whose fundamental rights are safeguarded has no grounds for complaint. I agree with Beitz against Brighouse: a political decision-making process that systematically shortchanges the preferences and interests of a persistent minority gives them a reasonable justification for rejecting it. Political equality in general, in my view, is also concerned with structural power. However, my explanation of the specifically democratic problem posed by persistent minorities—posed not for political equality in general but for democratic equality in particular—does not involve appealing to the pattern of outcomes as such. Recall: the democratic conception of political equality concerns equal agential power in decision making—power understood in dispositional terms as an equal capacity to effect outcomes by way of one’s intentional actions, not in success terms as what is actually effected. That individuals are systematically unsuccessful in actually realizing their preferences or interests may be a good sign of their relative impotence, but their lack of agential power cannot consist in lack of success: agential power, as a conditional-dispositional concept, concerns an agent’s hypothetical, not actual, preferences, and it may or may not be exercised.

The democratic problem posed by persistent minorities is that its members lack equal agential power in decision making. But neither the a priori conception of power nor the notion of informal power over those with formal-institutional standing can capture this power inequality. I shall now outline a conception of agential power that explains the democratic inequality suffered by persistent minorities.

A POSTERIORI Voting Power

One of the chief objections leveled against the concept of a priori voting power is that, in virtue of being abstracted from the actual distribution of preferences, it is based on decisiveness just as much in logically possible but practically impossible vote configurations as in actually likely ones; it consequently overestimates the power of agents with eccentric preferences (Garrett and Tsebelis Reference Garrett and Tsebelis1999; Tsebelis and Garrett Reference Tsebelis and Garrett1996). The point is that to know who can effect outcomes in the actual circumstances, one must draw on information about actors’ preferences, likely coalitions, etc. Because the phenomenon of persistent minorities is constituted by social structures that induce particular preference distributions, turning to the concept of a posteriori voting power, which incorporates such information, might help explain their members’ unequal power.

Felsenthal and Machover have responded to criticisms of a priori power, however, by arguing that whereas for empirical purposes the concept of a posteriori voting power is more useful, for normative purposes—such as constitutional design or assessing decision rules—one must appeal to a priori power. They provide two main reasons: first, that preferences are normally too transient or volatile to be a reliable basis for long-run evaluation and, second, that even if they were stable, “it would be unfair to tailor the decision rule to the specific interests, preferences and affinities of the voters … we must go ‘behind a veil of ignorance’ … and act impartially, considering only the voting power that the voters will derive from the decision rule itself” (Reference Felsenthal and Machover2002, 13; Reference Felsenthal and Machover2004, 15). Once one excludes information about preferences, the principle of insufficient reason is supposed to justify assuming equiprobable divisions.

Neither reply withstands scrutiny. First, there often does exist information providing some degree of justified belief about future preferences (Beisbart Reference Beisbart, Fara, Leech and Salles2014, 108). While specific policy preferences may be transient, the underlying social structures shaping those preferences can be remarkably stable. Even if agents’ specific preferences in the long run are not reliably predictable, one may nevertheless reliably anticipate persistent affinities or identities that restrict the range of future preferences or structure their distribution. Thus when over 150 years ago the fathers of Confederation anticipated the lasting political significance of a French-Canadian minority, although they could not anticipate the decline of identification with Roman Catholicism, they did foresee the persistence of a social identity and consequent patterns of correlation in preferences. Things may have turned out differently, but they were not completely in the dark.

Second, fairness or impartiality may require—rather than rule out—tailoring decision-making procedures in light of social structures that shape the distribution of preferences or interests. A decision-making procedure that has desirable formal properties but that, against a particular background of social cleavages, would systematically go against the preferences or interests of a persistent minority may be reasonably rejectable by them on the basis of such a pattern of outcomes. But such rejection would depend on information about structures that affect the distribution of preferences and interests, and avoiding the pattern would require tailoring the procedure in light of such information. While fairness or impartiality rules out cooking up procedures partial to the specific content of specific actors’ preferences or interests, it does not rule out taking account of social structures, such as class, gender, or ethnicity, that induce certain preference distributions—for example, preferences that are single-peaked or otherwise, independent or correlated, or overlapping or in conflict, etc.Footnote ²⁷

