A Quantitative Analysis of Multi-Winner Rules
To choose a suitable multi-winner rule, i.e., a voting rule for selecting a subset of alternatives based on a collection of preferences, is a hard and ambiguous task. Depending on the context, it varies widely what constitutes the choice of an “optimal” subset. In this paper, we offer a new perspective to measure the quality of such subsets and—consequently—multi-winner rules. We provide a quantitative analysis using methods from the theory of approximation algorithms and estimate how well multi-winner rules approximate two extreme objectives: diversity as captured by the (Approval) Chamberlin–Courant rule and individual excellence as captured by Multi-winner Approval Voting. With both theoretical and experimental methods we classify multi-winner rules in terms of their quantitative alignment with these two opposing objectives.
A multi-winner rule is a voting method for selecting a fixed-size subset of alternatives, a so-called committee. More formally, it is a function that given a set of objects, preferences of a population of voters over these objects, and an integer , returns a subset of exactly objects. Multi-winner rules are applicable to problems from and beyond the political domain, for instance to selecting a representative body such as a parliament or university senate [13, 18], to shortlisting candidates (e.g., in a competition) , designing search engines [37, 15] and other recommendation systems , and as mechanisms for locating facilities .
Ideally, a multi-winner rule should select the “best” committee, but the suitability of a chosen committee strongly depends on the specific context. For instance, if voters are experts (e.g., judges in a sport competition) whose preferences reflect their estimates of the objective qualities of candidates, then the goal is typically to pick individually best candidates, e.g., those candidates who receive the highest scores from judges. Intuitively and somehow simplified, in this and similar scenarios the quality of candidates can be assessed separately, and a suitable multi-winner rule should pick the best-rated ones. On the contrary, if the voters are citizens and the goal is to choose locations for public facilities (say, hospitals), then our goal is very different: assessing the candidates separately can result in building all the facilities in one densely populated area; yet, it is preferable to spread them in order to ensure that as many citizens as possible have access to some facility in their vicinity.
These two examples illustrate two very different goals of multi-winner rules, which can be informally described as follows : Diversity requires that a rule should select a committee which represents as many voters as possible; this translates to choosing a hospital distribution that covers as many citizens as possible. Individual excellence suggests picking those candidates that individually receive the highest total support from the voters; this translates to selecting a group of best contestants in the previous example. However, many real-life scenarios do not fall clearly into one of the two categories. For example, rankings provided by a search engine should list the most relevant websites but also provide every user at least one helpful link. In such cases, a mechanism designer would be interested in choosing a rule that guarantees some degree of diversity and individual excellence at the same time, putting more emphasis on either of them depending on the particular context. Consequently, to properly match rules with specific applications, it is essential to understand to which degree committees chosen by established multi-winner rules are diverse or individually excellent. In this paper we (1) develop a set of tools that allow one to better understand the nature of multi-winner rules and to assess the tradeoffs between their diversity and individual excellence, and (2) provide a classification that clarifies the behavior of these rules with respect to the two criteria. We focus on the case where voters express their preferences by providing subsets of approved candidates (the approval-based model), yet our approach is applicable to other preference models as well.
1.1 Methodology and Contribution
In our approach we identify two multi-winner rules, the Chamberlin–Courant rule (CC) and Multi-winner Approval Voting (AV), as distinctive representatives of the principles of diversity and individual excellence, respectively. Next, we measure how close certain rules are to AV and CC—we measure this distance by using the concept of the worst-case approximation. Thus, by investigating how well certain rules approximate AV (resp. CC), we provide guarantees of how individually excellent (resp. diverse) these rules are. Such guarantees could be viewed as quantitative properties that measure the level of diversity and individual excellence of the studied rules. This is quite different from the traditional axiomatic approach to investigating properties of voting rules, which is qualitative: a rule can either satisfy a property or not. Our approach provides much more fine-grained information and allows us to estimate the degree to which a certain property is satisfied. With these methods, we understand voting rules as a compromise between different (often contradictory) goals.
