In Arrow’s foundational book “Social Choice and Individual Values” , voting rules rank candidates according to their social merit and, if desired, this ranking can be used to select the best candidate(s). As these rules are concerned with “mutually exclusive” candidates, they can be seen as single-winner rules. In contrast, the goal of multi-winner rules is to select the best group of candidates of a given size; we call such a fixed-size set of candidates a committee. Multi-winner elections are of importance in a wide range of scenarios, which often fit in, but are not limited to, one of the following three categories [25, 29]. The first category contains multi-winner elections aiming for proportional representation. The archetypal example of a multi-winner election is that of selecting a representative body such as a parliament, where a fixed number of seats are to be filled; and these seats are ideally filled so as to proportionally represent the population of voters. Hence, voting rules used in parliamentary elections typically follow the intuitive principle of proportionality, i.e., the chosen subset of candidates should proportionally reflect the voters’ preferences. The second category comprises multi-winner elections with the goal that as many voters as possible should have an acceptable representative in the committee. Consequently, there is no or little weight put on giving voters a second representative in the committee. Ensuring a representative for as many voters as possible can be viewed as an egalitarian objective. This goal may be desirable, e.g., in a deliberative democracy , where it is more important to represent the diversity of opinions in an elected committee rather than to include multiple members representing the same popular opinion. Another example would be the distribution of facilities such as hospitals in a country, where voters would prefer to have a hospital close to their home but are less interested in having more than one in their vicinity. Voting rules suitable in such scenarios follow the principle of diversity. The third category contains scenarios where the goal is to choose a fixed number of best candidates and where ballots are viewed as expert judgments. Here, the chosen multi-winner rule should follow the individual excellence principle, i.e., to select candidates with the highest total support of the experts. An example is shortlisting nominees for an award or finalists in a contest where a nomination itself is often viewed as an achievement. We review further applications of multi-winner voting in Section 1.2.
In this paper, we consider multi-winner rules based on approval ballots, which allow voters to express dichotomous preferences. Dichotomous preferences distinguish between approved and disapproved candidates—a dichotomy. An approval ballot thus corresponds to a subset of (approved) candidates. A simple example of an approval-based election can highlight the distinct nature of proportionality, diversity, and individual excellence: There are 100 voters and 5 candidates : 66 voters approve the set , 33 voters approve , and one voter approves . Assume we want to select a committee of size three. If we follow the principle of proportionality, we could choose, e.g., ; this committee reflects as closely as possible the proportions of voter support. If we aim for diversity and do not consider it important to give voters more than one representative, we may choose the committee ; this committee contains one approved candidate of every voter. The principle of individual excellence aims to select the strongest candidates: , , and have most supporters and are thus a natural choice, although the opinions of 34 voters are essentially ignored. We see that these three intuitive principles give rise to very different committees. In this paper, we will explore these principles in a formal, mathematical framework.
For single-winner rules, one distinguishes between social welfare functions, i.e., voting rules that output a ranking of candidates, and social choice functions, i.e., rules that output a single winner or a set of tied winners. For multi-winner rules, an analogous classification applies: we distinguish between approval-based committee (ABC) ranking rules, which produce a ranking of all committees, and ABC choice rules, which output a set of winning committees. While axiomatic questions are well explored for both social choice and social welfare functions, far fewer results are known for multi-winner rules (we provide an overview of the related literature in Section 1.3). However, such an axiomatic exploration of multi-winner rules is essential if one wants to choose a multi-winner rule in a principled way. Axiomatic characterizations of multi-winner rules are of crucial importance because multi-winner rules may have very different objectives: as we have seen in the example, proportionality, diversity, and individual excellence may be conflicting goals. Also, as we will see in Section 2, many multi-winner rules have rather involved definitions and their properties often do not reveal themselves at first glance. An axiomatic characterization of such rules helps to categorize multi-winner rules, to highlight their defining properties, and to assess their applicability in different scenarios.
The main goal of this paper is to explore the class of consistent ABC ranking rules. An ABC ranking rule is consistent if the following holds: if two disjoint societies decide on the same set of candidates and if both societies prefer committee to a committee , then the union of these two societies should also prefer to . This is a straightforward adaption of consistency as defined for single-winner rules by Smith  and Young . Consistency plays a crucial role in many axiomatic characterizations of single-winner rules (we give a more detailed overview in Section 1.3). Our results highlight the diverse landscape of consistent multi-winner rules and their defining and widely varying properties.
1.1 Main results
The first main result of this paper is an axiomatic characterization of ABC counting rules, which are a subclass of ABC ranking rules. ABC counting rules are informally defined as follows: given a real-valued function (the so-called counting function), a committee receives a score of from every voter for which committee contains approved candidates and that approves candidates in total; the ABC counting rule implemented by ranks committees according to the sum of scores obtained from each voter. We obtain the following characterization of ABC counting rules.
An ABC ranking rule is an ABC counting rule if and only if it satisfies symmetry, consistency, weak efficiency, and continuity.