However, we are not here directly concerned with the impartiality or fairness of either outcomes or procedures in general; we are concerned with democratic equality and hence equal agential power in decision making. This provides the basis for a further objection against a posteriori voting power. A relatively powerless agent, unable to effect a broad range of desired outcomes, does not suddenly become more powerful by restricting his preferences to the few outcomes he is actually able to effect: he remains powerless to effect the alternative outcomes, even if he no longer actually prefers to do so. Conversely, an agent able to effect outcomes she does not actually prefer will be less inclined to exercise her power to effect those outcomes, but she nevertheless still has the power to do so. Because power is a conditional-dispositional concept, it must be measured by the agent’s hypothetical, not actual preferences—that is, by counterfactualizing over her preferences. The question is what she could effect if she so preferred (Braham and Holler Reference Braham and Holler2005; Goldman Reference Goldman1972, 223). True, a French Canadian inclined to vote for greater state support for French would have no prospect for success under majoritarian decision making if such policies were invariably supported only by the Francophone minority and opposed by the Anglophone majority. But this fact cannot illuminate the inequality of power she would face: agential power, as a dispositional concept, neither consists in prospects for success nor can be analyzed in terms of the agent’s actual preferences.

The thought that the dispositional concept of power rules out appealing to a posteriori considerations, however, rests on a conflation of two separate distinctions: between a priori and a posteriori power, on one hand, and the ex ante and ex post perspective, on the other. The ex ante perspective on an agent’s voting power is the perspective one takes independently of her actual preferences—that is, counterfactualizing over her preferences and possible actions within the voting structure. Both a priori and a posteriori conceptions of power must take this ex ante perspective. The ex post perspective, by contrast, is the perspective one takes subsequent to the actual outcome on the basis of all agents’ actual preferences and actions. The ex post perspective is the perspective one takes to analyze the effect of agents’ exercise of power, not the agents’ power itself.Footnote ²⁸ However, while we cannot formulate a dispositional concept of power from an ex post perspective, this does not rule out an a posteriori concept appealing to structures beyond those constituted by formal-procedural rules. This is because in assessing someone’s agential power, one is required to counterfactualize only over her preferences: appealing to other voters’ actual preferences does not abandon the ex ante perspective.

The problem is that appealing directly to the content of other voters’ preferences, but not the power-assessee’s, is insufficient for explaining the inequality faced by a persistent-minority member: if the persistent-majority and persistent-minority groups are defined in terms of members’ substantive preferences, then counterfactualizing over the power-assessee’s preferences implies abstracting from her membership of the persistent minority. Her power inequality would consequently fail to register.

The solution lies in two related considerations. First, the problem itself implies that, for the purpose of assessing power-of-numbers, it would be incoherent to define a persistent minority directly in terms of shared substantive preferences (such as for state French-language support). If group members were picked out directly in virtue of shared preferences, then we could not assess the power-of-numbers of anyone qua member: because we must counterfactualize over the power-assessee’s preferences, we would have to abstract away from any power-assessee’s membership of a group so defined. A persistent minority must therefore be defined and picked out in terms of its members’ position in other social structures, such as language, ethnicity, race, class, or geography—a position that induces a correlation in preferences.

Second, adopting the ex ante perspective, and hence refraining from appealing to the content of the power-assessee’s actual preferences, does not rule out taking account of the abstract structure of the distribution of preferences and her position in it. Once we drop the voting independence assumption, we can, while still counterfactualizing over the power-assessee’s preferences, appeal to information about the correlation between her preferences and other voters’. This is because such information is based on information, not about the content of her preferences but about the social structures that shape both her and others’ preferences. On the approach adopted here, we abstract from the agenda-set of alternatives (such as the proposal for increased support for French) and ask, For any alternative over some range of issues, if the power-assessee supports it, what is the probability other voters will support or oppose it? If the power-assessee is a persistent-minority member, then the preferences and votes of other persistent-minority members will be correlated positively with hers, those of persistent-majority members negatively. We can on this basis formulate a Generalized Penrose–Banzhaf measure of voting power:Footnote ²⁹

$$ {GPB}_i\hskip0.5em {=}_{\mathrm{def}}\hskip0.5em {\sum}_{dj\in D}{e}_i\left({d}_j\right)\hskip-0.1em \cdotp \hskip-0.1em p\left({d}_j\right) $$