Our main contribution lies in developing a new method for evaluating multi-winner rules. Specifically, we provide two types of analyses for a number of multi-winner rules:
In Section 3, we derive theoretical upper bounds on how much an outcome of the considered multi-winner rules can differ from the outcomes of CC and AV. We call these bounds CC-guarantee and AV-guarantee. These can be interpreted as worst-case (over all possible preference profiles) guarantees for diversity and individual excellence. Our guarantees are given as functions of the committee size and return values between 0 and 1. Intuitively, a higher CC-guarantee (resp. AV-guarantee) indicates a better performance in terms of diversity (resp. individual excellence), where 1 denotes that the rule performs as good as CC (resp., AV). Table 1 summarizes our results. We also prove bounds on how well proportional rules can approximate AV and CC.
In Section 5, we complement our results with an analysis of the axiom of efficiency, which can be viewed as an incarnation of Pareto efficiency, in the context of multi-winner elections. We say that a committee dominates a committee if each voter approves as many members of as of and some voter approves strictly more members of than of . Efficiency says that a rule should never select a dominated committee; thus efficiency could be viewed as a basic axiom for individual excellence. Since efficiency appears to be very fundamental, it may come as a surprise that many known rules (in particular, the Monroe rule and all sequential rules) do not satisfy this property. The result of this analysis is also summarized in Table 1.
Our most important findings can be summarized as follows. Proportional Approval Voting (PAV) achieves the best compromise between AV and CC; this can be observed both from theoretical and experimental results. The sequential rules seq-PAV and Phragmén’s rule, however, achieve almost the same quality while being polynomial-time computable (in contrast to PAV, which is computationally intractable [35, 4]). Also the 2-Geometric rule achieves a very good compromise, but is slightly leaning towards diversity. More generally, we show that the -Geometric rule spans the whole spectrum from AV to CC, controlled through the parameter . Hence, by adjusting the parameter , one can obtain any desired compromise between AV and CC.
1.2 Related Work
The normative study of multi-winner election rules typically focuses on axiomatic analysis. For approval-based rules a number of axioms describing proportionality have been recently identified and explored, in particular in the context of the rules that we study in this paper [1, 32, 10, 37, 3, 33, 22]. Similar properties for the ordinal model have been discussed by Dummett , Elkind et al. , Aziz et al.  and in the original works by Monroe  and Chamberlin and Courant ; and for the model with weak preferences by Baumeister et al. . For a survey on properties of multi-winner rules, with the focus on the ideas of individual excellence, diversity, and proportionality, we refer the reader to the book chapter by Faliszewski et al. .
Another approach to understanding the nature of different multi-winner rules is to analyze how these rules behave on certain subdomains of preferences, where their behavior is much easier to interpret, e.g., on two-dimensional geometric preferences , on party-list profiles , or on single-peaked and single-crossing domains . Other approaches include analyzing certain aspects of multi-winner rules in specifically-designed probabilistic models [23, 21, 34, 30], quantifying regret and distortion in utilitarian models , assessing their robustness , and evaluating them based on data collected from surveys [39, 31].
For each , we let . For a set , we write to denote the powerset of , i.e., the set of all subsets of . By we denote the set of all -element subsets of .
Let and be sets of candidates and voters, respectively. Voters reveal their preferences by indicating which candidates they like: by we denote the approval set of voter (that is, the set of candidates that approves of). For a candidate , by we denote the set of voters who approve . Given a set of candidates , we write to denote the set of voters who approve at least one candidate in , that is . We call the collection of approval sets , one per each voter, an approval profile. We use the symbol to represent the set of all possible approval profiles.
We call the elements of size- committees. Hereinafter, we will always use the symbol to represent the desired size of the committee to be elected. An approval-based committee rule (in short, an ABC rule) is a function that takes as an input an approval profile and an integer (the required committee size), and returns a set of size- committees.111Rules which for some profiles return multiple committees as tied winners are often called irresolute. In practice, one usually uses some tie-breaking mechanism to single out a winning committee. Below, we recall the definitions of ABC rules which are the objects of our study.
- Multi-winner Approval Voting (AV).
This rule selects candidates which are approved by most voters. More formally, for a profile the AV-score of committee is defined as , and AV selects committees that maximize .
- Approval Chamberlin–Courant (CC).
For a profile we define the CC-score of a committee as ; CC outputs . In words, CC aims at finding a committee such that as many voters as possible have their representatives in (a representative of a voter is a candidate she approves of). The CC rule was first mentioned by Thiele , and then introduced in a more general context by Chamberlin and Courant .