The axioms used in this theorem can be intuitively described as follows: We say that a rule is symmetric if the names of voters and candidates do not affect the result of an election. Weak efficiency informally states that candidates that no one approves cannot be “better” committee members than candidates that are approved by some voter. Continuity is a more technical condition that states that a sufficiently large majority can dictate a decision. As weak efficiency is satisfied by every sensible multi-winner rule and continuity typically only rules out the use of certain tie-breaking mechanisms [65, 71], Theorem 1 essentially implies that ABC counting rules correspond to symmetric and consistent ABC ranking rules. Furthermore, we show that the set of axioms used to characterize ABC counting rules is minimal.
Theorem 1 gives a powerful technical tool that allows to obtain further axiomatic characterizations of more specific rules. Indeed, building upon this result, we further explore the space of ABC counting rules, and obtain our second main result—the axiomatic explanation of the differences between three important ABC counting rules: Multi-Winner Approval Voting (AV), Proportional Approval Voting (PAV), and Approval Chamberlin–Courant (CC), which are defined by the following counting functions:
Note that these three specific example of counting functions do not depend on the parameter ; we discuss this fact in Section 3.1. These three well-known rules are prime examples of multi-winner systems following the principle of individual excellence, proportionality, and diversity, respectively. Our results imply that the differences between these three voting rules can be understood by studying how these rules behave when viewed as apportionment methods. Apportionment methods are a well-studied special case of approval-based multi-winner voting, where the set of candidates can be represented as a disjoint union of groups (intuitively, each group can be viewed as a political party), and where each voter approves all candidates within one of these groups (which can be viewed as voting for a single party)—we refer to preference profiles that can be represented in such way as party-list profiles. For these mathematically much simpler profiles it is easier to formalize the principles of individual excellence, proportionality and diversity:
Disjoint equality states that if each candidate is approved by at most one voter, then any committee consisting of approved candidates is a winning committee. One can argue that the principle of individual excellence implies disjoint equality: if every candidate is approved only once, then every approved candidate has the same support, their “quality” cannot be distinguished, and hence all approved candidates are equally well suited for selection.
D’Hondt proportionality defines in which way parliamentary seats are assigned to parties in a proportional manner. The D’Hondt method (also known as Jefferson method) is one of the most commonly used methods of apportionment in parliamentary elections.
Disjoint diversity states that as many parties as possible should receive one seat and, if necessary, priority is given to stronger parties. In contrast to D’Hondt proportionality, there are no guarantees for strong parties to receive more than one seat.
We show that Multi-Winner Approval Voting is the only ABC counting rule that satisfies disjoint equality, Proportional Approval Voting is the only ABC counting rule satisfying D’Hondt proportionality and that Approval Chamberlin–Courant is the only ABC counting rule satisfying disjoint diversity. Together with Theorem 1, these results lead to unconditional axiomatic characterizations of AV, PAV, and CC. In particular, our results show that Proportional Approval Voting is essentially the only consistent extension of the D’Hondt method to the more general setting where voters decide on individual candidates rather than on parties. Our proof strategy for this result is general and can be applied to other forms of proportionality, e.g., square-root proportionality as proposed by Penrose .
Our results illustrate the variety of ABC ranking rules. Even within the class of consistent ABC ranking rules, we encounter two extremes: Multi-Winner Approval Voting chooses maximally approved candidates and disregards any considerations of diversity, whereas the Approval Chamberlin–Courant rule maximizes the number of voters with at least one approved representative but possibly denies even large majorities a second approved candidate in the committee. In between lies Proportional Approval Voting, which satisfies strong proportional requirements and thus achieves a balance between respecting majorities and (sufficiently sizeable) minorities. This variety is due to their defining counting function ; see Figure 1 for a visualization. Our results indicate that counting functions that have a larger slope than put more emphasis on majorities and thus become less egalitarian, whereas counting functions that have a smaller slope than treat minorities preferentially and thus approach the Approval Chamberlin–Courant rule. In particular, we show that counting functions that are not “close” to (all those not contained in the gray area around ) implement ABC ranking rules that violate a rather weak form of proportionality called lower quota.
All characterizations mentioned so far hold for ABC ranking rules. We demonstrate the generality of our results by proving that the characterizations of Proportional Approval Voting and Approval Chamberlin–Courant also hold for ABC choice rules. The method used for this proof is however not applicable to Multi-Winner Approval Voting. Thus, characterizing ABC counting rules within the class of ABC choice rules remains as important future work.
Finally, we note that Theorem 1 can be used to obtain many further axiomatic characterizations of interesting multiwinner rules. For instance, we have shown that by using axioms relating to strategy-proofness we can characterize the class of Thiele methods, the class of dissatisfaction counting rules, and Multi-Winner Approval Voting (using an alternative set of axioms) .
1.2 Relevance of Multi-Winner Rules
In the following we will discuss the relevance of multi-winner voting, in particular the relevance of approval-based multi-winner rules.