Here the probability distribution p gives the probability for each division d_j (where $ \sum p\left({d}_j\right) $ = 1), and summarizes information about other voters’ preferences and strategies on the assumption that the power-assessee v_i ’s preference for each alternative is equiprobable. This generalized measure covers both a priori and a posteriori voting power: PB is simply a special case of GPB that assumes equiprobable divisions (for all j, p(d_j ) = $ \frac{1}{\mid D\mid } $ ). But once we drop the voting independence assumption, and take into account correlations between voters’ votes, GPB no longer reduces to PB.

Like PB, GPB can be interpreted as representing the probability the power-assessee will be decisive. But we now calculate p(d_j ) on the basis of the probability of other voters voting for alternatives conditional on the power-assessee voting as she does in the division d_j whose probability we are calculating. If votes are not independent, then this conditional probability will no longer equal the unconditional probability used for a priori voting power. Hence the generalized measure no longer measures voting power based on equiprobable divisions. The probability of being decisive is now an a posteriori probability: a probability in virtue of not only the formal-procedural decision-making structure but also the structure of the distribution of preferences (and the potential effects of strategic interaction). While the equiprobable voting assumption for the power-assessee models the ex ante, first-personal perspective from which she deliberates about what to do, and from which her preferences and vote are an open question, the correlation with others’ preferences models the third-personal fact that their preferences—and hence the way they, including her, decide—are shaped by the same social structures.

To illustrate, consider again majority-rule voting with five voters. There are 32 logically possible divisions and each voter is decisive in 12. PB assumes each division is equiprobable—it weights each instance of decisiveness equally—yielding, on the classic model, PB = ⅜ for each voter. But now assume voters v ₁ and v ₂ compose a persistent minority and voters v ₃, v ₄, and v ₅ a persistent majority—in virtue of the position they occupy in a social structure such as race, ethnicity, class, primary tongue, etc. Assume further that group members have perfectly correlated preferences: conditional on v ₁ voting yes, v ₂ will vote yes and voters v ₃, v ₄, and v ₅ will vote no. Thus the division in which v ₁, v ₂, and v ₃ vote yes and v ₄ and v ₅ vote no—a division in which v ₁ is decisive—will occur with zero probability. In fact, all divisions in which v ₁ would be decisive occur with zero probability. Under these assumptions, v ₁’s voting power as measured by GPB, like v ₂’s, would be zero. Conversely, the voting power of each of the three persistent-majority members would be 1.

Therefore, it looks as if the generalized measure, which abandons the voting independence assumption, can account for the inequality in voting power faced by persistent-minority members under majoritarian democracy. The problem with the majoritarian conception, therefore, is that political equality requires equalizing power subjects have in virtue of not only structures constituted by formal-procedural decision-making rules but also those that shape preferences.

This conclusion invites at least two objections. First, once we move to more realistic cases, with a large number of voters in which the sizes of the minority and majority differ by more than one voter, it seems the generalized measure fails to show any difference in voting power between members. Consider five million voters divided into persistent minority and majority in the same proportion as above—two million in the former, three million in the latter. Again assume binary majority rule and perfectly correlated votes. Now consider any persistent-minority power-assessee. Conditional on her voting yes, there will be two million yes-votes, three million no-votes. She is not decisive in this vote configuration. Conditional on her voting no, the division will be two million yes, three million no. Again, she is not decisive. Therefore, she seems to have no voting power: she is decisive in none of the actually possible divisions. But the same conclusion holds for any persistent-majority member. Conditional on him voting yes, three million will vote yes, and although he will get his way, he will not be decisive. The same holds if he votes no. So again it looks as if the only difference between members of persistent minorities and majorities is that whereas the latter get their way, the former do not—but without any difference in voting power. This is essentially the criticism Kolodny (Reference Kolodny2014) levels against concepts of voting power seeking to capture, as I have done, an a posteriori, rather than a priori, probability a voter will be decisive: because the probability any particular individual will in reality be decisive is vanishingly small, Kolodny suggests, political equality cannot be analyzed in terms of equal a posteriori voting power: everyone effectively has zero probability of actually being decisive.

