Consistent Approval-Based Multi-Winner Rules

Transcription

1 Consistent Approval-Based Multi-Winner Rules arxiv: v3 [cs.gt] 20 Jul 2017 Martin Lackner University of Oxford Oxford, UK Piotr Skowron Technische Universität Berlin Berlin, Germany Abstract This paper is an axiomatic study of consistent approval-based multi-winner rules, i.e., voting rules that select a fixed-size group of candidates based on approval ballots. We introduce the class of counting rules, provide an axiomatic characterization of this class and, in particular, show that counting rules are consistent. Building upon this result, we axiomatically characterize three important consistent multiwinner rules: Proportional Approval Voting, Multi-Winner Approval Voting and the Approval Chamberlin Courant rule. Our results demonstrate the variety of multi-winner rules and illustrate three different, orthogonal principles that multiwinner voting rules may represent: excellence, diversity, and proportionality. Keywords: voting, axiomatic characterizations, approval voting, multi-winner elections, dichotomous preferences, apportionment 1 Introduction In Arrow s foundational book Social Choice and Individual Values [6], 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 thebestgroupofcandidatesofagivensize; wecallsuchafixed-sizeset ofcandidates 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 [30,34]. 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, 1

2 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 [22], 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 prefertohaveahospitalclosetotheirhomebutarelessinterestedinhavingmorethanone 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 multiwinner rule should follow the 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 excellence: There are 100 voters and 5 candidates {a,b,c,d,e}: 66 voters approve the set {a,b,c}, 33 voters approve {d}, and one voter approves {e}. Assume we want to select a committee of size three. If we follow the principle of proportionality, we could choose, e.g., {a, b, d}; 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 {a, d, e}; this committee contains one approved candidate of every voter. The principle of excellence aims to select the strongest candidates: a, b, and c 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 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 2

3 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 W 1 to a committee W 2, then the union of these two societies should also prefer W 1 to W 2. This is a straightforward adaption of consistency as defined for single-winner rules by Smith [90] and Young [97]. 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 result of this paper and the main technical tool to obtain further results 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 f(x,y) (the so-called counting function), a committee W receives a score of f(x,y) from every voter for which W contains x out of their y approved candidates; the ABC counting rule implemented by f ranks committees according to the sum of scores obtained from each voter. We obtain the following characterization of ABC counting rules. Theorem 1. 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 (symmetry is also often called impartiality [71]). Weak efficiency means that voters do not gain utility by having fewer approved candidates in the committee. Continuity (also known as Archimedean Property or Overwhelming Majority) is a more technical condition that states that a sufficiently large majority can dictate a decision. In Section 3, we discuss all axioms and their justification in detail. As weak efficiency is satisfied by every sensible multi-winner rule and continuity typically only rules out the use of certain tie-breaking mechanisms [90, 98], 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. Building upon this result, we further explore the space of ABC counting rules. To this end, we characterize three important ABC ranking rules, each of which being a representative for one of the three principles mentioned earlier: Multi-Winner Approval Voting for excellence, Proportional Approval Voting for proportionality, and the Approval Chamberlin Courant rule for diversity. Note in the following that these three rules are defined by counting functions f(x,y) which do not depend on y; we will remark further on this fact later on. 3

