The existing literature on the justice of electoral systems – both popular and academic – is based on surprisingly illiberal premises. Both the advocates of proportional representation (PR) and their critics nearly always base their arguments on fairness or equality between groups. Indeed the different conceptions of fairness can be analysed in terms of which groups (partisan, social or geographic) are considered relevant. By contrast, we show that a strictly liberal conception of fairness, based only on equal treatment of individuals, logically implies PR. This allows us to clarify the assumptions upon which much of the debate on electoral fairness rests.
Since arguing that PR is fair or egalitarian is commonplace, it is important to emphasize how the argument presented here is different. It is not the usual argument that PR is fair because it treats voters fairly in the sense that the same number of votes for each party is required to elect a representative. This is really a rephrasing of the argument that PR treats parties fairly. It is phrased in terms of individual voters, but the individual voters are conceived in terms of a group (i.e. partisan) identity. It is thus liable to the objection that if we imposed a different group identity on voters (perhaps place of residence), we would get different conclusions. By contrast, our argument is not based on equality or fairness to individuals as party voters, as residents of a district or as members of any other group identity, but rather on the equality of individual voters per se. That is, it relies on the concept of liberal equality, the requirement that all individual voters be treated equally.
Liberal equality requires that an electoral system be anonymous (does not discriminate between voters on the basis of their names) and neutral (does not discriminate between candidates, lists or parties on the basis of their identities). Taking this axiomatization of liberal equality, we can use a social choice theoretic argument to show that this implies a result essentially identical to PR. The mechanics of the proof are similar to May’s theorem that shows that anonymity, neutrality, decisiveness and positive responsiveness imply majority rule when deciding between two alternatives.Footnote 1 For this reason, it is surprising that PR does not appear to have been previously treated in such an axiomatic fashion. Indeed, McLean, reviewing the literature in 1991, argued that PR had no basis in the theory of voting.Footnote 2 In fact, proportional representation and majority rule have frequently been presented as incompatible principles.Footnote 3 Our result shows that precisely the same principle of liberal equality that recommends majority rule as a decision rule within parliaments, also recommends proportional representation as an electoral system.
Our formal results assume single-vote voting systems. However, they can be extended to multiple-vote systems, where voters rank the various alternatives. We can show that to satisfy liberal equality, any such system must be compatible with proportionality when voters only care about lists or parties (as opposed to individual candidates). While an ordinal voting system that satisfied liberal equality would be extremely desirable, it is notable that the most common ordinal voting systems proposed violate our axioms in one way or another, although Single Transferable Vote applied to a national district only violates non-negative responsiveness.
Of course, equality – liberal or otherwise – is not the only criterion for assessing electoral systems. Indeed, Schumpeter argued that democracy per se was not a value worth considering at all, and that democracy should be considered instrumental in terms of producing other values.Footnote 4 The degree to which there is a trade-off between liberal equality and other values, such as stability, minority protection or accountability – if indeed there is a trade-off at all – is an empirical issue. The question of how liberal equality should be balanced against these other values is also beyond the scope of this article. Finally, this article deals only with liberal equality as it applies to electoral systems. Electoral systems are part of a broader constitutional system that makes socially binding decisions. Arguments as to why political equality in terms of social decisions requires equality in representation are presented in McGann’s The Logic of Democracy, and Christiano’s The Rule of the Many.Footnote 5
Liberal versus group-based arguments
It is initially surprising that the normative case both for and against PR in nearly all of the existing literature is not based on a concept of liberal equality, but rather some concept of fairness to political parties or social groups. Actually, the justice of proportionality is frequently asserted in an intuitionist manner, as if it were a self-evident moral fact. As Lijphart concludes, for many PR supporters, proportionality is simply a goal in itself, ‘virtually synonymous with political justice’.Footnote 6 However, when justifications are given for the principle of proportionality, these are nearly always in terms of fairness to political parties or social groups, or in terms of the desirable instrumental effects of PR. For example, McLean argues that the case for PR rests on the idea that a legislature should be a microcosm of the population, as did Black.Footnote 7 Similarly, Pitkin identifies the case for PR in terms of ‘descriptive representation’ (the fair representation of every salient group), while Still uses the similar concept of ‘group representation’.Footnote 8 The political electoral reform discourse frequently emphasizes the unfairness of non-proportional representation to certain political parties, who win far fewer seats than their vote share would entitle them to under proportionality.
Of course, arguments for PR can be made in terms of fairness to political parties or social groups, and these arguments are convincing to a considerable number of people. However, these arguments are often taken to be the only arguments for PR, with the implication that if these arguments are refuted, the case for PR collapses. One notable recent example of this argument occurs in the US Supreme Court case of Vieth et al. v. Jubilirer.Footnote 9 Justice Scalia, together with Chief Justice Rehnquist and Justices O’Connor and Thomas, argues that claims of partisan gerrymandering cannot be judicially determined because there is no manageable standard for judging such gerrymandering.Footnote 10 They reject the standard that a majority of voters should be able to elect a majority of representatives on the ground that this assumes a right to proportional representation, which the Constitution does not provide. It is argued that the Constitution only guarantees equal protection to individuals, not to social groups. The assumption here is that a right to proportional representation cannot be derived from equal protection of individuals, but requires an argument in terms of group rights. This assumption is precisely what we refute in this article.
A number of theorists have recently analysed the concepts of fairness and equality, and have argued that different concepts of fairness imply different electoral systems, thus implicitly or explicitly critiquing the assumption that PR is intrinsically fair.Footnote 11 Opponents of PR have made a similar case with a far more polemical intent.Footnote 12 It is argued that proportional representation assures one particular type of equality – mathematical equality in the treatment of political parties – but that there are other kinds of equality or fairness that are equally or more compelling (for example, fairness to geographical communities, candidates or voters). These literatures draw welcome attention to the lack of rigour in the use of the terms fairness and equality. However, the effect is frequently relativist. If an equally valid form of fairness or equality can be found to justify any electoral system, then arguments based on fairness are nothing more than statements of personal preference.