- Proportional Approval Voting (PAV).
This rule selects committees with the highest PAV-scores, defined as , where is the -th harmonic number, i.e., . By using the harmonic function , voters who already have more representatives in the committee get less voting power than those with fewer representatives. While using other concave functions instead of would give similar effects, the harmonic function is particularly well justified—it implies a number of appealing properties of the rule , and it allows one to view PAV as an extension of the famous d’Hondt method [11, 22].
This rule, introduced by Skowron et al. , can be described similarly to PAV. The difference is that it uses an exponentially decreasing function instead of the harmonic function to describe the relation between the voting power of individual voters and the number of their approved representatives in the committee. Formally, for a given parameter the -geometric rule assigns to each committee the score , and picks the committees with the highest scores. It is easy to see that the -geometric rule is simply AV.
- Sequential CC/AV/PAV/-Geometric.
For each rule , we define its sequential variant, denoted as seq-, as follows. We start with an empty solution and in each of the consecutive steps we add to a candidate that maximizes , i.e., the candidate that improves the committee’s score most. We break ties lexicographicly.
Monroe’s rule , similarly to CC, aims at maximizing the number of voters who are represented in the elected committee. The difference is that for calculating the score of a committee, Monroe additionally imposes that each candidate should be responsible for representing roughly the same number of voters. Formally, a Monroe assignment of the voters to a committee is a function such that each candidate is assigned roughly the same number of voters, i.e., that . Let be the set of all possible Monroe assignments to . The Monroe-score of is defined as ; the rule returns .
- Greedy Monroe .
This is a sequential variant of the Monroe’s rule. It proceeds in steps: In each step it selects a candidate and a group of or not-yet removed voters222To be precise, for , the first groups of voters to be removed have size and the remaining have size . so that is maximal; next candidate is added to the winning committee and the voters from are removed from the further consideration.
- Phragmén’s Sequential Rule (seq-Phragmén).
Perhaps the easiest way to define the family of Phragmén’s rules [27, 28, 29, 20, 10] is by describing them as load distribution procedures. We assume that each selected committee member is associated with one unit of load that needs to be distributed among those voters who approve (though it does not have to be distributed equally). Seq-Phragmén proceeds in steps. In each step it selects one candidate and distributes its load as follows: let denote the total load assigned to voter just before the -th step. In the -th step the rule selects a candidate and finds a load distribution that satisfies the following three conditions: (1) implies that , (2) (3) the maximum load assigned to a voter, , is minimized. The new total load assigned to a voter after the -th step is .
3 Worst-Case Guarantees of Multi-winner Rules
The Chamberlin–Courant Rule and Approval Voting represent two extreme points in the spectrum of multi-winner rules [11, 16, 18, 22]. Specifically, CC and AV are prime examples of rules aiming at diversity, and at individual excellence, respectively. For a detailed discussion on these two principles we refer the reader to the book chapter of Faliszewski et al. , but below we also include a simple example which illustrates the difference between AV and CC. In short, AV cares about selecting candidates who receive the highest total support from the population of voters, and CC cares mostly about representing the minorities in the elected committee.
Consider a profile where 30 voters approve candidates , 20 voters approve , and 5 voters approve . Let . For this profile AV selects candidates , while CC selects the committee (among others).
In this section we analyze the multi-winner rules from Section 2 with respect to how well they perform in terms of diversity, and individual excellence. In our study we use the established idea of approximation from computer science, but in a novel way: by estimating how well a given rule approximates CC (resp., AV), we quantify how performs with respect to diversity (resp., individual excellence). This differs from the typical use of the idea of approximation in the following aspects: (1) We do not seek new algorithms approximating a given objective function as well as possible, but rather analyze how well the existing known rules approximate given functions (even if it is apparent that better and simpler approximation algorithms exist, these algorithms might not share other important properties of the considered rules). (2) We are not approximating computationally hard rules with rules easier to compute. On contrary, we will be investigating how computationally hard rules (such as PAV, Monroe, etc.) approximate AV, which is easy to compute.