4 Multi-Winner Approval Voting (AV). Multi-Winner Approval Voting [14] is defined by the counting function f AV (x,y) = x, i.e., a committee obtains a score of x for every voter with x approved candidates in the committee. Thus, candidates obtain a score equal to the number of voters that approve them and the top-scoring candidates are chosen for the committee. Multi-Winner Approval Voting is the prime example of a multi-winner rule following the principle of excellence. We obtain two axiomatic characterizations. The first uses the disjoint equality axiom, which is a slightly adapted version of disjoint equality as used by Fishburn [40] and Sertel [85] to characterize Approval Voting as a single-winner rule. 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 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. Theorem 2. Multi-Winner Approval Voting is the only ABC ranking rule that satisfies symmetry, consistency, continuity, and disjoint equality. If we accept the argument that excellence implies disjoint equality, then Theorem 2 can be understood as showing that Multi-Winner Approval Voting is the only ABC counting rule that follows the excellence principle. The second characterization highlights a different aspect of Multi-Winner Approval Voting. It is the only rule that satisfies two weak forms of strategyproofness, namely independence of irrelevant alternatives and monotonicity. Independence of irrelevant alternatives, an adaption of the homonymous axiom of Arrow [6], states that the relative orderoftwocommittees intheoutputofanabcrankingruledoesnotdependonwhether candidates outside of these committees are approved or not; monotonicity states that approving a candidate in a committee cannot weaken this committee (see Section 3.2 for more detailed descriptions). Hence, this theorem singles out Multi-Winner Approval Voting among consistent ABC ranking rules as the one that is immune against some basic forms of strategic voting. Theorem 3. Multi-Winner Approval Voting is the only non-trivial ABC ranking rule that satisfies symmetry, consistency, continuity, independence of irrelevant alternatives, and monotonicity. Proportional Approval Voting (PAV). Proportional Approval Voting was first proposed by Thiele [92] in the late 19th century. It is an ABC ranking rule defined by the counting function f PAV (x,y) = x i=1 1 /i, i.e., a committee W receives a score of 1 from every voter with at least one approved candidate in W, an additional score of 1 /2 for every voter with at least two approved candidates in W, etc. We show that Proportional Approval Voting is exceptionally well-suited for proportional representation: it is the only ABC ranking rule that satisfies D Hondt proportionality. D Hondt proportionality was defined for apportionment problems, i.e., the proportional distribution of seats to parties 4

5 in parliamentary elections. The apportionment problem is a special case of approval-based multi-winner voting: a vote for a party corresponds to an approval of all candidates of this party. Hence D Hondt proportionality also applies to our setting, although it only applies to so-called party-list profiles, i.e., profiles where voters approval sets are either disjoint or identical. We obtain the following strong characterization: Theorem 4. Proportional Approval Voting is the only ABC ranking rule that satisfies symmetry, consistency, continuity and D Hondt proportionality. This theorem shows 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., non-linear methods such as the Penrose method [74]. Approval Chamberlin Courant (CC). The third ABC ranking rule we consider is the Approval Chamberlin Courant rule. This rule was also first mentioned by Thiele [92] and then studied in the context of ordinal preferences by Chamberlin and Courant [22]. It is defined by the counting function { 0 if x = 0, f CC (x,y) = 1 if x 1. This definition implies that CC chooses a committee W that maximizes the number of voters that have at least one approved candidate in W. Our characterization of CC is based on independence of irrelevant alternatives and a newly introduced axiom called disjoint diversity. Disjoint diversity states that in party-list profiles with at most as many parties as desired committee members, every winning committee must contain at least one candidate of each party (again, we refer the reader to Section 3 for a formal statement). This axiom aligns with our claim that CC is a prime example of a multi-winner rule aiming for diverse committees. Theorem 5. The Approval Chamberlin Courant rule is the only ABC ranking rule that satisfies symmetry, consistency, weak efficiency, continuity, independence of irrelevant alternatives, and disjoint diversity. 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 5

6 score f(x,.) Multi-Winner Approval Voting Proportional Approval Voting Appr. Chamberlin Courant f AV f PAV number of approved candidates in committee (x) Figure 1: Different counting functions and their corresponding ABC counting rules. Counting functions outside the gray area fail the lower quota axiom; see Section 5.3 for a formal statement. f CC defining counting function f(x, y); see Figure 1 for a visualization. Our results indicate thatcountingfunctionsthathavealargerslopethanf PAV putmoreemphasisonmajorities and thus become less egalitarian, whereas counting functions that have a smaller slope than f PAV treat minorities preferentially and thus approach the Approval Chamberlin Courant rule. In particular, we show that counting functions that are not close to f PAV (all those not contained in the gray area around f PAV ) implement ABC ranking rules that violate even a rather weak form of proportionality called lower quota; see Section 5.3 for a formal statement. As mentioned before, it is noteworthy that all three counting functions f AV, f PAV, and f CC do not depend on y. Since the class of ABC ranking rules with this property was first discussed by Thiele [92], we refer to such rules as Thiele methods; they are also known as Generalized Approval Procedures [60]. The following characterization of Thiele methods is based on independence of irrelevant alternatives, which already made an appearance in characterizations of the Approval Chamberlin Courant rule and Multi- Winner Approval Voting, but as the following theorem implies it is also satisfied by Proportional Approval Voting. Theorem 6. Thiele methods are the only ABC ranking rules that satisfy symmetry, consistency, weak efficiency, continuity, and independence of irrelevant alternatives. Finally, we note that all theorems mentioned so far hold for ABC ranking rules. We demonstrate the generality of our results by proving that the characterizations of Proportional Approval Voting (Theorem 4) and the Approval Chamberlin Courant rule (Theorem 5) also hold for ABC choice rules. The method used for this proof is not applicable to Multi-Winner Approval Voting. Thus, characterizing the Multi-Winner Approval Voting 6