It is notable that the arguments put forward by these authors for the fairness of systems other than PR are also based on conceptions of fairness to groups or political parties. For example, the winner-takes-all principle considers fairness in terms of parties or candidates, not voters. First-past-the-post is fair, it is argued, because the party that wins a fair contest gets the prize. Similarly, the constituency principle is based on fairness to geographically defined groups of people. Blau’s ‘populist’ conception of fairness (the idea that ‘the people’ and not party leaders should determine who the government is) is also clearly group-based – ‘the people’ is a collective noun, after all.Footnote 13 Of course, advocates of these principles prefer to phrase them in individualist terms (the rights of individual residents of a constituency, or of voters of the plurality party), just as advocates of PR do. However, as with the usual argument for PR, these arguments are still group-based in that they conceive individual voters in terms of a preconceived group identity, instead of conceiving them as individuals per se in the liberal sense.
The normative ambiguity in the literature is a direct result of there being multiple conceptions of fairness between groups. First, there are many different ways to divide society into groups. Secondly, there are many plausible principles of fairness and impartiality to arbitrate between groups. If we consider parties to be the relevant group and we consider fairness to require the equal treatment of each vote won, then proportional representation is fair. However, if we consider that parties are treated fairly if the party that wins the most votes in a fair contest gets the prize, we get the winner-take-all conception of fairness (the horse racing analogy in the British use of ‘first-past-the-post’ to describe single-member district plurality elections captures this sense of fairness). Similarly, if we start with predefined social groups as the relevant units, we can generate arguments for the fairness of different electoral systems. If we consider geographical constituencies to be the most relevant unit, then we can produce an argument for the constituency principle and single-member district elections. However, if we consider other social groups to be more salient (perhaps ethnic groups or social classes), then we can make an argument that PR is fairer in that it represents these groups better, although seats reserved for the relevant group or simply a random selection of the legislature may do better still.
We argue that liberal equality offers a way out of the ambiguity resulting from group-based arguments on electoral justice. It is therefore surprising that arguments based on the fair treatment of individuals are so uncommon. There are, however, a few exceptions. Dodgson advocates a PR system primarily on grounds of individual fairness. The first two desiderata of an electoral system he gives are that every voter has the same chance of being represented, and that every represented voter be represented by the same fraction of a member.Footnote 14 Ward uses computer simulations of one-dimensional party competition to argue that PR is the electoral system most likely to produce policy choices close to those of the median voter, and would thus be the system chosen by individual voters behind a Rawlsian veil of ignorance.Footnote 15 Hout et al. show that the anonymity, neutrality, consistency, faithfulness and ‘topsonlyness’ imply the plurality ranking property.Footnote 16 Our results draw on this insight.
The concepts of liberal equality and proportional representation
Our central result is that the axioms of liberal equality imply a seat allocation rule that produces results essentially identical to those produced by pure proportional representation. It is therefore necessary to define the concepts of liberal equality and proportional representation.
Liberal Equality
By liberal equality we mean the requirement that procedures treat every individual voter equally. This may seem the most obvious kind of equality to apply to political institutions, especially in countries with a long tradition of liberal democracy. However, as we have seen, the electoral reform and electoral systems literatures nearly always rely on some conception of equality between groups. We will see that starting strictly from the liberal equality of individuals allows us to understand many of the contradictions of the existing literature.
When considered in terms of individual voters, the concepts of equality (the demand that individuals are treated the same), fairness (the demand that every individual is treated as they ought to be) and impartiality (the demand that how someone is treated should not depend on inappropriate factors, such as personal friendship, class, race or gender) are practically equivalent. When we consider the rights of individual citizens in a liberal democratic state, fairness implies equality, unless we believe that some citizens are entitled to more political rights or better treatment than others. This follows from the standard liberal assumptions that every individual’s interest is worthy of equal consideration and that every individual is the best judge of their own interests.Footnote 17 However, even if we accepted that some individuals were more worthy or more qualified than others, we would still have to find a mutually acceptable procedure to determine which citizens were better qualified or more deserving. Given that this is likely to be impossible, equality is the only option we have remaining that can be described as fair. Similarly, if we treat individual voters in different ways, we can hardly argue that we are acting impartially, unless the different treatment is handed out in a strictly random way. It is hard to see how we can justify such discrimination in terms of individual voters. Of course, many reasons have been given as to why different voters should be treated differently. However, all these arguments rely on fairness and impartiality in terms of groups, and thus are not framed in terms of liberal equality.
However, if we consider groups instead of individual voters, this equivalence of equality, fairness and impartiality no longer holds. It is not clear that it is fair to treat differently sized groups equally (this certainly violates equal treatment of the individual members of the groups). Neither is it obvious that impartiality is violated if we treat differently sized groups unequally, as to argue this it would be necessary to show that the size of the group is an unacceptable criterion for discriminating. In addition to size differences, some groups may be politically salient, while others may not. Furthermore, it is possible to divide a population into groups in many different ways. These different sets of groups may overlap or have cross-cutting membership. Therefore, to make an argument about fairness based on groups, it is necessary first to justify why fairness should be seen in terms of a particular partition of the voters into groups. This involves both justifying why fairness should be seen in terms of these groups rather than in terms of individuals, and justifying why these groups are more appropriate than other possible sets of groups. Secondly, it is necessary to justify a particular conception of fairness between groups.