Electing a representative body such as a parliament is perhaps the most classic example of a multi-winner election. Most contemporary democracies use closed party-list systems to elect their parliaments, i.e., citizens vote for political parties rather than for individual candidates and an apportionment method is used to distribute parliamentary seats between different parties [8, 56]. Closed party-list systems have a number of drawbacks. For instance, it has been argued that in closed party-list systems the elected candidates have a stronger obligation to their party than to their electorate, and it can be the case that candidates focus on campaigning within their parties rather than for the citizens’ votes (see, e.g., [22, 3, 2, 19] for a more elaborate discussion of these issues). To counteract these disadvantages, some countries use election systems that allow to vote for candidates rather than only for parties, such as various open-list systems (used, e.g., in national-wide elections in Belgium, Finland, Latvia, Luxembourg, Netherlands, Sweden and in elections in some districs in France, Switzerland etc.), and STV (used in Australia and Republic of Ireland, and for some elections in India, Malta, and New Zealand). For a comparative description of modern electoral systems we refer the reader to the book of Farrell , and for the discussion on personalization in voting to the book of Renwick and Pilet .
Applications beyond political elections.
In recent years, there has been an emerging interest in multi-winner elections from the computer science community. In this context, multi-winner election rules have been analyzed and applied in a variety of scenarios: personalized recommendation and advertisement [47, 48], group recommendation , diversifying search results 
, improving genetic algorithms, and the broad class of facility location problems [30, 61]. In all these settings, multi-winner voting either appears as a core problem itself or can help to improve or analyze mechanisms and algorithms. For a brief overview of this literature we refer the reader to a recent survey by Faliszewski, Skowron, Slinko, and Talmon .
The use of approval ballots has several advantages : On the one hand, approval ballots allow voters to express more complex preferences than plurality ballots, where voters can only choose a single, most-preferred candidate. On the other hand, providing dichotomous preferences requires less cognitive effort than providing an ordering of all candidates as in the Arrovian framework . Furthermore, Brams and Herschbach  suggest that approval ballots may encourage voters to participate in elections and, at the same time, reduce negative campaigning. The positive impact of approval voting on participation is also argued for by Aragones, Gilboa, and Weiss . As a consequence, voting based on approvals is being advocated as a desirable electoral reform: Brams and Fishburn  analyze the use of Approval Voting, mostly in scientific societies; Brams and Fishburn  provide an extensive discussion on the possible, generally positive impact of using Approval Voting for political elections. Furthermore, Laslier and Van der Straeten  conducted a real-life experiment during the 2002 presidential elections in France and proved the feasibility of approval balloting in political elections. They also studied multi-winner voting systems with approval ballots in the context of elections in Switzerland [46, 67].
Preferences over committees.
In principle, it is possible to use single-winner rules instead of multi-winner rules by requiring voters to express preferences over committees instead of over candidates. However, in all aforementioned scenarios it is generally not possible to elicit preferences over committees since the size of such a preference relation grows exponentially with the committee size. Hence, this relation would be far too large for all practical purposes, and—even for a relatively small number of candidates—eliciting preferences over committees from voters would be infeasible. Thus, in practice, in all the aforementioned scenarios it is common to assume separable preferences of voters and to ask them to compare individual candidates rather than whole committees. If, however, separability is not a valid assumption (e.g., in combinatorial auctions), more expressive models of preferences are required.
Fixed-size vs. variable-sized committees.
Multi-winner elections are concerned with electing a fixed-size committee, such as a parliament. However, the analysis of axiomatic properties of multi-winner election rules is also relevant for understanding the problem of selecting a variable-sized committee. Consider a scenario where the goal is to select the “best” committee with no fixed constraint on its size. Observe that in such a case the selected committee must—in particular—outperform all other committees of the same size. Hence, axioms concerned with comparing committees of different sizes may be generalizations of axioms describing how to compare committees of the same size. In general, some constraints on the size of the committee exist (too large or too small committees may not be practical), which leads to considerations similar to the fixed-size setting. Thus, a solid understanding of multi-winner elections is likely to be necessary for the analysis of variable-sized committees.
ABC ranking rules vs. ABC choice rules.
In this paper, we mostly study ABC ranking rules, i.e., multi-winner analogues of social welfare functions. Understanding such rules is important also when the goal is to simply select a winning committee rather than to establish a full ranking over all possible committees. Each ABC ranking rule naturally defines an ABC choice rule by returning all top-ranked committees. Thus, by considering ABC ranking rules we simply assume that there exists a collective preference ranking over all committees which allows us to formulate certain axioms. These axioms are relevant even when the goal is simply to select a winning committee rather than to compute a collective preference ranking. Furthermore, most multi-winner rules have an intuitive interpretation both as ABC ranking and ABC choice rules. In Appendix C we provide a more formal discussion, explain the relation between the ABC ranking rules and ABC choice rules, and show how to transfer some of the results for ABC ranking rules to the world of ABC choice rules.