7 and, more generally, ABC counting rules within the class of ABC choice rules remains as important future work. 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. The first three points concern the diverse applications of multi-winner voting and advantages of relying on approval ballots. Then we argue that single-winner voting cannot be a practical substitute in multi-winner scenarios. Furthermore, we discuss the relation between fixed-size committees (the topic of this paper) and variable-sized committees. Finally, we elaborate on our focus on ABC ranking rules and their relation to ABC choice rules. Political elections. 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 [10, 79]. 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., [2,3,23,27] for a more elaborate discussion of these issues). To counteract these disadvantages, there is a movement for promoting personalization in voting [81]. Some countries, such as Finland and Latvia, use open party-list systems that allow voters to influence in which order members of a party are assigned seats in the parliament. Under the panachage systems voters are allowed to vote for candidates from different parties (these systems are used, e.g., for some elections in Luxembourg, France, and Switzerland). Multi-winner voting can be seen as an overarching concept that generalizes these systems. For an extensive discussion on current trends towards personalization in voting, we refer the reader to the recent book of Renwick and Pilet [81], and to the seminal theoretical works of Chamberlin and Courant [22] and Monroe [70]. 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 [66, 67], group recommendation [86], diversifying search results [88], improving genetic algorithms [31], and the broad class of facility location problems [35,86]. 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 [34]. 7

8 Approval ballots. The use of approval ballots has several advantages [62]: 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 [6]. Furthermore, Brams and Herschbach [16] 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 [4]. As a consequence, voting based on approvals is being advocated as a desirable electoral reform: Brams and Fishburn [15] analyze the use of Approval Voting, mostly in scientific societies; Brams and Fishburn [14] provide an extensive discussion on the possible, generally positive impact of using Approval Voting for political elections. Furthermore, Laslier and Van der Straeten [63] conducted a real-life experiment during the 2002 presidential elections in France and proved the feasibility of approval balloting in political elections. 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 8

9 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 candidates. 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 F 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 Our work deals with axiomatic characterizations of multi-winner rules. The study of axiomatic properties of single-winner voting rules was initiated by Arrow [6]. Arrow s classical theorem can be seen as a negative result since it excludes the existence of voting rules satisfying three desirable axioms: unanimity, non-dictatorship, and independence of irrelevant alternatives. Yet, Arrow s contribution was much more fundamental he created a framework that allows for the normative comparison of voting rules. To some extent, this was foreshadowed by May s Theorem [68], which is an axiomatic characterization of the majority rule limited to two candidates (see also the work of Fishburn [39] for another characterization of the majority rule). The seminal work of Arrow was followed by impossibility results of Gibbard [50] and Satterthwaite [84], which show that every sensible rule is susceptible to strategic voting: there is no strategyproof, nondictatorial voting rule in which every candidate may win. Both key ingredients of these results, independence of irrelevant alternatives for Arrow s theorem and strategyproofness for the Gibbard Satterthwaite Theorem, are also essential for our axiomatic study of multi-winner rules. 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 a over candidate b, then the union of these two societies should also prefer a over b. Smith [90] and Young [97] 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 [98] also proved an analogous result for social choice functions, i.e., voting rules that return the set of winning candidates. Furthermore, Myerson [72] and Pivato [78] 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. These include several different characterizations of the Borda rule [45, 48, 54, 73, 90, 96], the Plurality rule [25, 52, 82], the Antiplurality rule [12], and Approval Voting [9, 40, 41, 52, 85, 94]. For a more comprehensive overview of these and other characterizations of single-winner rules, we refer the reader to the survey by Chebotarev and Shamis [24]; an 9