Our interest in liberal equality is primarily analytic. However, we will summarize the normative arguments made in favour of preferring liberal equality over some form of group-based fairness. First, there is the liberal argument that the rights of individuals should enjoy normative priority over the rights of groups that these individuals make up. However, even if we do not accept this doctrinal liberalism, a case can still be made for liberal equality as a fair means to arbitrate between different conceptions of group-based fairness. As we have seen, there are multiple forms of group-based fairness depending on which set of groups and which principle of allocation we choose. Who has a right to impose their conception of fairness over everyone else? Liberal equality offers a way out in that it merely demands that all individuals be treated equally. Individual voters can then decide what group identities are salient to them, as opposed to certain forms of group fairness being imposed by the electoral system. We can compare this to the role that political liberalism plays in Rawls.Footnote 18 There political liberalism provides a means for arbitrating between competing ‘comprehensive doctrines’ that lay out the values that define the good life. Here liberal equality provides a means of arbitrating between different conceptions of group-based fairness.
We now turn to how the concept of liberal equality can be operationalized in electoral systems. We conceive electoral systems as seat allocation rules. A seat allocation rule is a mechanism that takes the votes and individuals and translates them into seat shares for individual alternatives. The alternatives presented to the voters may be individual candidates, lists of candidates or parties. This generality is important. It is often assumed that PR elections assume political parties. The argument in this article, however, makes no such assumption. By using the general concept of a seat allocation rule, we can make a liberal argument that can accommodate, but does not rely on, political parties.
The most efficient way to operationalize liberal equality is to proceed negatively, excluding those considerations whose inclusion would violate equality. This is what we do when we consider equality or fairness in terms of impartiality.Footnote 19 It is also the basis of Williams’s ‘relevant reasons’ approach to equality – discrimination has to be justified by reasons, and some kinds of reasons cannot be used to morally justify such discrimination.Footnote 20 Instead of trying to define what equality is in a specific context, we derive axioms by excluding various considerations.
First, and most obviously, if an electoral system treats all voters equally, then the outcome should not depend on the names of the voters, or their identities. That is to say, if we permute the names of the voters, this should not change the outcome. Thus the electoral system needs to satisfy the axiom of anonymity. Secondly, to treat all voters equally, the electoral system cannot discriminate on the basis of the name or identities of the alternatives. If it did, it would give favourable treatment to the supporters of one alternative over another. Furthermore, such an electoral system would violate the idea of popular sovereignty. If changing the names of the candidates changes the result, then the result depends on a built-in bias in the electoral system, and not solely on the preferences of the voters. Thus equality also requires the condition of neutrality.
It should be noted that anonymity is a direct consequence of a liberal conception of equality, not a separate stronger requirement as some have suggested.Footnote 21 For this reason liberal equality is incompatible with district-based elections, unless voter preferences are distributed randomly. Indeed, both Rogowski and Christiano have shown that single-member district plurality elections violate anonymity, and they have argued that for this reason proportional representation is the only system compatible with political equality.Footnote 22 Liberal equality requires that all individual voters be treated identically, while a district-based system treats voters differently depending on where they reside. If I live in a safe district, my vote makes no difference to the result. However, if I change places with someone in a district that may produce a dead-heat, my vote could change the result not only in that district, but in the national election.
It is true that the violation of individual equality in district-based systems does not depend on the individual characteristics of the voter, but on the distribution of opinion in the district in which that voter lives.Footnote 23 If voter preferences were distributed randomly across the electorate, this inequality would not be problematic. In such a case, a district-based system could satisfy a weaker form of anonymity – swapping voters around would not change the expected result, and everyone would have an equal chance of being the decisive voter. However, in reality voter preferences are usually not distributed randomly. We know that in many countries some districts are far more likely to be marginal than others, and these districts tend to be the same ones in repeated elections. Furthermore, candidates know this and adapt their campaigning accordingly. This leads to systematic biases in district-based systems that favour some voters over others, violating the principle of liberal equality.
Liberal equality, however, does not imply that geography cannot affect the result. It only requires that the electoral system not discriminate between voters. It is a requirement that all people be treated equally, and this is incompatible with guaranteeing that all places be treated equally. However, if individual voters choose to vote on the basis of geographical identification, then an electoral system satisfying liberal equality will produce a considerable degree of geographical representation. This, indeed, is what Latner and McGann find in the case of Israel and the Netherlands.Footnote 24 Although these countries have national list PR electoral systems, legislators are drawn from all over the countries (and especially from peripheral regions), apparently because parties know that voters care about this.
It should also be noted that the requirement of neutrality does not correspond to the demand that parties should be treated ‘equally’ or ‘fairly’ as this is normally understood. Neutrality is merely the condition that alternatives are not discriminated against on the basis of their identity. Neutrality does not stop a seat allocation rule from favouring large parties over small or geographically concentrated parties over dispersed ones. Neutrality is a minimal condition that virtually all currently existing electoral systems (including single-member district plurality) respect. Thus our conclusions are not the result of us assuming that candidates or parties must be treated fairly, except in the minimal sense that they cannot be given special treatment because of their names.
These axioms of liberal equality are applied not just to the choice between alternatives, but also to that between coalitions of alternatives. This is important because political power is not distributed on the basis of the relative seat share of candidates or parties, but on the relative size of coalitions – or more precisely in terms of which coalitions have a majority of seats and are thus able to elect a government or pass legislation. Applying anonymity to coalitions is straightforward – if the electoral system is anonymous with regards to alternatives, it is automatically anonymous with regard to coalitions of alternatives. However, neutrality to coalitions does not follow automatically from neutrality to alternatives, and actually represents a considerably stronger requirement. Coalition neutrality implies that the electoral system cannot discriminate between different kinds of coalition. Of course, a case can be made that the electoral system should reward coalitions made up of a small number of alternatives, because this promotes consolidation and stability; or that coalitions of a large number of alternatives should be encouraged because this allows for the representation of a greater number of points of view. The liberal defence of coalition neutrality is that this question should not be settled by the electoral system, but by the voters themselves. The electoral system should be neutral between coalitions, because the seat share allocated to each coalition should depend only on the voters, not on a bias built into the system.