1.3 Related Work
The most important axiomatic concept in our study is consistency. In the context of social welfare functions, this axiom informally states that if two disjoint societies both prefer candidate over candidate , then the union of these two societies should also prefer over , i.e., consistency refers to consistent behavior with respect to varying populations, refered to as population-consistency in other contexts. Smith  and Young  independently introduced this axiom and characterized the class of positional scoring rules as the only social welfare functions that satisfy symmetry, consistency, and continuity. Subsequently, Young  also proved an analogous result for social choice functions, i.e., voting rules that return the set of winning candidates. Furthermore, Myerson  and Pivato  characterized positional scoring rules with the same set of axioms but without imposing any restriction on the input of voting rules, i.e., ballots are not restricted to be a particular type of order. Extensive studies led to further, more specific, characterizations of consistent voting rules , such as the Borda rule [52, 69, 40, 36, 65, 38], Plurality [58, 21, 39], Antiplurality , Approval Voting [34, 60, 7, 68, 39, 35], and the probabilistic maximal lottery rule .
This impressive body of axiomatic studies shows that single-winner voting is largely well-understood and characterized. Axiomatic properties of multi-winner rules are considerably fewer in number. Debord  characterized the -Borda rule using similar axioms as Young . Felsenthal and Maoz  and Elkind, Faliszewski, Skowron, and Slinko  formulated a number of axiomatic properties of multi-winner rules, and analyzed which multi-winner voting rules satisfy these axioms; however, they do not obtain axiomatic characterizations. Elkind et al.  also defined the class of committee scoring rules, which aims at generalizing single-winner positional scoring rules to the multi-winner setting. This broad class contains, among others, the Chamberlin–Courant rule . In recent work, Skowron, Faliszewski, and Slinko  showed that the class of committee scoring rules admits a similar axiomatic characterization as their single-winner counterparts—this result plays a major role in the proof of Theorem 1. Faliszewski, Skowron, Slinko, and Talmon [27, 28] further studied the internal structure of committee scoring rules and characterized several multi-winner rules within this class.
A central concept in the study of multi-winner elections is proportionality. Proportionality is well-understood for party-list elections; the corresponding mathematical problem is called the apportionment problem. It typically arises when allocating seats to political parties based on the number of votes. For an overview of the literature on apportionment we refer the reader to the comprehensive books by Balinski and Young  and by Pukelsheim .
The concept of proportionality in arbitrary multi-winner elections (i.e., in the absence of parties) is more elusive. The first study of proportional representation in multi-winner voting dates back to Black , who informally defined proportionality as the ability to reflect shades of a society’s political opinion in the elected committee. Later, Dummett  proposed an axiom of proportionality for multi-winner rules that accept linear orders as input; it is based on the top-ranked candidates in voters’ rankings. An insightful discussion on the need for notions of proportionality that are applicable to linear order preferences can be found in the seminal work of Monroe ; he referred to such concepts as fully proportional representation since they are to take “full” preferences into account. More recently, axiomatic properties for approval-based rules have been proposed that aim at capturing the concept of proportional representation [6, 59]. This body of research demonstrates that the concept of proportionality can be sensibly defined and discussed in the context of multi-winner rules, even though this setting is more intricate and mathematically complex than the party-list setting. It is noteworthy that the results in our paper (in particular Theorem 2) show that for obtaining axiomatic characterizations it is in general not necessary to rely on proportionality definitions considering the full domain; proportionality defined in the restricted party-list setting—i.e., proportionality as defined for the apportionment problem—may be sufficient for characterizing multi-winner rules.
1.4 Structure of the Paper
This paper is structured as follows: We briefly state preliminary definitions in Section 2. Section 3 contains a formal introduction of ABC counting rules, their defining set of axioms, and the statement of our main technical tool (Theorem 1). In Section 4, we discuss and prove theorems that explore how different axioms of (dis)proportionality yield specific ABC counting rules: Section 4.1 contains the axiomatic characterization of Proportional Approval Voting based on D’Hondt proportionality, Section 4.2 the characterization of Multi-Winner Approval Voting based on disjoint equality, and Section 4.3 the characterization of Approval Chamberlin–Courant via disjoint diversity. Section 4.4 makes the statement precise that only functions “close” to can satisfy the lower quota axiom. Finally, in Section 5 we summarize the big picture of this paper and discuss further research directions.
As this paper contains a large number of proofs, we have moved substantial parts into appendices. Appendix A contains the main technical and most complex part of this paper, the proof of Theorem 1. Appendix B contains omitted proofs for non-essential lemmas and propositions. Finally, in Appendix C, we show how to translate some of our results from the setting of ABC ranking rules to ABC choice rules.
We write to denote the set and to denote for . For a set , let denote the powerset of , i.e., the set of subsets of . Further, for each let denote the set of all size- subsets of . A weak order of is a binary relation that is transitive and complete (all elements of are comparable), and thus also reflexive. A linear order is a weak order that is antisymmetric. We write to denote the set of all weak orders of and to denote the set of all linear orders of .
Approval profiles. Let be a set of candidates. We identify voters with natural numbers, i.e., is the set of all possible voters. For each finite subset of voters , an approval profile over , , is an -tuple of subsets of . For , let denote the subset of candidates approved by voter . We write to denote the set of all possible approval profiles over and to be the set of all approval profiles (for the fixed candidate set ). Given a permutation and an approval profile , we write to denote the profile .
Let and . For and , we write to denote the profile defined as . For a positive integer , we write to denote , times.
Approval-based committee ranking rules. Let denote the desired size of the committee to be formed. We refer to elements of as committees. Throughout the paper, we assume that both and (and thus ) are arbitrary but fixed. Furthermore, to avoid trivialities, we assume .
An approval-based committee ranking rule (ABC ranking rule), , maps approval profiles to weak orders over committees. Note that and are parameters for ABC ranking rules but since we assume that and are fixed, we omit them to alleviate notation. For an ABC ranking rule and an approval profile , we write to denote the weak order . For , we write if and not , and we write if and . A committee is a winning committee if it is a maximal element with respect to .
An approval-based committee choice rule (ABC choice rule), , maps approval profiles to sets of committees, again referred to as winning committees. As before, and are parameters for ABC choice rules but we omit them from our notation.
An ABC ranking rule is trivial if for all and it holds that . An ABC choice rule is trivial if for all it holds that .
Let us now list some important examples of ABC ranking rules and ABC choice rules. For some of these rules it was already mentioned in the introduction that they belong to the class of ABC counting rules; we discuss this classification in detail in Section 3 and also give their defining counting functions. The definitions provided here are more standard and do not use counting functions.
- Multi-Winner Approval Voting (AV).
In AV each candidate obtains one point from each voter who approves of . The AV-score of a committee is the total number of points awarded to members of , i.e., . Multi-Winner Approval Voting considered as an ABC ranking rule ranks committees according to their score; AV considered as an ABC choice rule outputs all committees with maximum AV-scores.
- Thiele Methods.
In 1895, Thiele  proposed a number of ABC ranking rules that can be viewed as generalizations of Multi-Winner Approval Voting. Consider a sequence of weights and define the -score of a committee as , i.e., if voter has approved candidates in , receives a score of from . The committees with highest -score are the winners according to the -Thiele method. Thiele methods can also be viewed as ABC ranking rules, where committees are ranked according to their score.
Thiele methods form a remarkably general class of multi-winner rules: apart from Multi-Winner Approval Voting which is defined by the weights , PAV and CC also fall into this class.
- Proportional Approval Voting (PAV).
PAV was first proposed by Thiele ; it was later reinvented by Simmons , who introduced the name “proportional approval voting”. PAV is a Thiele method defined by the weights . These weights being harmonic numbers guarantee proportionality—in contrast to Multi-Winner Approval Voting, which is not a proportional method. The Proportionality of PAV is illustrated in the example below.
Consider a population with 100 voters; 75 voters approve of candidates and 30 voters of candidates . Assume . For such a profile Multi-Winner Approval Voting selects a single winning committee . The PAV-score of is equal to . We can obtain a better committee by (proportionally) selecting three candidates from the set and one from the set ; the PAV-score of such committees is equal to . This is the highest possible score and hence such committees are the winning committees according to PAV.
- Approval Chamberlin–Courant (CC).
Also the Approval Chamberlin–Courant rule was suggested and recommended by Thiele . It closely resembles the Chamberlin–Courant rule , which was originally defined for ordinal preferences but can easily be adapted to the approval setting. The Approval Chamberlin–Courant rule is a Thiele method defined by the weights . Consequently, CC chooses committees so as to maximize the number of voters which have at least one approved candidate in the winning committee.
For a general overview of approval-based multi-winner rules, we refer the reader to a handbook chapter by Kilgour .
3 ABC Counting Rules
In this section we define a new class of multi-winner rules, called ABC counting rules. ABC counting rules can be viewed as an adaptation of positional scoring rules [65, 70] to the world of approval-based multi-winner rules. ABC counting rules can also be seen as a generalization of Size Approval Voting111ABC counting rules generalize Size Approval Voting in two aspects: Size Approval Voting is defined as a single-winner rule (i.e., for ), and it requires that , a constraint that does not apply to ABC counting rules. Size Approval Voting contains, e.g., the single-winner variants of Approval Voting and Satisfaction Approval Voting., introduced by Alcalde-Unzu and Vorsatz . Furthermore, ABC counting rules can be viewed as analogous to the class of (multi-winner) committee scoring rules as introduced by Elkind, Faliszewski, Skowron, and Slinko  but defined for approval ballots instead of linear order preferences.
In this section, after formally defining ABC counting rules and some basic axioms, we present our main technical result: an axiomatic characterization of the class of ABC counting rules. This result forms the basis for our subsequent axiomatic analysis.
3.1 Defining ABC Counting Rules
A counting function is a mapping satisfying whenever . The intuitive meaning is that denotes the score that a committee obtains from a voter that approves of members of and candidates in total. We define the score of a committee in as
We say that a counting function implements an ABC ranking rule if for every and ,
Analogously, we say that a counting function implements an ABC choice rule if for every ,
i.e., returns all committees with maximum score. An ABC (winner) rule is an ABC counting rule if there exists a counting function that implements .
Several ABC ranking rules that we introduced earlier are ABC counting rules. As we have seen in the introduction, AV, PAV, and CC can be implemented by the following counting function:
Note that only Satisfaction Approval Voting is implemented by a counting function depending on . As can easily be verified, Thiele methods are exactly those ABC counting rules that can be implemented by a counting function not dependent on .
It is apparent that not the whole domain of a counting rule is relevant; consider for example or —these function values will not be used in the score computation of any committee, cf. Equation (1). The following proposition provides a tool for showing that two counting rules are equivalent. It shows which part of the domain of counting rules is relevant and that affine transformations yield equivalent rules.
Let and let be counting functions. If there exist and such that for all then yield the same ABC counting rule, i.e., for all approval profiles and committees it holds that if and only if .
3.2 Basic Axioms
In this section, we provide and discuss formal definitions of the axioms used for our main characterization result (Theorem 1). All axioms are natural and straightforward adaptations of the respective properties of single-winner election rules and will be stated for ABC ranking rules. In Appendix C, where we extend some of our results to ABC choice rules, we explain how these axioms should be formulated for ABC choice rules. Similar axioms have also been considered in the context of multi-winner rules based on linear-order preferences [25, 62].
Anonymity and neutrality enforce perhaps the most basic fairness requirements for voting rules. Anonymity is a property which requires that all voters are treated equally, i.e., the result of an election does not depend on particular names of voters but only on votes that have been cast. In other words, under anonymous ABC ranking rules, each voter has the same voting power.
Anonymity. An ABC ranking rule is anonymous if for such that , for each bijection , and for and such that for each , it holds that .
Neutrality is similar to anonymity, but enforces equal treatment of candidates rather than of voters.
Neutrality. An ABC ranking rule is neutral if for each bijection and with it holds for that if and only if .
Due to their analogous structure and similar interpretations, anonymity and neutrality are very often considered together, and jointly referred to as symmetry.
Symmetry. An ABC ranking rule is symmetric if it is anonymous and neutral.
Consistency was first introduced in the context of single-winner rules by Smith  and then adapted by Young . In the world of single-winner rules, consistency is often considered to be the axiom that characterizes positional scoring rules. Similarly, consistency played a crucial role in the recent characterization of committee scoring rules , which can be considered the equivalent of positional scoring rules in the multi-winner setting. Consistency is also the main ingredient for our axiomatic characterization of ABC counting rules.
Consistency. An ABC ranking rule is consistent if for , and for , if and , then , and if and , then .
Next, we describe efficiency, which captures the intuition that voters prefer to have more approved candidates in the committee.
Efficiency. An ABC ranking rule satisfies efficiency if for and where for every voter we have , it holds that .
For , i.e., in the single-winner setting, efficiency is the well-known Pareto efficiency axiom, which requires that if a candidate is unanimously preferred to candidate , then in the collective ranking .
For the purpose of our axiomatic characterization, a significantly weaker form of efficiency suffices. Weak efficiency only requires that candidates that are approved by no voter are at most as desirable as any other candidate.
Weak efficiency. An ABC ranking rule satisfies weak efficiency if for each and where no voter approves a candidate in , it holds that .
If we consider the single-winner case here, we see that the axiom reduces to the following statement: if no voter approves candidate , then any other candidate is at least as preferable as .
The following lemma shows that efficiency in the context of neutral and consistent rules is implied by its weaker counterpart.
A neutral and consistent ABC ranking rule that satisfies weak efficiency also satisfies efficiency.
The final axiom, continuity  (also known in the literature as the Archimedean property  or the Overwhelming Majority axiom ), describes the influence of large majorities in the process of making a decision. Continuity enforces that a large enough group of voters is able to force the election of their most preferred committee. Continuity is pivotal in Young’s characterization of positional scoring rules  as it excludes specific tie-breaking mechanisms.
Continuity. An ABC ranking rule satisfies continuity if for each and where there exists a positive integer such that .
3.3 A Characterization of ABC Counting Rules
The following axiomatic characterization of ABC counting rules is a powerful tool that forms the basis for further characterizations of specific ABC counting rules. This result resembles Smith’s and Young’s characterization of positional scoring rules [70, 65] as the only social welfare functions satisfying symmetry, consistency, and continuity. Our characterization additionally requires weak efficiency, which stems from the condition that a counting function must be weakly increasing in . If a similar condition was imposed on positional scoring rules (i.e., that positional scores are weakly decreasing), an axiom analogous to weak efficiency would be required for a characterization as well.
An ABC ranking rule is an ABC counting rule if and only if it satisfies symmetry, consistency, weak efficiency, and continuity.
It is easy to verify that ABC counting rules satisfy symmetry, consistency, weak efficiency, and continuity; all this follows immediately from the definitions in Section 3.1. Proving the other implication, however, requires a long and complex proof, which can be found in Appendix A. Furthermore, the set of axioms used in Theorem 1 is minimal, i.e., any subset of axioms is not sufficient for the characterization statement to hold. A detailed argument can be found in Appendix A.3.