10 insightful exposition of these results has also been given by Merlin [69]. The Kemeny rule was characterized by Young and Levenglick [99] using the reinforcement axiom, which is comparable to consistency but generally weaker: informally speaking, reinforcement requires that whenever a voting rule returns the same ranking for two disjoint elections, it should also return this ranking for a merged election. Probabilistic social choice [38, 51,100] considers voting rules that output a lottery, i.e., the output is a probability distribution over candidates. Also in this setting, consistency is of major importance: Brandl, Brandt, and Seedig [19] characterized Fishburn s rule of maximal lotteries [44] as the only probabilistic social choice function that satisfies population-consistency (which corresponds to the consistency axiom mentioned before) and composition-consistency. 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 [28] characterized the k-borda rule using similar axioms as Young [96]. Elkind, Faliszewski, Skowron, and Slinko [30] formulated a number of axiomatic properties of multi-winner rules, and analyzed which multi-winner voting rules satisfy these axioms. Elkind et al. [30] 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 [22]. In recent work, Skowron, Faliszewski, and Slinko[87] 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 [32, 33] further studied the internal structure of committee scoring rules and characterized several multi-winner rules within this class. Felsenthal and Maoz [37] have studied how properties of single-winner rules transfer to their multi-winner counterparts and vice versa. Another interesting line of research focuses on the Condorcet principle and its generalization to multi-winner voting [11, 42, 43, 49, 57, 80]. 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. The literature on apportionment is vast, and in particular, precise axiomatizations of apportionment methods exist. Perhaps the most famous result pertaining to the axiomatization of apportionment methods is Balinski and Young s impossibility theorem [10], which states that there is no apportionment method that simultaneously stays within the quota and satisfies population monotonicity: staying within the quota means that a party with an α proportion of votes obtains between α k (lower quota) and α k (upper quota) of the k available seats, and population monotonicity says that a party cannot be allocated fewer seats due to receiving more votes. At the same time, Balinski and Young characterized the class of divisor methods as those which satisfy population monotonicity. The D Hondt method (which is a divisor method) can uniquely be characterized as the only apportionment method that satisfies population monotonicity and respects the lower quota. For an overview of the literature on apportionment we refer the reader to the comprehensive books by Balinski and Young [10] and by Pukelsheim [79]. For a more general, comparative analysis of different 10

11 electoral systems and a discussion of proportional representation, we refer the reader to the book of Lijphart and Grofman [65] and to a recent review by Grofman [53]. The concept of proportionality in arbitrary multi-winner elections (i.e., in the absence of parties) is much more elusive. The first study of proportional representation in multiwinner voting dates back to Black [13], who informally defined proportionality as the ability to reflect shades of a society s political opinion in the elected committee. Later, Dummett [29] 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. A variant of this axiom has been used in a discussion on the proportionality of Single Transferable Vote (STV) [30, 93, 95]; another normative criterion ( free association ) justifying proportionality of STV was studied by Sugden [91]. An insightful discussion on the need for notions of proportionality that are applicable to linear order order preferences can be found in the seminal work of Monroe [70]; he referred to such concepts as fully proportional representation since they are to take full preferences into account. Feld and Grofman gave a formal justification that even large societies can be well represented by committees of reasonable size, provided that such committees assign appropriate weights to their members for the final decision making process [36]. More recently, axiomatic properties for approval-based rules have been proposed that aim at capturing the concept of proportional representation [7, 83]; in particular, Proportional Approval Voting has been found to distinguish itself as the only known rule satisfying Extended Justified Representation [7]. 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 4) 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: After preliminary definitions in Section 2, we introduce and discuss in Section 3 all major axioms used in this paper. Section 4 contains a formal introduction of ABC counting rules, our main object of study, and the statement of our main technical tool, the axiomatic characterization of ABC counting rules (Theorem 1). In Section 5, we discuss and prove theorems that explore how different axioms of (dis)proportionality yield specific ABC counting rules: Section 5.1 contains the axiomatic characterization of Proportional Approval Voting based on D Hondt proportionality(theorem 4) and Section 5.2 contains the axiomatic characterization of Multi-Winner Approval Voting (Theorem 2) based on disjoint equality. Section 5.3 makes the statement precise that only functions close to f PAV can be proportional. Section 6 explores axioms of strategyproofness and their impact on ABC counting rules. In particular, we characterize Thiele methods (Theorem 6), the Approval Chamberlin Courant rule (Theorem 5), and 11