There is a further requirement necessary for liberal equality to apply in a substantive, as opposed to purely formal sense. This is that the citizens are able to create their own electoral alternatives and place them on the ballot. If parties have a monopoly on proposing alternatives, voters may be formally equal, in the fact that each vote has equal influence, but they could receive very different treatment in that one voter can vote for their ideal alternative while another hates all the alternatives on offer. However, if citizens are able to propose their own electoral alternatives at very low cost, this problem disappears. Dissatisfied citizens can simply create their own candidates and lists, and the electoral system does not discriminate against them. To restate the same point, liberal equality also requires that the voters have ultimate control over the alternatives with which they are presented.
Our results require a third axiom, positive (or at least nonnegative) responsiveness. This is not derived from the concept of liberal equality, but is necessary to exclude perverse voting systems that penalize alternatives for winning more votes. With single-vote systems, this is simply a technical requirement, in that there would appear to be no conceivable benefit to be gained from violating it. However, with ordinal systems (that is where voters rank order candidates) there is controversy about whether non-negative responsiveness is a strict requirement of a reasonable electoral system, or whether its violation is a necessary cost of obtaining other desirable characteristics. This will be considered in more detail below in the section on multiple-vote electoral systems. We can define positive responsiveness as follows: if alternative A gains votes – either from other parties or from people who previously abstained – while everyone else votes in the same way, then the seat share of alternative A must increase. This is a strong assumption, as it assumes that the seat allocation function is perfectly responsive – if an alternative wins even one more vote, it must get a greater seat share as a result. However, we can prove very similar results assuming only non-negative responsiveness – the assumption that if an alternative wins more votes and everything else remains equal, it does not lose seat share.
Proportional representation
Pure proportional representation is defined as a seat allocation rule that assigns seat share to alternatives in proportion to their vote totals (as before, alternatives can be candidates, lists or parties). Of course, as seats are not infinitely divisible, this is an abstraction that serves as a counterfactual ideal.Footnote 25 There are, however, some empirical systems that approximate it quite well. Electoral systems usually classified as ‘PR’ vary considerably in their degree of proportionality. The division of the electorate into electoral districts can reduce proportionality, especially if the districts are small or unevenly sized, although in many systems this is mitigated by national compensatory seats that are distributed to restore proportionality. Electoral thresholds also reduce proportionality, in that parties that win fewer votes than the threshold get no representation. Thus the systems that come closest to pure proportionality are national list systems with very low thresholds, such as the systems in the Netherlands and Israel.Footnote 26
In terms of seat allocation rules, Balinski and Young show that the only apportionment schemes that are monotonic (that is, satisfy non-negative responsiveness) are divisor schemes, and that the only divisor scheme that is unbiased between large and small parties is the Webster or Sainte-Laguë.Footnote 27 Therefore the Webster/Sainte-Laguë rule applied to a national electoral district is the closest approximation to pure PR, although the difference in outcome with other PR methods is likely to be small if the district magnitude is large.Footnote 28
However, we still need the concept of pure PR in our proof. Even though the Webster/Sainte-Laguë method is generally unbiased between large and small parties, when there are very small parties severe disproportionalities can occur. Suppose that we have a large number of very small, identically sized parties. Any anonymous and neutral seat allocation rule has to either assign each of them zero seats (in which case they are collectively underrepresented) or assign all of them one seat each (in which case they are collectively overrepresented). This phenomenon can even lead to examples where there are manufactured majorities (a coalition with a minority of votes has a majority of seats).Footnote 29 Such situations have seldom, if ever, occurred in practice with large district PR systems, but the possibility draws attention to the fact that pure PR is an ideal that can only be imperfectly realized when a finite number of seats need to be allocated. The closest we can get would probably be a probabilistic system where remainder seats were distributed randomly so that the expected seat share of each alternative equals its vote share.
It is often assumed that PR must be based on political parties and party lists. However, PR systems can be based on lists of citizen candidates (who may or may not be affiliated with a party) or even on individual candidates. This is the case with the system proposed by Dodgson.Footnote 30 Each voter casts one vote in a large electoral district and every candidate who receives a quota of votes is elected.Footnote 31 Candidates are then able to distribute any surplus votes they have received in any way they wish. The intuition is that the votes a candidate receives are their property to dispose of at will. Dodgson defends this in terms of the concept of representation – if I am willing to choose someone to make legislative decisions on my behalf, I surely trust this person enough to choose their own deputies to carry out this legislative task. List PR can be seen as a set of restrictions on the Dodgson PR system. If we simply require candidates to state in advance how they would distribute their surplus votes, then we have a rudimentary system of list PR. Of course, all existing list PR systems place considerably more constraints on candidates. Generally, candidates are only allowed to have their name on one list, and all candidates on the list are required to distribute their excess votes in the way defined by the list. In practice, most systems of list PR force candidates to compete as mutually exclusive teams.
This, however, does not mean that candidates have to compete as political parties, or that parties are privileged by the system. This is important, because we have argued that for liberal equality to exist in a substantively meaningful sense it is necessary for citizens to be able to create electoral alternatives with low entry costs. It is true that some list PR systems, such as that in Germany, require that lists be affiliated with registered political parties, and thus can be referred to as party-list PR systems. However, other systems, such as that in the Netherlands, afford parties very little in the way of special treatment. Rather elections are organized in terms of lists of candidates, which any group of citizens can propose. Furthermore, the fact that 0.66 per cent of the vote is usually sufficient to win a seat means that new lists have a realistic chance of gaining representation. It is true that most lists are affiliated with political parties. However, while the list PR system in countries such as the Netherlands accommodates political parties, it does not mandate them. Indeed, hastily assembled citizens’ lists have been quite successful in some recent elections in the Netherlands.