4 Proportional and Disproportional ABC Counting Rules
In this section we consider axioms describing winning committees in party-list profiles and that capture a specific variant of proportionality, individual excellence, or diversity. In party-list profiles, voters and candidates are grouped into clusters; such clusters can be intuitively viewed as political parties.
An approval profile is a party-list profile with parties if the set of voters can be partitioned into and the set of candidates can be partitioned into such that, for each , every voter in approves exactly .
We will show that axioms for party-list profiles are sufficient to characterize certain ABC counting rules: PAV, AV, and CC. Using the axiomatic characterization of ABC counting rules (Theorem 1), we obtain full axiomatic characterizations of these three rules. Finally, we consider a weaker form of proportionality (lower quota) and show that any ABC counting rule satisfying lower quota is implemented by a counting function that is “close” to , the PAV counting function.
4.1 D’Hondt Proportionality
In party-list profiles, we intuitively expect a proportional committee to contain as many candidates from a party as is proportional to the number of this party’s supporters. There are numerous ways in which this concept can be formalized—different notions of proportionality are expressed through different methods of apportionment [8, 56]. In this paper we consider one of the best known, and perhaps most commonly used concept of proportionality: D’Hondt proportionality [8, 56].
The D’Hondt method is an apportionment method that works in steps. It starts with an empty committee and in each step it selects a candidate from a set (party) with maximal value of ; the selected candidate is added to .
Consider an election with four groups of voters, , , , and with cardinalities equal to 9, 21, 28, and 42, respectively. Further, there are four groups of candidates , , , and . Each voter in a group approves exactly of the candidates in . Assume and consider the following table, which illustrates the ratios used in the D’Hondt method for determining which candidate should be selected.
In this example the D’Hondt method will select a candidate from first, next a candidate from , next from or (their ratios in the third step are equal), etc. Eventually, in the selected committee there will be one candidate from , two candidates from , three from , and four from ; the respective ratios are printed in bold.
An important difference between the apportionment setting and our setting is that we do not necessarily assume an unrestricted number of candidates for each party. As a consequence, a party might deserve additional candidates but this is impossible to fulfill. Taking this restriction into account, we see that if the D’Hondt method picks a candidate from and adds it to , then, for all , either or , i.e., all candidates from party are already in the committee. Note that if and , then the D’Hondt method would rather select a candidate from than from . These observations allow us to give a precise definition of D’Hondt proportional committees.
Let be a party-list profile with parties. A committee is D’Hondt proportional for if for all one of the following conditions holds: (i) , or (ii) , or (iii) .
D’Hondt proportionality. An ABC ranking rule satisfies D’Hondt proportionality if for each party-list profile , is a winning committee if and only if is D’Hondt proportional for .
Note that this axiom is weak in the sense that it only describes the expected behavior of an ABC ranking rule on party-list profiles. As we will see, however, it is sufficient to obtain an axiomatic characterization of PAV in the more general framework of ABC ranking rules.
Proportional Approval Voting is the only ABC counting rule that satisfies D’Hondt proportionality.
To see that PAV satisfies D’Hondt proportionality, let be the PAV counting function defined by . Consider a party-list profile with parties, i.e., we have a partition of voters and their corresponding joint approval sets . For the sake of contradiction let us assume that is a winning committee and that there exists such that , and . Let and . We define . Let us compute the difference between PAV-scores of and :
Thus, we see that has a higher PAV-score than , a contradiction.
To show the other direction, let be an ABC ranking rule satisfying all the above axioms. By Lemma 2, also satisfies weak efficiency. Now Theorem 1 implies that is an ABC counting rule. Let be the corresponding counting function. We intend to apply Proposition 1 to show that is equivalent to the PAV counting function . Hence we have to show that there exists a constant and a function such that for all . W.l.o.g., we can focus on the case when .
We first consider the case when and . Now, let us fix such that , , and . Let us consider the following party-list profile. There are three groups of voters: with , and ; their corresponding approval sets are . Let , , and . Consider the two following committees: we choose such that , , and ; we choose such that , , and . It is straight-forward to verify that both and are D’Hondt proportional. Thus, and are winning committees and hence have the same scores. Their respective scores are
Since we have
As we show this statement for and , we can expand this equation and obtain
Obviously, the above equality also holds for .
Now, we move to the case when and . Since , we have that . Consider the party-list profile with parties with and , for ; the corresponding candidate sets are and for . Note that . Consider the two following committees of size : consists of candidates from , and a single candidate from for each ; consists of candidates from , and a single candidate from each , .
Similar to before, it can be shown that both and are D’Hondt proportional and hence . The PAV-scores of and are
We conclude that for we have
Hence we have shown that indeed for and and, by Proposition 1, is PAV. ∎
We can now use Theorem 2 to obtain an axiomatic characterization within the class of ABC ranking rules.
An ABC ranking rule that satisfies neutrality, consistency, and D’Hondt proportionality also satisfies weak efficiency.