The flaw in this criticism is that it relies on the assumption that a voter efficaciously exercises power in a decision-making procedure only if she is fully decisive over the outcome—that is, only if she could change the outcome by altering her action. This assumption also underlies the classic Penrose–Banzhaf model, which assumes that a voter’s division efficacy score e_i (d_j ) for any given division can only be 1 (if she is fully decisive) or 0 (if not). This assumption is mistaken. Return to the case of five voters using majority-rule voting and consider the division in which all five vote yes. The outcome is yes. Since no voter would be able singlehandedly to change the outcome by altering her vote, the classic Penrose–Banzhaf model assumes that, in this division, each voter’s division efficacy score is 0. It assumes, in other words, that ex post no individual voter has efficaciously exercised any power to effect the outcome. But this is false: in contrast to a yes-outcome under random rule, the ex post outcome here is caused by the yes-voters.

More generally, the classic assumption that voting power is solely a function of full decisiveness falters over cases of overdetermined causation. Consider a case in which five assassins together push their victim to his death, even though any three of them would have been sufficient to effect the outcome (and at least three are necessary). No assassin’s action alone constitutes a necessary condition for the outcome—none is decisive—and none constitutes a sufficient condition. Yet this does not imply no individual played a causal role in, and efficaciously exercised power in effecting, the murder. We can explain their causal efficacy by appealing to, not their full decisiveness or necessity, but the fact each satisfies a NESS test: each assassin’s action constitutes a necessary element of a sufficient set of actions to realize the outcome (which is a subset of the actual set of actions effecting the outcome) (Wright Reference Wright1985; Reference Wright1988). No assassin on her own is fully decisive for the outcome, but each is partially decisive or efficacious. Similarly, each of the five voters is partially efficacious for the outcome in the unanimous division (in particular, each is ⅗ efficacious).Footnote ³⁰

Once we incorporate partial efficacy in the measure of voting power—once we allow e_i (d_j ) to range between 0 and 1—the problem we encountered with larger, more realistic numbers vanishes. Return to the persistent minority of two million and persistent majority of three million with perfectly correlated preferences. An important power asymmetry now arises between members of the two groups. Begin with any persistent-minority power-assessee. Conditional on her voting yes, the division will be two million yes, three million no, and the outcome no; conditional on her voting no, the division will be two million no, three million yes, and the outcome yes. She is obviously never decisive, nor is she ever partially efficacious, because her action is never a necessary element of a sufficient set of actions for the actual outcome. Now consider a persistent-majority power-assessee. Conditional on him voting yes, the division will be three million yes, two million no, and the outcome yes; conditional on his voting no, the division will be three million no, two million yes, and the outcome no. He is partially efficacious in each division because his action forms a necessary element of many of the sufficient sets of actions for the actual outcome.

The Power of Numbers

Thick majoritarianism furnishes the second objection to my proposed solution. Thick majoritarians might argue that taking into account social structures that induce correlations between voters’ preferences does not require abandoning the concept of a priori voting power. Rather, they might argue, violations of independence are best accounted for, not via the concept of voting power but by appeal to informal power over those with a priori voting power. If someone has power to shape others’ preferences in his own image, then their votes will of course be correlated: conditional on his voting yes, those whose preferences he shapes will also be inclined to vote yes—not because he has greater voting power, but because he has the informal power to shape preferences. This is why political equality requires, on the thick majoritarian conception of democracy, equalizing not only (a priori) voting power but also the informal power to shape preferences.