It should be noted that there is a considerable literature on proportionality in contexts other than electoral systems. Examples include the distribution of shared costs within firms and the distribution of assets between claimants in cases where the sum of the claims exceeds the assets.Footnote 32 In these cases proportionality has an important quality of being strategy-proof. For example, in the case of a bankrupt firm, any non-proportional scheme for distributing the assets creates an incentive for claimants to reallocate their claims amongst themselves. These problems are similar to the seat distribution problem in some ways. However, we are not concerned with preventing strategic behaviour from parties, but simply with ensuring that the electoral system respects the equality and sovereignty of the voters.
Results
Taking the definitions of liberal equality and proportional representation from the last section, we show that liberal equality (operationalized as the axioms of anonymity and neutrality) implies a single-vote seat allocation rule essentially equivalent to pure list proportional representation. First, we show that anonymity, neutrality and positive responsiveness imply the strong plurality-ranking property – if alternative A wins more votes than alternative B, then it must receive a greater seat share. If we only require non-negative responsiveness, then anonymity and neutrality imply the weak plurality-ranking property (alternatives that get more votes must win at least the same number of seats).
PROPOSITION 1: Any seat allocation rule that is anonymous, neutral and positively (non-negatively) responsive satisfies the strong (weak) plurality-ranking property. (The proof is to be found in the Appendix.)
The intuition here is straightforward. If two alternatives have the same number of votes, then by anonymity and neutrality, they must have the same seat share. If in this case one alternative receives more seats, then either the vote allocation system is inherently biased in its favour (violating neutrality) or some voters’ votes count for more than others (violating anonymity). If one alternative then increases its vote at the expense of the other or by gaining the votes of people who previously abstained, by positive responsiveness it must receive a greater seat share than the other alternative. Therefore, if one alternative receives more votes than another it must receive more seats (the strong plurality-ranking property). If we only assume non-negative responsiveness, then when an alternative gains votes, it must at least not lose seats. This implies that an alternative that wins more votes than another must get at least an equal seat share (the weak plurality-ranking property).
We can extend Proposition 1 to apply not only to seat allocation for individual alternatives, but to seat allocation for coalitions of alternatives. We can partition the alternatives into non-overlapping coalitions, in any way we wish. We can then apply the various axioms to these coalitions. Strictly speaking, we define a seat allocation rule as anonymous, neutral and positively responsive to coalitions if, once we have partitioned the alternatives into coalitions, the allocation of seats to coalitions always satisfies anonymity, neutrality and positive responsiveness. We can then proceed by treating the coalitions resulting from our partition just as we treated individual alternatives.
PROPOSITION 2: Any seat allocation rule that is anonymous, neutral and positively (non-negatively) responsive for coalitions satisfies the strong (weak) plurality-ranking property for coalitions. (The proof is to be found in the Appendix.)
The proof of this proposition is essentially identical to that of Proposition 1. If one coalition of alternatives receives more votes than another, it must receive a greater total seat share if we assume positive responsiveness, and at least an equal total seat share if we only assume non-negative responsiveness.
In representative bodies, governments are typically formed by a process of majority-rule coalition formation. Which coalition forms is a result of a bargaining process. However, the bargaining situation is defined in terms of which coalitions have sufficient seats to win a majority-rule vote of investiture (or confidence) and form a government. For example, suppose that there are four alternatives. Suppose Alternative 1 has sufficient seats to form a majority coalition if it combines with either Alternative 2, 3 or 4, while Alternatives 2, 3 and 4 can form a winning coalition if they all unite. Then the set of winning coalitions is [{1, 2}, {1, 3}, {1, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {1, 2, 3, 4}, {2, 3, 4}]. This information is all we need to know about the bargaining situation – exactly how many seats each alternative has makes no difference to what winning coalitions are possible.Footnote 33 We can show that any seat allocation function that satisfies the coalitional strong plurality-ranking property defines a coalition formation game identical to that defined by pure PR. Therefore anonymity, neutrality and positive responsiveness imply a single-vote seat allocation that produces a parliamentary outcome to all intents and purposes identical to that produced by pure PR.
PROPOSITION 3: Any seat allocation function that is anonymous, neutral and positively (non-negatively) responsive for coalitions defines a majority rule coalition game with a set of winning coalition that is identical to (a subset of) that defined by seat allocation by proportionality. (The proof is to be found in the Appendix.)
The intuition behind the proof comes from the fact that under majority rule a coalition is winning if it has a greater seat share than all the alternatives excluded from it. By Proposition 2, anonymity, neutrality and positive responsiveness require that if a coalition has more votes than another coalition, it must receive a greater seat share. Therefore, the set of winning coalitions must be the set of coalitions whose members have more votes than all the alternatives excluded by them. This is exactly the same as the set of winning coalitions under pure PR. If we only assume non-negative responsiveness, it is possible for a coalition to win a majority of the vote, but only to receive exactly half the seats and thus be a blocking, but not winning, coalition. However, it is impossible to have a ‘manufactured majority’ (a situation where an alternative or coalition with a minority of the vote gets a majority of the seats) without violating anonymity or neutrality. The intuition behind this is similar to the case assuming positive responsiveness. By Proposition 2, anonymity, neutrality and non-negative responsiveness require that if a coalition has more votes than another coalition, it must receive at least an equal seat share. It is possible for a coalition to have an equal number of seats to its complement, so it is possible for a coalition to win a majority of the vote, but only to receive exactly half the seats. Any winning coalition under our seat allocation rule is a winning coalition under proportionality, but not vice versa.