12 Multi-Winner Approval Voting (Theorem 3). Finally, in Section 7 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 lemmasandpropositions. InAppendixD,wediscuss whether thesetofaxiomsusedinthe characterization theorems are minimal, i.e., to which degree these axioms are independent. Appendix C contains omitted proofs showing that some rules satisfy or fail certain axioms; Appendix Econtains a technical noteonthe disjoint equality axiom asused in Theorem 2. Finally, in Appendix F, we show how to translate some of our results from the setting of ABC ranking rules to ABC choice rules. 2 Preliminaries We write [n] to denote the set {1,...,n} and [i,j] to denote {i,i+1,...,j} for i j N. For a set X, let P(X) denote the powerset of X, i.e., the set of subsets of X. Further, for each l let P l (X) denote the set of all size-l subsets of X. A weak order of X is a binary relation that is complete and transitive; a linear order is a weak order that is antisymmetric. We write W (X) to denote the set of all weak orders of X and L(X) to denote the set of all linear orders of X. Approval profiles. Let C = {c 1,...,c m } be a set of candidates. We identify voters with natural numbers, i.e., N is the set of all possible voters. For each finite subset of voters V = {v 1,...,v n } N, an approval profile over V, A = (A(v 1 ),...,A(v n )), is an n-tuple of subsets of C. For v V, let A(v) C denote the subset of candidates approved by voter v. We write A(C,V) to denote the set of all possible approval profiles over V and A(C) = {A(C,V): V N and V is finite} to be the set of all approval profiles (for the fixed candidate set C). Given a permutation σ: C C and an approval profile A A(C,V), we write σ(a) to denote the profile (σ(a(v 1 )),...,σ(a(v n ))). For each l [0,m], we say that an approval profile A is l-regular if each voter in A approves of exactly l candidates. We say that A is l-bounded if each voter in A approves of at most l candidates. Let V = {v 1,...,v s } N and V = {v 1,...,v t} N. For A A(C,V) and A A(C,V ), we write A + A to denote the profile B A(C,[s + t]) defined as B = (A(v 1 ),...,A(v s ),A (v 1),...,A (v t)). For a positive integer n, we write na to denote A+A+ +A, n times. Sometimes we want to ignore multiplicities of votes and write set(a) to denote {A(v) : v V}. Approval-based committee ranking rules. Let k denote the desired size of the committee to be formed. We refer to elements of P k (C) as committees. Throughout the paper, we assume that both k and C (and thus m) are arbitrary but fixed. Furthermore, to avoid trivialities, we assume k < m. Anapproval-basedcommittee rankingrule (ABCrankingrule),F: A(C) W (P k (C)), maps approval profiles to weak orders over committees. Note that C and k are param- 12

13 eters for ABC ranking rules but since we assume that C and k are fixed, we omit them to alleviate notation. For an ABC ranking rule F and an approval profile A, we write F(A) to denote the weak order F(A). For W 1,W 2 P k (C), we write W 1 F(A) W 2 if W 1 F(A) W 2 and not W 2 F(A) W 1, and we write W 1 = F(A) W 2 if W 1 F(A) W 2 and W 2 F(A) W 1. Acommittee isawinning committee ifit isamaximal element with respect to F(A). An approval-based committee choice rule (ABC choice rule), F: A(C) P(P k (C))\ { }, maps approval profiles to sets of committees, again referred to as winning committees. As before, C and k are parameters for ABC choice rules but we omit them from our notation. An ABC ranking rule is trivial if for all A A(C) and W 1,W 2 P k (C) it holds that W 1 = F(A) W 2. An ABC choice rule is trivial if for all A A(C) it holds that F(A) = P k (C). Sometimes we associate an approval set S C with the single-voter profile A A(C,{1}) such that A(1) = S; in such a case we write F(S) as a short form of F(A) for appropriately defined A. 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 4 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 c C obtains one pointfromeachvoter whoapprovesofc. TheAV-scoreofacommitteeW isthetotal number of points awarded to members of W, i.e., v V A(v) W. 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 [92] proposed a number of ABC ranking rules that can be viewed as generalizations of Multi-Winner Approval Voting. Consider a sequence of weights w = (w 1,w 2,...) and define the w-score of a committee W as v V W A(v) j=1 w j, i.e., if voter v has x approved candidates in W, W receives a score of w 1 + w 2 + +w x from v. The committees with highest w-score are the winners according to the w-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 w AV = (1,1,1,...), the following three rules also fall into this class. Proportional Approval Voting (PAV). PAV was first proposed by Thiele [92]; it was later reinvented by Simmons [58], who introduced the name proportional approval voting. PAV is a Thiele method defined by the weights w = (1, 1 /2, 1 /3,...). These 13

14 weights being harmonic numbers guarantee proportionality in contrast to Multi- Winner Approval Voting, which is not a proportional method. This is a main result of our paper and is illustrated in the example below. Example 1. Consider a population with 100 voters; 75 voters approve of candidates c 1,...,c 4 and 30 voters of candidates c 5,...,c 8. Assume k = 4. For such a profile Multi-Winner Approval Voting selects a single winning committee W AV = {c 1,...c 4 }. The PAV-score of W AV is equal to 75(1+ 1 /2+ 1 /3+ 1 /4) = We can obtain a better committee by (proportionally) selecting three candidates from the set {c 1,...,c 4 } and one from the set {c 5,...,c 8 }; the PAV-score of such committees is equal to 75(1+ 1 /2+ 1 /3)+25 1 = 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[92]. It closely resembles the Chamberlin Courant rule [22], which was originally defined for ordinal preferences but easily can be adapted to the approval setting. The Approval Chamberlin Courant rule is a Thiele method defined by the weights w CC = (1,0,0,...). In other words, CC chooses committees so as to maximize the number of voters which have at least one approved candidate in the winning committee. Constant Threshold Methods. Fishburn and Pekeč [46] proposed Constant Threshold Methods as a class of ABC ranking rules similar to the Approval Chamberlin Courant rule. For a fixed threshold t with 1 t k, we define a sequence of weights w = (0,...,0,1,0,0,...)with theoneinthet-thposition. Asaconsequence, acommittee W receives a score of 1 from each voter with at least t approved candidates in the committee. Note that Constant Threshold Methods generalize CC as CC is the Constant Threshold Method with threshold t = 1. Another natural choice would be t = k /2 if the committee makes decisions based on majorities; voters with fewer than k /2 approved candidates in the committee have to fear that their representatives are overruled. Satisfaction Approval Voting is a natural example of an ABC counting rule which is not a Thiele method: Satisfaction Approval Voting (SAV). Brams and Kilgour [17] proposed SAV as a variation of AV that chooses committees representing more diverse interests. The difference to AV is that each voter has only one point and distributes it evenly among all approved candidates. Consequently, the SAV-score of a committee W is equal to W A(v) v V. As for Thiele methods, these scores define both an ABC A(v) ranking rule and an ABC choice rule. Note, however, that SAV is not a Thiele method since the number of approved candidates influences the score. While all previous rules can be viewed both as ABC ranking rules and ABC choice rules, the following two do not fit well into the framework of ABC ranking rules because they do not allow to compare non-winning committees. 14

15 Sequential Thiele Methods. Consider a sequence of weights w = (w 1,w 2,...). The sequential w-thiele method starts with an empty committee W = and works in k steps; in the i-th step, 1 i k, it adds to the committee W a candidate c which maximizes the w-score of committee W {c}. Reverse-Sequential Thiele Methods. Reverse-sequential Thiele methods are similar to sequential Thiele methods but start with the committee W = C and remove candidates iteratively until it has the desired size k. Let w = (w 1,w 2,...). In each step the method removes a candidate c from the committee W whose removal reduces the w-score of W the least. Note that AV, Sequential AV and Reverse-Sequential AV are the same rule; all three rules select k candidates with the largest number of approving votes. For all other Thiele methods this does not hold; their sequential and reverse-sequential variants are different from the respective original methods. We omit a few notable approval-based multi-winner rules such as Monroe s approvalbased rule [70], Minimax Approval Voting [18, 59], and those invented by Phragmén [20, 56, 75 77]; however, none of these rules are consistent and thus not immediately relevant for our study. For a general overview of approval-based multi-winner rules, we refer the readertoahandbookchapterbykilgour[58]andtoasurveybykilgourandmarshall[60]. 3 Axioms In this section, we provide and discuss formal definitions of the axioms used for our characterization results. All axioms will be phrased for ABC ranking rules as most of our results apply to these. In Appendix F, where we extend some of our results to ABC choice rules, we explain how these axioms should be formulated for ABC choice rules. Most of the axioms that we consider are natural and straightforward adaptations of the respective properties of single-winner election rules. Similar axioms have also been considered in the context of linear order-based multi-winner election rules, i.e., committee selection rules which take as input voters preferences expressed as rankings over candidates [30, 87]. 3.1 Basic Axioms We start by describing two properties, anonymity and neutrality, which perhaps enforce 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 F is anonymous if for V,V N such that V = V, for each bijection ρ : V V, and for A A(C,V) and A A(C,V ) such that A(v) = A (ρ(v)) for each v V, it holds that F(A) = F(A ). 15

16 Neutrality is similar to anonymity, but enforces equal treatment of candidates rather than of voters. Neutrality. An ABC ranking rule F is neutral if for each bijection σ: C C and A,A A(C,V) with σ(a) = A it holds for W 1,W 2 P k (C) that W 1 F(A) W 2 if and only if σ(w 1 ) F(A ) σ(w 2 ). Due to their analogous structure and similar interpretations, anonymity and neutrality are very often considered together, and jointly referred to as the symmetry (sometimes symmetry is also referred to as impartiality [71]). Symmetry. An ABC ranking rule is symmetric if it is anonymous and neutral. The next axiom, consistency, was first introduced in the context of single-winner rules by Smith [90] and then adapted by Young [97]. 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 [30], 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 F is consistent if for A,A A(C), and for W 1,W 2 P k (C), (i) if W 1 F(A) W 2 and W 1 F(A ) W 2, then W 1 F(A+A ) W 2, and (ii) if W 1 F(A) W 2 and W 1 F(A ) W 2, then W 1 F(A+A ) W 2. Next, we describe efficiency, which captures the intuition that voters prefer to have more approved candidates in the committee. Efficiency. An ABC ranking rule F satisfies efficiency if for W 1,W 2 P k (C) and A A(C,V) where for every vote v V we have A(v) W 1 A(v) W 2, it holds that W 1 F(A) W 2. For k = 1, i.e., in the single-winner setting, efficiency is the well-known Pareto efficiency axiom, which requires that if a candidate c is unanimously preferred to candidate d, then d c in the collective ranking [71]. 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. 16

17 Weak efficiency. An ABC ranking rule F satisfies weak efficiency if for each W 1,W 2 P k (C) and A A(C,V) where no voter approves a candidate in W 2 \W 1, it holds that W 1 F(A) W 2. If we consider the single-winner case here, we see that the axiom reduces to the following statement: if no voter approves candidate d, then any candidate c is at least as preferable as candidate d. The following lemma shows that efficiency in the context of neutral and consistent rules is implied by its weaker counterpart. Hence the following lemma allows us to use the efficiency axiom instead of weak efficiency and thus simplifies the technical discussion in further proofs. Lemma 1. A neutral and consistent ABC ranking rule that satisfies weak efficiency also satisfies efficiency. Our next axiom, continuity (also known in the literature as the Archimedean property[90] or the Overwhelming Majority axiom[72]) 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 [98] as it excludes specific tie-breaking mechanisms. Continuity. AnABC ranking rulef satisfies continuity if for each W 1,W 2 P k (C) and A,A A(C) where W 1 F(A ) W 2 there exists a positive integer n such that W 1 F(A+nA ) W Axioms Barring Forms of Strategic Voting Independence of irrelevant alternatives is one of the axioms used in Arrow s impossibility theorem [5]: it requires that the order of two candidates, as obtained from a social welfare function, does only depend on the relative order of these two candidates in voters individual preferences and not on any other candidate. This axiom can be considered an incentive for voters to truthfully reveal preferences, since a certain form of strategic voting is impossible (i.e., altering the position of a third candidate to influence the order of two candidates). Independence of irrelevant alternatives appears to be a very strong axiom, and is often perceived as the primary axiom leading to Arrow s impossibility theorem. With dichotomous preferences, the situation changes however: independence of irrelevant alternatives is satisfied by (single-winner) Approval Voting rule. Independence of irrelevant alternatives can be naturally extended to the multi-winner setting informally speaking, it requires that the relative order between committees W 1 and W 2 does not depend on candidates which do not belong to either of these two committees. Also in this setting, this property can be viewed as barring a certain form of strategic voting, i.e., rules satisfying independence of irrelevant alternatives are resistant 17