Recall that for a profile , and denote the AV-score and CC-score of committee , respectively. The AV-guarantee of an ABC rule is a function that takes as input an integer , representing the size of the committee, and is defined as:
Analogously, the CC-guarantee of is defined by
The AV and CC-guarantees can be viewed as quantitative properties of multi-winner rules. In comparison with the traditional qualitative approach (analyzing properties which can be either satisfied or not), a quantitative analysis provides much more fine-grained information regarding the behavior of a rule with respect to some normative criterion. In the remaining part of this section we evaluate the previously defined rules against their AV- and CC-guarantees.
3.1 Guarantees for CC and AV
Clearly, the AV-guarantee of Approval Voting and the CC-guarantee of the Chamberlin–Courant rule are the constant-one function. Below we establish the AV-guarantee of CC and vice versa.
The CC-guarantee of AV is .
Consider an approval profile , and let be an AV-winning committee for . We know that contains a candidate who is approved by most voters—let us call such a candidate . Clearly, it holds that . Further, for any size- committee we have that , which proves that the AV-guarantee of CC is at least . To see that the guarantee cannot be higher than consider a family of profiles where the set of voters can be divided into disjoint groups: , with and for , for some large value . Assume that and that all voters from approve candidates . For this profile AV selects committee with the CC-score equal to . The optimal CC committee is e.g., , with the CC-score equal to . ∎
The AV-guarantee of CC and sequential CC is .
Proposition 1 and Proposition 2 give a baseline for our further analysis. In particular, we would expect that “good” rules implementing a tradeoff between diversity and individual excellence, should have AV and CC-guarantees better than .
We conclude this section by noting that the CC-guarantee of the sequential Chamberlin–Courant rule is (which approaches for large ). This is the result of the fact that sequential CC is a -approximation algorithm for CC .
3.2 An Optimal Proportional Compromise
Next, we examine what are the possible AV- and CC-guarantees that a proportional rule could achieve. We consider a very weak definition of proportionality, called lower quota. This axiom is widely used  in the context of apportionment methods (which are special cases of approval-based multi-winner rules) and is strictly weaker than proportionality axioms typically used in the context of approval-based multiwinner rules (such as extended and proportional justified representation [32, 1]).
We call a profile a party-list profile if for each pair of voters it holds that either or that . For a given committee size we say that a group of voters is -cohesive, if and .
An ABC rule satisfies lower quota if for each party-list profile , each and each -cohesive group of voters it holds that at least members of each winning committee from are approved by the members of .
We obtain the following two upper bounds on the guarantees of proportional rules.
The AV-guarantee of a rule that satisfies lower quota is at most .
Let us fix , and consider the following approval-based profile with voters divided into equal-size groups: , with for each . All the voters from the first groups approve candidates denoted as . For each all the voters from approve a single candidate .
Let be a rule that satisfies lower quota. Let and denote the committees returned by and by AV, respectively. Lower quota ensures that for each . Thus,
On the other hand, one can observe that , and so . As a result we have:
The CC-guarantee of a rule that satisfies lower quota is at most .
3.3 Guarantees for Monroe and Greedy Monroe
Let us turn our attention to the Monroe rule and its greedy variant. Since Monroe is often considered a proportional rule, as it satisfies proportionality axioms such as proportional justified representation . Hence, one could expect that in terms of AV and CC-guarantees this rule is between AV and CC. Surprisingly, this is not the case and in fact it does not offer a better AV-guarantee than CC.
The AV-guarantee of Greedy Monroe and Monroe is .
The CC-guarantee of Monroe and greedy Monroe is between and .
3.4 Guarantees for PAV
Let us now move to multi-winner voting systems offering asymptotically better guarantees than the (greedy) Monroe rule. As we will see, the examination of such rules requires a more complex combinatorial analysis. We start with Proportional Approval Voting.
The AV-guarantee of PAV is between and .
First, we show that the AV-guarantee of PAV is at least equal to . Consider an approval profile and a PAV-winning committee ; let denote the number of voters who approve some member of . For each we set . Let be a committee with the highest AV-score. W.l.o.g., we can assume that . Now, consider a candidate with the highest AV-score, and let denote the number of voters who approve . If we replace a candidate with , the PAV-score of will change by:
Let us now compute the sum:
We know that for each we have , thus and . We now use the inequality between harmonic and arithmetic mean to get:
This can be reformulated as:
Now, let us consider two cases. If , then we observe that:
On the other hand, if , then:
In either case we have that . This yields the required lower bound.