Proportional representation and multiple-vote rules
We have shown that pure list proportional representation is the only single-vote electoral system that satisfies anonymity, neutrality and positive responsiveness for coalitions. However, it could be argued that an anonymous, neutral multiple-vote system would be better still, in that it would satisfy the requirement of liberal equality and make more use of the voters’ preferences. Characterizing multiple-vote systems is beyond the scope of this article. However, two things should be noted. First, any anonymous and neutral multiple-vote system must be a ‘proportional’ system, in the sense that it produces proportional outcomes when voters’ preferences can be summarized in a vote for a single list. Secondly, it is not clear what an anonymous, neutral and non-negatively responsive multiple vote electoral system would look like. The most commonly proposed alternatives all violate at least one of our axioms.
When considering multiple-vote electoral systems, it is important to distinguish between ordinal systems (where voters get to rank order the alternatives, as in single transferable vote) and mixed systems. It is trivial to extend our analysis to so-called mixed electoral systems, because these are really a combination of two separate single-vote elections.Footnote 34 Each voter has two votes, one for a candidate in a plurality election in the voter’s district, and the other for a party in a national or regional PR election. If there is no compensation between the results of two elections, the allocation of party seats will respect proportionality, but the district seats will not, so the overall results will violate proportionality and thus the axioms of liberal equality. If, however, there is compensation between the two processes, as in Germany, then the results will be approximately proportional and potentially compatible with liberal equality.Footnote 35 In practice, however, such systems may be problematic from the point of view of liberal equality if they force candidates to be associated with a registered political party or make entry by new parties more difficult.
Ordinal voting rules (where voters get to rank order the alternatives, as in the single transferable vote system) are more complex, in that they rely on more information on the voters’ preferences than single-vote systems. However, any ordinal rule that respects liberal equality must be compatible with proportionality when voters have list preferences (they all rank the candidates on one electoral list in the order of the list, and are indifferent between all the candidates not on that list). If this is the case, the preferences of every voter can be summarized by a voting correspondence (‘Voter 1 supports List y’). If this is so, the results of the previous section can be applied to show that liberal equality implies proportionality in this case. Although intended here as a thought experiment, list voting may not be that far from reality in many countries. For example, in the Netherlands over 90 per cent of voters routinely vote for the national leader of their preferred party, in spite of having the option of voting for anyone on the party list.Footnote 36 However, the argument here does not rest on the realism of list preferences. Rather the point is to provide a test of whether a particular ordinal system satisfies the value of liberal equality. Clearly, ordinal systems give voters many more choices than simply voting for a straight list. However, if all voters were to have list preferences, and an ordinal system fails to deliver proportionality in this simple hypothetical case, then this ordinal system definitely violates liberal equality.
We cannot characterize in this article all multiple-vote electoral systems that respect liberal equality. We can, however, show that certain systems violate it. District-based systems typically violate anonymity, so any system that respects liberal equality will need to be based on large districts or ideally a single national district. Seat allocation by Borda count, or indeed any scoring system, violates liberal equality in that it does not produce a proportional outcome even when all voters have list preferences.Footnote 37 This is due to the fact that any scoring rule that considers more than the first preference of voters is vulnerable to fragmentation, as shown by Hout et al.Footnote 38 Consider the following example. Seats are allocated in proportion to the relative Borda scores of the alternatives. There are two lists, Left and Right, each of which has the support of half the voters. Using our Borda-based seat allocation rule, each list receives half the seats. However, suppose that Left fragments into two lists, First Left and Second Left, with everyone ranking Second Left just below First Left (thus maintaining list preferences). Now, with our Borda seat allocation rule, First Left and Second Left between them receive two-thirds of the seats. Right receives only one-third of the seats, even though it is still supported by half the population. The problem is that with a scoring system the outcome depends not only on voter preferences, but also on the number of alternatives of each type there are.Footnote 39
One ordinal electoral system that does respect proportionality when voters have list preferences, as well as being anonymous and neutral (as long as it is applied to a single national district) is single transferable vote (STV).Footnote 40 The axiom that STV violates is non-negative responsiveness, in that ranking an alternative higher can sometimes disadvantage that alternative, as recognized by Dodgson and demonstrated by Doron and Kronick.Footnote 41 However, as we argued earlier, the axiom of non-negative responsiveness is not derived from the value of liberal equality, but is an additional ‘rationality’ requirement. Some theorists, from Dodgson to Dummett, have argued that violating non-negative responsiveness is a serious, even fatal, flaw in STV.Footnote 42 However, it could be argued that the violation of non-negative responsiveness is a price worth paying for the greater choice offered by STV over list PR, especially if instances of these violations are few and far between.Footnote 43 In any case, we cannot reject STV on grounds of liberal equality. However, to be compatible with liberal equality, STV would have to be applied to very large districts (ideally a single national district) as opposed to small districts (as in Ireland and Malta), as district-based systems violate anonymity. In practice, national STV is likely to produce outcomes similar to list PR.Footnote 44 Indeed, Monroe shows that both STV and list PR are both restrictions of a more general class of systems he refers to as ‘fully proportional representation’.Footnote 45
We are not able to consider here the relative merits of list PR and STV (or indeed other ordinal systems that may be compatible with liberal equality). However, the argument that a good election rule should consider the entire preference schedule of the voters is not obvious, although it has been stated as such by many authors.Footnote 46 Following Dodgson, we can make a strong normative case for considering only the first preference of voters in allocating seats in legislatures.Footnote 47 Under a pure proportional system (an abstraction, of course, given that seats are not divisible in reality), everyone gets a representative of the list they chose, however small. There is no need to consider second place preferences because everyone gets their first preference. Furthermore, it can be argued that the fact that my preferred representative is your very least preferred is irrelevant – their job is to represent me, not you. We can conclude that the choice between list PR and ordinal systems that respect proportionality when necessary (such as national STV) cannot be made on formal grounds of liberal equality, but rather depends on a more substantive theory of the meaning of representation.