The thick majoritarian response is insufficient. Imagine a society in which the correlated preferences within persistent majority and minority are explained by members’ shared position in a particular social structure—say, ethnicity. Imagine further that the reason ethnicity shapes political preferences is that elites in each group—what Laitin (Reference Laitin2007) calls “ethnic” and “political entrepreneurs”—exercise their informal power over other members to confer salience to ethnicity as a social category. We have a model with four groups: majority elites who subject their coethnic masses to their power, shaping their preferences and thereby forging a persistent majority; and the corresponding minority elites and masses. The appeal to informal power does indeed register the power of agents in fashioning a persistent majority for themselves. It therefore registers a power asymmetry between, say, majority elites and minority masses.

But it cannot explain the power asymmetry between majority masses and minority masses, neither of whom—by hypothesis—have such informal power. In other words, thick majoritarianism remains incapable of explaining the independent phenomenon of the power of numbers. It cannot explain the greater agential power of one who finds himself the member of a persistent majority through no power of his own. A member of the majority who would persistently defeat voters under majoritarian decision making may find himself in a persistent majority despite not having any power over anyone’s preferences. This difference in power-of-numbers is what the a posteriori model above, which refers to correlations between preferences caused by social structures—but not to the particular agents who play a causal role in forging those correlations—captures.

The persistent-majority masses here illustrate the phenomenon of empowering subjection. Recall: power is not constant sum. A corollary: being subject to another’s agential power does not necessarily reduce your power. Of course it may do—as when others restrict your options by interfering. But when others exercise power over you by shaping your preferences (Lukes Reference Lukes2005, 27) or, indeed, your subjectivity (Foucault Reference Foucault, Defert, Ewald and Lagrange1994, 277), they do not necessarily reduce your agential power because, recall, we analyze agential power by counterfactualizing over the agent’s preferences. Indeed, those who exercise power over you may, by helping to constitute your subjectivity in a particular way, increase your agential power. As Foucault recognized, the effects of power can be productive, not just constraining. One way they might increase your power is by helping to enhance your subjective capacities. But another way is by forging a group with whom you can act in concert. Power-of-numbers does not always stem from empowering subjection in this way, but the fact it can explains a familiar social phenomenon: sometimes people, by subjugating themselves and their preferences to other group members, may nevertheless (correctly) feel empowered in virtue of their consequent ability to act in concert with them—as when a populist’s followers become politically empowered in being unified even though they are mere pawns of their leader.

To be sure, it is commonly thought that one is merely lucky, not powerful, if others over whom one has no power are inclined to help satisfy one’s preferences or interests (Barry Reference Barry1980; cf. Goldman Reference Goldman1972, 225). On this view, the fact someone always gets his way because he just happens to find himself a member of a persistent majority and another always loses because she is a persistent-minority member does not amount to his having more power than her; he’s merely lucky, and she’s unlucky.

I have already shown why this common view is mistaken; it is incapable of explaining the power individual agents efficaciously exercise in effecting overdetermined outcomes.Footnote ³¹ Moreover, its underlying intuition is readily explained. It is grounded in the erroneous thought that agential power is an intrinsic property of agents and/or that luck and (agential) power are opposites—that, whereas in exercising agential power one effects outcomes by way of intentional actions, what occurs because of factors extrinsic to one’s intentional actions is a matter of luck. This is a false dichotomy: outcomes effected by way of one’s intentional actions occur always also because of extrinsic factors. Intentional actions effect outcomes always within a specific environmental or structural context. Because agential power is not the agent’s intrinsic property, holding her intrinsic features constant but varying the context may very well vary her power: one may have the power to effect the movement of boulders in one world, but not in an alternative world identical save for much heavier boulders. Certainly, assuming one has no say over which world one finds oneself in, it would be lucky boulder-wise to find oneself in the first, but one would nevertheless have a power in the first lacking in the second. Similarly, one may have the power to effect boulders’ movement in a world in which many are inclined to assist but not one in which no one is so inclined. Therefore, one may indeed be lucky to have numbers on one’s side. But that is just to say one is lucky to be so powerful. Having numbers on one’s side is a kind of power-with, not merely a matter of luck (Abizadeh Reference AbizadehN.d. b).