Proportional Approval Voting is the only ABC ranking rule that satisfies symmetry, consistency, continuity, and D’Hondt proportionality.
The characterization of Corollary 1 also hold for ABC choice rules, i.e., for approval-based multi-winner rules returning a set of winning committees. Some of the axioms have to be slightly adapted to be suitable for ABC choice rules, e.g., consistency can only distinguish between winning and losing committees. We refer the reader to Appendix C for precise descriptions of the relevant axioms and a proof of Theorem 2 (and Corollary 1) for ABC choice rules.
4.2 Disjoint Equality
In some scenarios we might not want a multi-winner rule to be proportional. For example, if our goal is to select a set of finalists in a contest based on a set of recommendations coming from judges or reviewers (a scenario that is often referred to as a shortlisting), candidates can be assessed independently and there is no need for proportionality. For instance, if our goal is to select 5 finalists in a contest, and if four reviewers support candidates and one reviewer supports candidates then it is very likely that we would prefer to select candidates as the finalists—in contrast to what, e.g., D’Hondt proportionality suggests.
Disjoint equality is a property which might be viewed as a certain type of disproportionality. Intuitively, it requires that each approval of a candidate has the same power: a candidate approved by a voter receives a certain level of support from which does not depend on what other candidates approves or disapproves of; in particular it does not depend on whether there are other members of a winning committee which are approved by . Disjoint equality was first proposed by Fishburn  and then used by Sertel  as one of the distinctive axioms characterizing single-winner Approval Voting. The following axiom is its natural extension to the multi-winner setting.
Disjoint equality. An ABC ranking rule satisfies disjoint equality if for every profile with and where each candidate is approved at most once, the following holds: is a winning committee if and only if .
In other words, disjoint equality asserts that in a profile consisting of disjoint approval ballots every committee wins that consists of approved candidates. Note that disjoint equality applies to an even more restricted form of party-list profiles.
Multi-Winner Approval Voting is the only ABC counting rule that satisfies disjoint equality.
It is straightforward to verify that Multi-Winner Approval Voting satisfies disjoint equality. For the other direction, consider an ABC counting rule satisfying disjoint equality that is implemented by a counting function . As in previous proofs we rely on Proposition 1 to show that and implement the same ABC counting rule. It is thus our aim to show that for it holds that for some and . More specifically, we will show that for with it holds that . It then follows from induction that and thus we will be able to conclude that implements Multi-Winner Approval Voting.
Let with and . We construct a profile with and . All voters have disjoint sets of approved candidates. Hence this construction requires candidates. Since , it holds that and hence ; we see that a sufficient number of candidates is available. Let contain candidates from and one candidate from each. Let contain candidates from and one candidate from each. Note that . By disjoint equality both and are winning committees. Hence
and thus . ∎
4.3 Disjoint Diversity
The disjoint diversity axiom is strongly related to the diversity principle, as it states that there exists a winning committee in which the strongest parties receive at least one seat—without consideration of their actual proportions.
Disjoint diversity. An ABC ranking rule satisfies disjoint diversity if for every party-list profile with parties and , there exists a winning committee with for all .
Note that disjoint diversity is a slightly weaker axiom in comparison to D’Hondt proportionality and disjoint equality since it does not characterize all winning committees for party-list profiles—it only guarantees the existence of one specific winning committee.
The Approval Chamberlin–Courant rule is the only non-trivial ABC counting rule that satisfies disjoint diversity.
The Approval Chamberlin–Courant rule maximizes the number of voters that have at least one approved candidate in the committee. In a party-list profile, this implies that the largest parties receive at least one representative in the committee and hence disjoint diversity is satisfied.
For the other direction, let be an ABC counting rule implemented by a counting function . Recall Proposition 1 and the relevant domain of counting functions . In a first step, we want to show that for and . Let us fix such that , , and . Furthermore, let us fix a committee and consider a set with and . We construct a party-list profile as follows: contains votes that approve (intuitively, is a large natural number); further for each candidate , profile contains a single voter who approves . This construction requires candidates. Since , we have .
If we apply disjoint diversity to profile , we obtain a winning committee with and . By efficiency (which holds due to Lemma 1), is winning as well. Let be the committee we obtain from by replacing one candidate in with a candidate in (such a candidate exists since ). By disjoint diversity is not a winning committee. Consequently, and thus
The above condition can be written as . Since this must hold for any , we get that . Efficiency implies that ; thus we get that for . By Proposition 1 we can set for each . We conclude that is also implemented by the counting function
As a next step we show that for the counting function we can additionally assume that , for each . Observe that if , then for each committee , a voter who approves candidates in total, approves at least one member of . By our previous reasoning, each committee gets from such a voter the same score, and so such a voter does not influence the outcome of an election. Consequently, we can assume that for . Now, for , we also show that . Towards a contradiction assume that and further, without loss of generality, . To this end, let be natural number large enough so that . Consider a party-list profile consisting of voters approving , and, for , voters each that approves candidate . The committee obtains a score of , whereas obtains a score of . Since by choice of it holds that , committee is winning. This contradicts disjoint diversity and hence