Conclusion
We have presented a justification of PR that is strictly within the limits of liberal theory, that is, relying only on the equal treatment of individual voters. We use axiomatic social choice theory to determine what political institutions are required to meet the normative requirements of liberal equality. We show that any single-vote electoral system that satisfies liberal equality (axiomatized as the properties of anonymity and neutrality) must produce results essentially identical to PR. The result is in principle very similar to May’s theorem that anonymity, neutrality and positive responsiveness require majority rule as a decision-making rule, which is somewhat ironic given that majority rule and proportional representation have often been portrayed as competing principles. It turns out that the same principle of liberal equality underlies both.
This result does not rely on the existence of political parties, and can be applied to systems of PR based on lists of candidates or even individual candidates. However, it does apply only to single-vote electoral systems. Nevertheless, we can show that ordinal electoral systems must be compatible with proportionality if voter preferences happen to correspond to the electoral lists, so any ordinal system that satisfies liberal equality must be ‘proportional’ in this sense. Of course, we have only argued that PR is the only electoral system that can satisfy liberal equality, not that PR is a ‘good’ system for all circumstances. Such a judgment would depend on how well PR produces other values such as minority protection, stability and accountability. This would require empirical evidence.
The normative literature on electoral systems for the most part rests on illiberal premises. Advocates of PR and of non-proportional systems both rely on arguments in terms of fairness to parties or social groups, although these are often phrased in terms of fairness to the individual members of the group in question. This multiplicity of conception of fairness can easily lead to relativist conclusions, where the grounds for preferring one conception of fairness over another are taken as purely subjective preferences. This is not surprising – anyone familiar with the literature on fair division in economics and philosophy (or indeed, with small children) is aware that with sufficient ingenuity a very wide range of claims can be phrased in terms of fairness. When we consider group-based claims about the fairness of electoral systems, we have two variables to manipulate: the type of groups that are salient, and the principle to arbitrate between the claims of these groups. Given these two variables, we can generate arguments ranging from the proportional representation of partisans, to the winner-take-all conception of fair competition, to the constituency principle, and no doubt many more besides.
If we consider electoral systems in terms of liberal equality, however, we get determinate results. If we consider individual voters, the concepts of equality, fairness and impartiality become equivalent, and we can derive axioms that imply precise institutional requirements. This begs the question of why we should consider liberal equality as opposed to some group-based conception of fairness. Two justifications can be given. First, there is the doctrinally liberal premise that justice between individuals has normative priority over justice between the groups that these individuals make up. Secondly, there is the argument that liberal equality can provide a political means of adjudicating between competing concepts of justice. If we do not accept the doctrinally liberal position that individual equality is privileged, we can still argue that the only fair way to decide between alternative, contested conceptions of justice is a form of political liberalism in which every individual’s position is treated equally. However, even if we do not accept the case for liberal equality at all, our results still have analytic value, forcing us to confront and justify the group-based conceptions of fairness we rely upon when arguing about electoral systems.
APPENDIX: PROOFS
Let us define the set of eligible voters as N, with voters numbered 1 … n, and the set of alternatives A, numbered 1 … a.
The voting correspondence V is defined over the Cartesian product N × A, with 
reading, ‘i votes for alternative j’. Assigning the value 1 for true and 0 for false: 
. (Each individual either votes for one alternative or does not vote.)
The function T maps the voting correspondence into the total vote for each alternative: 
.
The function E maps the voting correspondence into the seat share for each alternative: 
. We will assume that seats are infinitely divisible, to abstract from rounding problems.
We can define the following properties of the seat share function E.
Let V′ be a vote pattern over N × A.
Anonymity: Let σ be a function that permutes N. Then E is anonymous if E(V′)=E(σV′).
Neutrality: Let π be a function that permutes A. Then E is neutral if πE(V′) = E(πV′).
Cancellation property: If Tj = Tk, then Ej = Ek, where Ej is the seat share of party j.
(If the vote share for two alternatives are the same, then their seat shares must be the same.)
Non-negative (positive) responsiveness:
Let V′′ = V′, except that some voters or abstainers have switched to alternative j (
)
(If i votes for j under V′, then I votes for j under V″; there is someone who votes for j under V″ who did not vote for j under V″; anyone who did not vote for j under V″, voted for k under V″ iff they voted for k under V″.)
E is non-negatively responsive iff
E is positive responsive iff
Weak plurality ranking property: 
(If alternative j wins more votes than alternative k, alternative j receives a greater or equal seat share to alternative k.)
Strong plurality ranking property: 
(If alternative j wins more votes than alternative k, alternative j receives a greater seat share than alternative k.)
LEMMA 1: Any anonymous and neutral seat allocation function satisfies the cancellation property.
Proof:
The proof is isomorphic to the proof of Lemma 2 in Van der Hout et al.Footnote 48
We need to show that
If Tj = Tk, we can permute all voters so that the voters of alternatives j and k change places. We will call the permutation function σ′.
By anonymity, the seat shares stay the same: 
.
We can permute the parties so that alternatives j and k change places. We call this permutation function π′.
By neutrality, the seat share again remains the same: 
.
However, by construction 
. (We have swapped the supporters of alternatives j and k, and then swapped the names of the parties, so we are back to the original situation.)
Therefore, 
QED.
PROPOSITION 1: Any seat allocation rule that is anonymous, neutral and positively (non-negatively) responsive satisfies the strong (weak) plurality-ranking property.