Normative Implications

The power-of-numbers thesis is indispensable for explaining the inequality suffered by persistent-minority members under majoritarian decision making. Its normative implication is that democratic equality sometimes requires counter-majoritarian, formal-procedural inequalities in decision making to compensate for inequalities in power-of-numbers. These departures are precisely what federalist and representative institutions sometimes enable. Whereas some have defended federalism’s compatibility with democracy on political agency grounds—which overlooks the distinction between federalism and mere subsidiarity—the present argument appeals instead to political equality.

The particular institutions best suited to redressing inequalities in power-of-numbers—and whether such inequality can or, all things considered, ought to be compensated—will depend on context and vary from one minority-type to the next. The present theoretical argument cannot determinately tell us which particular counter-majoritarian and representative institutions will best serve democratic equality in general: that greatly depends on the polity’s contextual, empirical circumstances. The argument here provides, rather, a conceptual and normative framework for making such judgements in light of those circumstances.

To flesh out this framework, we begin with an initial abstract model, then relax its simplifying assumptions in reverse order. Consider a hypothetical polity in which the formal-procedural power of all subjects is equal and (a) no one has informal power over others’ political preferences; (b) there is only one social structure that induces a persistent majority/minority cleavage, and this is the only such cleavage; and (c) the persistent minority is territorially concentrated. Majoritarians fail to register the inequality here, but the clear upshot of the power-of-numbers thesis is that, in this simple case, counter-majoritarian federalism would reduce political inequality.

While federalist institutions may be effective in compensating territorially concentrated minorities, if the minority is territorially dispersed—relaxing (c)—then rather different institutional remedies may be required, remedies centered not so much on jurisdictional boundaries and regional vetoes as on the electoral and party systems. For example, one way to compensate for the unequal power-of-numbers of the working class may be via representative institutions favoring the presence and legislative power of a labor party, which are often favored by multiparty rather than two-party representative systems, which in turn are favored by larger, multimember (rather than small, single-member) electoral districts and proportional-representation voting systems.

Once we relax (b), and confront potentially many social structures that produce distinct persistent cleavages, we may find that compensating inequalities along one axis cuts against doing so along another. Consider the rural-urban divide. If policy preferences are persistently divided along this axis, then rural voters may constitute a persistent minority whose members have less power-of-numbers than urban ones. But if the rural/urban cleavage is itself correlated with another persistent social structure, for example a white/non-white cleavage, then compensating rural voters may exacerbate inequality for non-white voters (if they disproportionately inhabit cities). Matters become even more complicated under an intersectional approach that identifies groups with reference to multiple axes (e.g., white rural voters).

Therefore, even which social structures are the relevant ones over which to compensate individuals will depend on contextual features—such as the distribution of other kinds of social power. There is no formula here: addressing inequalities in power-of-numbers via formal institutional arrangements such as federalism is inherently crude and always carries trade-offs and remainders along different axes. Because of these remainders, determining whether and which minorities ought to be compensated may require appealing beyond the concept of agential power and the formal theory of democratic equality outlined here to a broader theory of justice, domination, or oppression (Haslanger Reference Haslanger, Levine and Pataki2004) that takes structural power into account.

But our formal theory’s bite is not yet exhausted. Relaxing (a) reveals another reason why not every inequality in numbers ought to be compensated: sometimes, a minority already enjoys outsized power of a different kind. Consider the set of billionaires. They occupy a position in the class structure in virtue of which their policy preferences might predictably correlate. But even if billionaires constituted a persistent minority suffering from relatively low power-of-numbers, there would be no all-things-considered justification for compensating them because, in virtue of their financial resources, they already enjoy outsized informal political power. The same holds for propertied classes in general.

Moreover, to the great surprise of many nineteenth- and twentieth-century observers (Przeworski Reference Przeworski2010), the rich may not form a persistent minority at all. Because the causal arrow from social-structural positions to groups with shared preferences is mediated by the agential power of actors, not all shared social-structural positions yield—as they do in our four-group ethnic model above—isometric preference cleavages. The rich might end up forging a persistent policy majority for themselves—not a majority tracking people’s position in the class structure but one tracking the scope of those subject to their informal financial power.

Relaxing assumption (a) thus returns us to our four-group model. When some individuals’ persistent-minority position and hence lesser power-of-numbers arise from being subject to others with greater informal political power—as in the case of minority masses—their political equality is compromised by their lesser power of both kinds. Here contextually appropriate procedural inequalities in their favor may furnish justifiable redress. However, once we turn to minority elites, the present argument’s implications are indeterminate. True, procedural inequalities in favor of minority elites would redress their lesser power-of-numbers, but such inequalities would also exacerbate their relatively greater informal power. The ideal remedy, of course, is to equalize informal power, but we still confront indeterminacy in real-world conditions where doing so may be infeasible. An all-things-considered judgement will, therefore, again depend on not just contextual features but a broader theory of justice.

The theoretical argument here does, however, obviate two potential objections. The first charges that the power-of-numbers thesis absurdly requires compensating political eccentrics on democratic-equality grounds. Recall, however, that because in measuring agential power we must counterfactualize over the power-assessee’s preferences, we cannot define a persistent minority directly in terms of its members’ substantive preferences: the concept of the power-of-numbers of a member of a minority so-defined is incoherent. This implies that persistent-minority members with unequal power-of-numbers must be identified on the basis of social structures other than the distribution of policy preferences itself. A French Canadian may be specified as a persistent-minority member in virtue of her position in social structures constituted by shared primary language, subjective identification, etc. but not directly in terms of shared policy preferences: a French Canadian whose policy preferences do not strongly correlate with other members does not thereby cease to be French Canadian. This is why policy preferences will in practice be probabilistically rather than perfectly correlated. But democratic equality cannot be coherently concerned with an individual’s power-of-numbers in virtue of her membership of a group defined directly in terms of shared policy preferences. If libertarians or Bolsheviks are persistently outnumbered, democratic equality does not call for formal-procedural inequalities to compensate them.Footnote ³² (By contrast, if Bolshevism in the society correlates with being its most destitute, because occupying the lowest position in the class structure causally explains people’s Bolshevism, then democratic equality may very well do so—but in virtue of class position, not ideological eccentricity.)

The second objection resurrects the charge that power-of-numbers is a misnomer for merely good luck. Consider a bellwether voter who exercises no informal power over others but who nevertheless always agrees with a majority on any question regardless of cleavage. It seems counterintuitive to conclude that the bellwether enjoys outsized power-of-numbers. To reply, we must differentiate three scenarios. First, assume there is no social-structural reason for the coincidence in preferences: it results from pure chance. Then the present argument does not attribute to him greater power-of-numbers. True, there exists a set of subjects with whom his preferences correlate—the set of voters. But recall, we cannot specify the persistent majority of which he is supposedly a member by reference to the correlation between his preferences and others’. Nor can we arbitrarily pick out any independent category of people with whose preferences his just happen to correlate. Rather, we must define the persistent majority in light of his position in the social structure that causally explains this correlation. Ex hypothesi, his preferences do not correlate with other voters’ in virtue of his position qua voter: that social structure plays no explanatory role. The correlation here is a matter of sheer luck, not power—except, perhaps, his divine power.

Second, assume a social-structural explanation does exist—along each axis of correlation. He is in the majority on religious issues, because he’s Christian; on racial issues, because he’s white; on linguistic issues, because he’s Anglophone; etc. Then the power-of-numbers thesis does indeed attribute greater power to him by virtue of his privileged position in these social structures—which is exactly right.

Third, assume the correlation is mediated by the informal power to which others subject him: his preferences perfectly correlate with a majority in each case because, along each axis, he is subject to the elites’ total power. This is a species of the second scenario, and the present argument again attributes power-of-numbers to him. It may seem counterintuitive to attribute power to someone so thoroughly subjugated, but this is a limit case throwing up unreliable intuitions: as noted, subjugation can in other respects be empowering and—as with the billionaire minority—the pro tanto case for redress may be overwhelmed here by the massive inequality of informal power in the other direction.