The CC-guarantee of PAV is between and .
3.5 Guarantees for Sequential PAV
For sequential PAV we can prove qualitatively similar AV-guarantees to the ones for PAV.
The AV-guarantee of sequential PAV is between and .
Let us now discuss the CC-guarantee of sequential PAV. One can observe that the construction for PAV from Theorem 2 also works for sequential PAV, which shows that the CC-guarantee of seq-PAV is at most equal to . Proposition 7 below establishes a lower bound. In this case however, the gap between the lower and upper bounds on the CC-guarantee of the rule is large. Finding a more accurate estimate remains an interesting open question.
The CC-guarantee of sequential PAV is at least equal to .
3.6 Guarantees for -Geometric Rule
The following two theorems estimate the guarantees for the -geometric rule. These guarantees are visualized in Figure 1. We can see that -geometric rules, for , form a spectrum connecting AV and CC (with we approach AV and with we approach CC): by adjusting the parameter one can control the tradeoff between the diversity and individual excellence of the rule.
Let us recall that denotes the Lambert W function. For each it holds that . Intuitively, is a function that asymptotically increases slower than the natural logarithm .
The AV-guarantee of the -geometric rule is between:
The CC-guarantee of the -geometric rule is between and .
3.7 Guarantees for the Sequential Phragmén’s Rule
Finally we consider seq-Phragmén, another rule aimed at achieving proportionality of representation.
The AV-guarantee of seq-Phragmén is between and .
The next theorem shows that the CC-guarantee of seq-Phragmén is asymptotically equal to .
The CC-guarantee of seq-Phragmén is between and .
4 Average Guarantees: Experimental Analysis
To complement the theoretical analysis of Section 4, we have run experiments that aim at assessing AV-ratios and CC-ratios achieved by several voting rules. These two ratios are per-instance analogues of AV- and CC-guarantee and are defined as follows: Given a voting rule and a profile , the AV-ratio and the CC-ratio are defined as:
In these experiments, we have calculated the AV- and CC-ratios for real-world and randomly generated profiles and compared them for different voting rules. We have used two data sets: profiles obtained from preflib.org  and profiles generated via an uniform distribution (see details below).
We restricted our attention to profiles where both the AV-ratio of CC and the CC-ratio of AV is at most . This excludes profiles where an (almost) perfect compromise between AV and CC exists. The uniform dataset consists of 500 profiles with 20 candidates and 50 voters, each. Voters’ approval sets are of size – (chosen uniformly at random); the approval sets of a given size are also chosen uniformly at random. Experiments for the uniform dataset use a committee size of .
The preflib dataset is based on preferences obtained from preflib.org. Since their database does not contain approval-based datasets, we extracted approval profiles from ranked ballots as follows: for each ranked profile and , we generated an approval profile assuming that voters approve all candidates that are ranked in the top positions. As before, we excluded profiles that allowed an almost perfect compromise between AV and CC. For the preflib dataset we considered and obtained a total number of 243 instances.
We considered the following voting rules: AV, CC, seq-CC, PAV, seq-PAV, seq-Phragmén, Monroe’s rule, as well as the 1.5-, 2-, and 5-Geometric rule. Our results are displayed as boxplots in Figure 2 for the preflib dataset and in Figure 3 for the uniform dataset. The top and bottom of boxes represent the first and third quantiles, the middle red bar shows the median. The dashed intervals (whiskers) show the range of all values, i.e., the minimum and maximum AV- or CC-ratio. The results for the preflib and random dataset are largely similar; we comment on the differences later on.
The main conclusion from the experiments is that the classification obtained from worst-case analytical bounds also holds in our (average-case) experiments. PAV, seq-PAV, and seq-Phragmén perform very well with respect to the AV-ratio, beaten only by 1.5-Geometric and AV itself. This is mirrored by our theoretical results as only PAV, seq-PAV, and seq-Phragmén achieve a AV-guarantee. For the uniform dataset, however, seq-Phragmén has slightly lower AV-ratios, but still comparable to PAV and seq-PAV. Also the 2-Geometric rule achieves comparable AV-ratios. Even better AV-ratios are achieved only by 1.5-Geometric and—by definition—by AV.