Proof:
The proof is isomorphic to the proof of Theorem 2 in Van der Hout et al.Footnote 49 For the positively responsive case, we need to show that neutrality, anonymity and positive responsiveness imply 
.
For the negatively responsive case, we need to show that neutrality, anonymity and positive responsiveness imply 
.
By Lemma 1, anonymity and neutrality imply the cancellation property: 
.
Let V″ be a vote pattern where Tj > Tk.
Then we can derive a vote pattern V′ where Tj = Tk by having an appropriate number of voters for alternative j abstain, so that the vote for alternative j and k are equal.
By the cancellation property: 
.
By positive responsiveness, 
.
By non-negative responsiveness, 
.
QED.
Extension to Coalitions
We may arbitrarily partition the set of alternatives A into m non-overlapping coalitions, c 1…cm, where 1 ≤ m ≤ a.
Let us define the set of coalitions resulting from such a partition as C, where
Let us define the coalition vote correspondence K on N × C, in terms of the vote correspondence for alternatives, V. Let 
read, ‘i votes for an alternative in coalition c’. Thus 
.
Let us define the coalition vote total function for partition C as the sum of the votes received by each alternative in the coalition. Thus
Let us define the coalition seat share function for partition C as the sum of the seats allocated to each alternative in the coalition. Thus
We can redefine the properties used in Proposition 1 for use with coalitions:
A seat allocation function E is anonymous (neutral, positively responsive, non-negatively responsive, cancelling, weakly plurality ranking, strongly plurality ranking) for coalitions iff, for all partitions C, the coalition seat allocation function F derived from it is anonymous (neutral, positively responsive, non-negatively responsive, cancelling, weakly plurality-ranking, strongly plurality-ranking).
Let K′ be a vote pattern over N × C.
Anonymity: A coalition seat allocation function F is anonymous iff 
, where σ is a function that permutes N.
Neutrality: A coalition seat allocation function F is neutral iff τ F(K′)=F(τK′), where τ is a function that permutes C.
Cancellation property: A coalition seat allocation function is cancelling iff 
.
(If the vote totals for two coalitions are the same, then their seat shares must be the same.)
Non-negative (positive) responsiveness in coalition seat allocation function F can be defined as follows:
Let K″=K′, except that some voters have switched to parties in coalition j:
F is non-negatively responsive iff
F is positively responsive iff
Weak (strong) plurality-ranking property: F satisfies the weak plurality-ranking property iff 
.
(If the alternatives in coalition j win more votes than the alternatives in coalition k, coalition j receives a greater or equal aggregate seat share to coalition k.)
F satisfies the strong plurality ranking property iff 
.
(If the alternatives in coalition j win more votes than the alternatives in coalition k, coalition j receives a greater aggregate seat share than coalition k.)
LEMMA 2: Any anonymous and neutral coalition seat allocation function satisfies the cancellation property.
Proof: Isomorphic to Lemma 1.
LEMMA 3: Any coalition seat allocation rule that is anonymous, neutral and positively (non-negatively) responsive satisfies the strong (weak) plurality-ranking property.
Proof: Isomorphic to Proposition 1.
PROPOSITION 2: Any seat allocation rule that is anonymous, neutral and positively (non-negatively) responsive for coalitions satisfies the strong (weak) plurality-ranking property for coalitions.
Proof: If seat allocation function E is anonymous, neutral and positively (non-negatively) responsive for coalitions, then, for any partition C, by definition the coalition seat-share function F derived from it must be anonymous, neutral and positively (non-negatively) responsive.
By Lemma 3, if F is anonymous, neutral and positively (non-negatively) responsive, it must satisfy the strong (weak) plurality-ranking property.
If, for any partition C, F satisfies the strong (weak) plurality-ranking property, then by definition E satisfies the strong (weak) plurality-ranking property for coalitions. QED.
Majority rule coalition formation
Let us define pure list proportion representation as a seat allocation function that allocates a seat share (a real number in [0, 1]) to each alternative, which is equal to the share of the total vote that alternative won.
Coalition games can be defined in terms of the set of winning coalitions 
.
Under majority rule, a coalition is winning iff it has a majority of seat share: 
.
Under pure list proportional representation, a coalition wins a majority of seat share iff it has more than 50 per cent of the vote: 
.
PROPOSITION 3: Any seat allocation function that is anonymous, neutral and positively (non-negatively) responsive for coalitions defines a majority rule coalition game with a set of winning coalition that is identical to (a subset of) that defined by seat allocation by proportionality.
Proof:
Non-negative responsiveness:
Given that the set of majority-rule winning coalitions is defined as 
, and under pure list proportional representation the set of coalitions with a majority of seat share is 
, we need to show that if seat allocation function E is anonymous, neutral and non-negatively responsive for coalitions, then 
.
If the seat-allocation function E is anonymous, neutral and non-negatively responsive for coalitions, then by Proposition 2 it satisfies the weak plurality ranking property for coalitions, and thus the coalition seat allocation F derived from it must satisfy the weak plurality ranking property: 
.
Suppose 
, but 
. If 
then by the weak plurality ranking property, 
. Contradiction. If E is anonymous and neutral for coalitions, then F must be anonymous and neutral, and so by Lemma 2 must satisfy the cancellation property. If 
, then by the cancellation property 
. Contradiction. QED.
Positive responsiveness:
We need to show that if seat-allocation function E is anonymous, neutral and positively responsive for coalitions, then 
.
Given that positive responsiveness implies non-negative responsiveness, we have already shown that 
.
All that remains is to show that 
.
This is simply the strong plurality ranking property for F. If the seat allocation function E is anonymous, neutral and positively responsive for coalitions, then by Proposition 2 it satisfies the strong plurality ranking property for coalitions, and thus the coalition seat allocation F derived from it must satisfy the strong plurality-ranking property. QED.