Considering the CC-ratio, we see almost optimal performance of seq-CC, Monroe, and 5-Geometric, and good performance of PAV, seq-PAV, seq-Phragmén, and 2-Geometric. Minor variations within these groups seem to depend on the chosen dataset. We also observe that 5-Geometric is better than Monroe’s rule and seq-CC according to both criteria.
When looking at the three Geometric rules considered here, we see the transition from AV to CC as our theoretical findings predict (cf. Figure 1): 1.5-Geometric is close to AV, whereas 5-Geometric resembles CC; 2-Geometric performs very similarly to PAV, slightly favoring diversity over IE.
Our results indicate that PAV is the best compromise between AV and CC. Yet, seq-PAV, seq-Phragmén, and 2-Geometric achieve comparable ratios, and the former two are cheaper to compute.
5 A Pareto Efficiency Axiom
In this section, we provide a complementary axiomatic analysis concerning individual excellence. We formulate the axiom of efficiency, a form of Pareto efficiency with respect to the number of approved candidates in a committee. In other words, this axiom dictates that only committees can be chosen where a further improvement of the total AV-score implies that the AV-score of some individual voter is reduced. We analyze our rules with respect to this property, and, maybe surprisingly, show that many rules do not satisfy this basic axiom.
Consider a committee size , two committees and an approval profile . We say that dominates in a if for each voter we have that , and if there exists a voter such that .
An ABC rule satisfies efficiency if for each profile and each committee size there exists no committee that dominates each committee in .
We start aith the rather surprising observation that seq-Phragmén does not satisfy efficiency.
Consider the set of 36 voters, and five candidates, . By we denote the set of voters who approve . Assume that:
The sequential Phragmén’s rule will select first, second, and third, yet committee is dominated by . This example also works for the Open d’Hondt method, which can be viewed as another variant of the Phragmén’s rule .
We note that the violation of efficiency is not an artifact of the rule being sequential (and so, in some sense “suboptimal”). Indeed, consider the optimal Phragmén’s rule, which is the variant where the committee members and their associated load distributions are not chosen sequentially, but rather simultaneously in a single step. Similarly, as in the case of its sequential counterpart, the goal of the optimal Phragmén’s rule is to find a committee and an associated load distribution that minimizes the load of the voter with the highest load (for more details on this rule we refer the reader to the work of Brill et al. ). The following example shows that the optimal Phragmén’s rule does not satisfy efficiency. The same example shows that the Monroe rule does not satisfy efficiency.
Consider 24 voters, and four candidates, , with the following preferences:
The optimal Phragmén’s and the Monroe’s rule would select , which is dominated by .
Greedy Monroe, seq-CC, and seq-PAV do not satisfy efficiency either. Intuitively, this is due to their sequential nature.
Consider the following profile with 20 voters and 4 candidates, where:
For this profile and for the greedy Monroe rule first picks , who is approved by 10 voters, will remove these 10 voters, and will pick . However, committee is dominated by . The same example shows that seq-CC and seq-PAV do not satisfy efficiency.
All the remaining rules that we consider satisfy efficiency.
AV, CC, PAV, and -geometric satisfy efficiency.
6 Conclusion and Future Work
Our work demonstrates the flow of ideas from theoretical computer science to theoretical economics, in particular to social choice. We designed new tools that can be used to assess the level of diversity and individual excellence provided by certain rules. Our results help to understand the landscape of multiwinner rules, specifically how they behave with respect to two contradictory goals.
Our work can be extended in several directions. First, we have focused on approval-based multi-winner rules—a natural next step is to perform a similar analysis for multi-winner rules that take rankings over candidates as input. Second, we have excluded some interesting voting rules from our analysis, in particular reverse-sequential PAV  and Minimax Approval Voting ; it is unclear how they compare to rules considered in this paper. Finally, we have chosen AV and CC as extreme notions that represent diversity and individual excellence. Another natural approach would be to take a proportional rule (such as PAV) as a standard and see how well others rules approximate it.
Martin Lackner was supported by the European Research Council (ERC) under grant number 639945 (ACCORD) and by the Austrian Science Foundation FWF, grant P25518 and Y698. Piotr Skowron was supported by a Humboldt Research Fellowship for Postdoctoral Researchers.
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Appendix A Proofs Omitted from the Main Text
The AV-guarantee of CC and sequential CC is .
For an approval profile let and be committees winning according to CC and AV, respectively. We will first prove that . If it was not the case, then by the pigeonhole principle, there would exists a candidate such that . However, this means that a committee that consists of and any candidates has a higher CC-score than , a contradiction. Thus, the AV-guarantee of CC is at least . For seq-CC, the same argument by contradiction applies as this candidate would have been chosen in the first round.
To see that this guarantee cannot be higher than consider the following profile: assume there are voters ( is a large integer) who approve candidates . Further, for each candidate there is a single voter who approves only her. The CC-winning committee is with the AV-score of . However, the AV-score of committee is , and for large enough the ratio can be made arbitrarily close to . ∎
The CC-guarantee of a rule that satisfies lower quota is at most .
Let be a rule that satisfies lower quota. Consider a profile with voters for some . Each from the first voters approves candidates . The other voters are divided into equal-size groups, each approving a different candidate from the set . Lower quota ensures that at least candidates need to be chosen from . Thus, the CC-score of a committee selected by is at most equal to . By selecting one candidate from and candidates from we get a CC-score of . Thus, the CC-guarantee is at most equal to:
The AV-guarantee of Greedy Monroe and Monroe is .
First, let us consider the greedy Monroe rule. To see the lower bound of , let be an approval profile and let denote the candidate who is approved by most voters. For the sake of clarity we assume that divides ; the proof can be generalized to hold for arbitrary . Clearly, for any committee it holds that . If , then the greedy Monroe rule in the first step will select . Otherwise, it will select some candidate approved by at least voters, and will remove of them from . By a similar reasoning we can infer that in the second step the rule will pick a candidate who is approved by at least voters; and in general, that in the -th step the rule will pick the candidate who is approved by at least voters. As a result, we infer that number of voters that have at least one approved candidate in the chosen committee is at least
Hence the AV-guarantee of Greedy Monroe is at least .
To see that the same lower bound holds for the Monroe rule, we distinguish two cases; let be a winning committee. If , then and we are done. If and , then there is a committee with a higher Monroe-score that contains ; a contradiction.
Now, consider the following instance witnessing that the AV-guarantee of Greedy Monroe is at most . Let and let be a profile with voters. Let with and . We define profile as follows: we have voters that approve and one voter that approves only , we have voters that approve and one voter that approves only , etc. This defines in total voters. AV selects the committee with an AV-score of ; Greedy Monroe selects the committee with an AV-score of . We have a ratio of , which converges to for . The same instance shows that the AV-guarantee of the Monroe rule is at most . ∎
The CC-guarantee of Monroe and greedy Monroe is between and .
First, for the sake of contradiction let us assume that there exists a profile where the CC-guarantee of Greedy Monroe is below . Let and be the committees winning in according to CC and Greedy Monroe, respectively. Let be an assignment of the voters to the committee members obtained during the construction of ; we say that a voter is represented if it is assigned to a member of who she approves of. Since , by the pigeonhole principle we infer that there exists a candidate who is approved by unrepresented voters, where:
Similarly, by the pigeonhole principle we can infer that there exists a candidate who is represented by at most voters. Thus, Greedy Monroe would select rather than , a contradiction. A similar argument can be made to show that the CC-guarantee of the Monroe rule is .
Now, consider the following approval profile. There are candidates, , and disjoint equal-size groups of voters, . For each , candidate is approved by all voters from . Candidate is approved by all voters from . One of the winning committees according to the Monroe and Greedy Monroe rule is , which has a CC-score of . On the other hand, has a CC-score of . Thus, the CC-guarantee of Monroe and Greedy Monroe is at most:
This completes the proof. ∎
The CC-guarantee of PAV is between and .
We first prove a lower bound of for the CC-guarantee of PAV. Consider an approval-based profile and a PAV winning committee . Similarly as in the proof of Theorem 1, for each voter we set . Let be a committee winning according to the Chamberlin–Courant rule For each two candidates, and , let denote the change of the PAV-score of due to replacing with . By Inequality 1, we have:
Let us now consider an arbitrary bijection , matching members of with the members of . We compute the sum: