Phragmén's Voting Methods and Justified Representation
PPhragm ´en’s Voting Methodsand Justified Representation * Markus Brill † Svante Janson ‡ Rupert Freeman § Martin Lackner ¶ In the late 19th century, Swedish mathematician Lars Edvard Phragm´en proposeda load-balancing approach for selecting committees based on approval ballots. Weconsider three committee voting rules resulting from this approach: two optimiza-tion variants—one minimizing the maximal load and one minimizing the varianceof loads—and a sequential variant. We study Phragm´en’s methods from an ax-iomatic point of view, focusing on properties capturing proportional representation.We show that the sequential variant satisfies proportional justified representation ,which is a rare property for committee monotonic methods. Moreover, we showthat the optimization variants satisfy perfect representation . We also analyze thecomputational complexity of Phragm´en’s methods and provide mixed-integer pro-gramming based algorithms for computing them.
1. Introduction
While most of the social choice literature is focused on single-winner scenarios, recent yearshave witnessed an increasing interest in committee voting rules [e.g., 16, 15, 49, 13]. In thissetting, a fixed-size subset of alternatives has to be selected based on the preferences of a groupof agents (or voters). In this paper, we assume that the preferences of individual agents aregiven by approval ballots , specifying which alternatives are “approved” by the agents. For anoverview of research on approval-based committee elections, we refer to the recent survey byLackner and Skowron [23].A crucial issue in group decision making is (proportional) representation . Informally speak-ing, an outcome of a decision-making process is representative if it reflects the preferences of the * A preliminary version of this paper has appeared in the
Proceedings of the 31st AAAI Conference on ArtificialIntelligence (AAAI 2017) [8]. † M. Brill, TU Berlin, Germany, [email protected] ‡ S. Janson, Uppsala University, Sweden, [email protected] § R. Freeman, University of Virginia, USA, [email protected] ¶ M. Lackner, TU Wien, Austria, [email protected] a r X i v : . [ c s . G T ] F e b embers of the group. In the context of approval-based committee elections, reasoning aboutrepresentation is non-trivial. Since approval sets may overlap arbitrarily, there are many differ-ent ways in which the set of agents can be split into more or less “cohesive” subgroups. Whethera given subgroup has a justified claim to be represented in the committee depends on the size ofthe subgroup as well as on its level of cohesiveness.Aziz et al. [1] and S´anchez-Fern´andez et al. [45] have identified axiomatic properties captur-ing the intuitive notion that subgroups that are “large enough” and “cohesive enough” deserve tobe represented in the committee: justified representation (JR) , proportional justified representa-tion (PJR) , and extended justified representation (EJR) . While a number of standard committeevoting rules have been shown to satisfy the basic requirement of JR, it turns out that the moredemanding properties PJR and EJR are much harder to satisfy.In this paper, we consider committee voting rules that are due to Swedish mathematicianLars Edvard Phragm´en (we provide brief biographical information in Section 1.2). Phragm´enphrases committee elections as load balancing problems: Adding a candidate to the committeeincurs some load , and this load should be shared among the agents approving this candidate.Phragm´en suggests choosing committees in such a way that the corresponding load distributionsare as balanced as possible, and different ways of measuring balancedness result in differentoptimization objectives. This approach yields two optimization variants, one minimizing themaximal load and one minimizing the variance of loads, and one sequential variant, whichproceeds by greedily selecting candidates so as to keep the maximal load as small as possible.In addition to the load balancing rules, Phragm´en also proposed a rule that adapts the principlebehind STV to approval ballots (see Section 4.3).Although Phragm´en’s methods were proposed in the same era as Proportional Approval Vot-ing (PAV), they have received hardly any attention until very recently. Since the publication ofthe conference version of this paper [8] in 2017, Phragm´en’s methods became increasingly cen-tral in the analysis of approval-based multiwinner rules. In politics, variants of both Phragm´en’smethods and PAV have been used in Swedish parliamentary elections (for distribution of seatswithin parties), and a version of one of Phragm´en’s methods is still part of the election law,although in a minor role [20]. Further, Phragmen’s sequential method is used extensively inblockchain protocols as a way to guarantee a minimum level of voting power to minority inter-ests; we formalize much of the intuitive appeal of the rule in this paper.
After briefly reviewing related work in Section 2 and introducing some basic notation in Sec-tion 3, we formally define Phragm´en’s methods in Section 4. In Section 5, we analyze the com-putational complexity of Phragm´en’s methods and we provide algorithms for computing them.The algorithms for the optimization variants are based on mixed-integer linear and quadratic pro-gramming. In Section 6, we consider the representation axioms mentioned above. We show that Proportional Approval Voting is a prominent committee voting rule due to Danish polymath Thorvald N. Thiele[53]. For a detailed comparison between PAV and Phragm´en’s methods, we refer to the works of Janson [20] andPeters and Skowron [32]. Two notable studies that predate the conference version of this paper are a survey by Janson [19] (in Swedish) anda paper by Mora and Oliver [29] (in Catalan). perfect represen-tation (PR) , a further representation axiom introduced by S´anchez-Fern´andez et al. [45]. Thelatter result provides a contrast to PAV, which is known to violate PR.
Lars Edvard Phragm´en (1863–1937) was a Swedish mathematician, actuary and insurance exec-utive. He began his mathematical university studies in Uppsala in 1882, but transferred in 1883to Stockholm, where he became a student (and later confidant) of G¨osta Mittag-Leffler [52]. In1888, Edvard Phragm´en was appointed coeditor of Mittag-Leffler’s journal
Acta Mathematica ,where he immediately made an important contribution by finding an error in a paper by HenriPoincar´e on the three-body problem. The paper had been awarded a prize in a competition thatMittag-Leffler had persuaded King Oscar II to arrange, but Phragm´en found a serious mistakewhen the journal already had been printed; the copies that had been released were recalled anda new corrected version was printed.In 1892, Edvard Phragm´en became professor of Mathematics at Stockholm University. In1897, he additionally became actuary in a private insurance company. His interest in ActuarialScience and insurance companies appears to have grown in these years, as in 1904 he left hisprofessorship to become the first head of the Swedish Insurance Supervisory Authority. In 1908he became director of a private insurance company, which he remained until 1933. His involve-ment in mathematics is witnessed, e.g., by his attendance at the 1924 International MathematicalCongress in Toronto, where he was elected one of the vice-presidents of the International Math-ematical Union [11]. Phragm´en also continued to be an editor of Acta Mathematica until hisdeath in 1937.His best known mathematical work is the Phragm´en-Lindel¨of principle in complex analysis, ajoint work with the Finnish mathematician Ernst Lindel¨of [38]. His interest in election methodsis witnessed by his publications [33, 34, 35, 36, 37]. Moreover, he was a member of the RoyalCommission on a Proportional Election Method 1902–1903 and of a new Royal Commissionon the Proportional Election Method 1912–1913. For further information we refer the readerto a survey by Janson [20] and to the book by Stubhaug [52] (in particular for his relation withMittag-Leffler).
2. Related Work
Proportional representation is an important issue in committee voting (see the influential paperby Monroe [27] and the references therein) and methods ensuring representation often lead tointeresting computational problems [40, 41, 25, 5].The problem of choosing representative committees based on approval ballots can be seenas a generalisation of the classical apportionment problem [4]. The latter setting correspondsto the special case in which candidates are arranged into party lists and each voter chooses asingle list. In particular, every approval-based committee voting rule induces an apportionmentmethod [9]. Voting settings between apportionment and approval-based committee voting have3lso been studied recently [10].In the setting of approval-based committee voting [22, 23], Aziz et al. [1] proposed two repre-sentation axioms: justified representation (JR) and its strengthening extended justified represen-tation (EJR) . Later, S´anchez-Fern´andez et al. [45] observed that EJR is not compatible with whatthey call perfect representation (PR) and proposed an axiomatic property, proportional justifiedrepresentation (PJR) , that is compatible. EJR implies PJR, which in turn implies JR.Aziz et al. [1] and S´anchez-Fern´andez et al. [45] showed that most common committee votingrules fail EJR and PJR. A notable exception is Thiele’s PAV, which satisfies EJR (and thus PJR).Interestingly, variants of PAV based on different weight vectors fail both EJR and PJR (and evenweaker proportionality requirements [24]). Moreover, a greedy approximation algorithm forPAV known as sequential PAV or reweighted approval voting fails JR (and consequently PJRand EJR) [1].Computing the outcome of PAV is NP-hard [49, 2] and thus not feasible in polynomial timeunless P = NP. Therefore, it has remained an open question whether there exist polynomial-time computable rules satisfying EJR or PJR. S´anchez-Fern´andez et al. [45] have shown thatthe polynomial-time computable
Greedy Monroe rule satisfies PJR in the special case where thecommittee size divides the number of voters (but fails PJR in the general case). Phragm´en’ssequential rule, as we show in this paper, is polynomial-time computable and satisfies PJR.Recent work has established that even EJR can be guaranteed by a polynomial-time votingrule. This was first shown by Aziz et al. [3]. Later, Peters and Skowron [32] presented Rule X,which is also polynomial-time computable and satisfies EJR. Interestingly, Rule X is based onthe same principle as Phragm´en’s sequential rule and shares some of its desirable properties(such as laminar proportionality and priceability, as discussed in [32]).None of these rules, however, is committee monotonic, i.e., an increase in the committeesize may result in a completely different committee. In many settings, this property is highlydesirable (e.g., when generating rankings [50]), and thus Phragm´en’s sequential rule has gainedmuch attention in recent years. Phragm´en’s sequential rule also satisfies further monotonicityaxioms [20, 43].Finally, let us mention the maximin support method, introduced by S´anchez-Fern´andez et al.[46]. This rule is closely related to Phragm´en’s sequential rule and shares many of its axiomaticproperties. The optimization variant of the maximin support method coincides with with one ofthe optimization variants of Phragm´en’s methods, and yields an equivalent formulation of thelatter in terms of maximin support distributions [46].We refer the interested reader to the survey by Lackner and Skowron [23] for a more compre-hensive overview of the literature on approval-based committee voting.
3. Preliminaries
We consider a social choice setting with a finite set N = { , . . . , n } of voters and a finite set C of candidates . Throughout the paper we let m = | C | denote the number of candidates and An example showing that Rule X violates committee monotonicity can be found in the survey by Lackner andSkowron [23]. We are not aware of a formal proof that the rules by Aziz et al. [3] fail committee monotonicity,but the way they are defined makes this claim very plausible. = | N | the number of voters. The preferences of each voter i ∈ N are given by a subset A i ⊆ C , representing the subset of candidates that the voter approves of. We refer to the list A = ( A , . . . , A n ) as the preference profile . For a candidate c ∈ C , we let N c denote the set ofvoters approving c , i.e., N c = { i ∈ N : c ∈ A i } . To avoid trivialities, we assume that N c (cid:54) = ∅ for all c ∈ C .We want to select a subset consisting of exactly k candidates, for a given natural number k ≤ m . An approval-based multiwinner voting rule (henceforth simply rule ) maps an instance ( A, k ) to a subset S ⊆ C of size k , the committee . In general, there may be ties, and we thenallow the rule to yield several choices, so formally the rule is a map from instances to non-emptysets of committees.Finally, for a tuple of real numbers z = ( z , . . . , z n ) , we let z ( (cid:96) ) denote the (cid:96) -th largest elementin z .
4. Phragm ´en’s Methods
The main idea behind Phragm´en’s methods is to identify committees whose “support” is dis-tributed as evenly as possible among the electorate. Phragm´en used different formulations forexplaining his methods; we refer the reader to the survey by Janson [20] for an overview andmore details. In this paper, we adopt the formulation from the 1899 paper [37]. In this formu-lation, every candidate in the committee is thought of as incurring one unit of “ load ,” and theload incurred by candidate c needs to be distributed among the voters in N c . The goal is to finda committee of size k for which the corresponding load distribution is as balanced as possible.Formally, a load distribution is a two-dimensional array x = ( x i,c ) i ∈ N,c ∈ C satisfying thefollowing four conditions: ≤ x i,c ≤ for all i ∈ N and c ∈ C (1) x i,c = 0 if c / ∈ A i (2) (cid:88) i ∈ N (cid:88) c ∈ C x i,c = k (3) (cid:88) i ∈ N x i,c ∈ { , } for all c ∈ C (4)Here, x i,c corresponds to the load that voter i receives from candidate c . Condition (2) ensuresthat the load incurred by candidate c is distributed among voters in N c only, and Conditions (3)and (4) ensure that x corresponds to a size- k committee { c ∈ C : (cid:80) i ∈ N x i,c = 1 } .For a load distribution x , we let ¯ x i denote the total load of voter i ∈ N , i.e., ¯ x i = (cid:80) c ∈ C x i,c ,and we refer to (¯ x , . . . , ¯ x n ) as the vector of voter loads . Using this notation, Condition (3)reads (cid:80) i ∈ N ¯ x i = k . Note that Condition (3) implies that the average voter load is kn .There are different ways to measure how balanced a given load distribution is, each givingrise to a different optimization objective. One such objective is to minimize the maximal loadassigned to a voter, i.e., min x max i ∈ N ¯ x i . (This is equivalent to minimizing the maximal differ-ence between a voter load and the average voter load.) Obviously, the average voter load kn isa lower bound on the maximal voter load, and we call a load distribution x perfect if ¯ x i = kn i ∈ N . Another objective is to minimize the variance of voter loads, i.e., the sum ofsquared distances from the average voter load. Again, a perfect load distribution is optimal forthis objective.We further distinguish between “optimization” methods, where we solve a global optimizationproblem to find a load distribution optimizing the objective, and “sequential” methods, where weiteratively construct a load distribution, in each round greedily choosing a candidate optimizingthe objective at that iteration.In this paper, we consider three rules: the optimization methods leximax-Phragm´en and var-Phragm´en —minimizing the maximal voter load and the variance of voter loads, respectively—and the sequential method seq-Phragm´en , which greedily minimizes the maximal voter load.For completeness, we also consider the Enestr¨om-Phragm´en method (see Section 4.3).The method seq-Phragm´en was introduced by Phragm´en [34, 35, 36, 37], and it is the variantthat he proposed to be used in actual elections. Phragm´en defined this method as a generaliza-tion of D’Hondt’s apportionment method to the case without party lists: for every instance ofthe party-list setting, seq-Phragm´en and D’Hondt’s method coincide [35, 20, 9]. Optimizationvariants and the objective of minimizing the variance are discussed in the 1896 paper [36]. We start by defining the optimization variants. The first optimization variant selects committeescorresponding to load distributions minimizing the maximal voter load. In case that two ormore committees have the same (minimal) maximal load, we employ a specific way of breakingties. This is because it might be the case that for two load distributions x and y , although max i ∈ N ¯ x i = max i ∈ N ¯ y i , one load distribution is clearly preferable to the other. Example 1.
Let C = { a, b, c } , k = 2 , and A = ( { a } , { a } , { b } , { c } ) . Any committee of size2 contains either b or c , which are approved by only one voter each, so the maximum load is1 for all committees. However, the committees containing a represent three voters, while thecommittee { b, c } only represents two. In order to refine the set of winning committees, we compare two vectors of voter loadsaccording to the leximax ordering. Definition 1.
For y = ( y , . . . , y n ) and z = ( z , . . . , z n ) , y is leximax-smaller than z , denoted y ˙ < z , if there exists j ≤ n such that y ( j ) < z ( j ) and y ( i ) = z ( i ) for all i ≤ j − . We are now ready to define the first optimization variant. leximax-Phragm´en:
The rule leximax-Phragm´en selects all committees corresponding to loaddistributions x such that (¯ x , . . . , ¯ x n ) is leximax-optimal, i.e., minimal with respect to ˙ < .As we will see in Section 6.3, leximax tie-breaking is necessary in order to guarantee strongrepresentation properties.The second optimization variant is based on a different optimization objective. The leximax ordering is defined analogously to the more commonly used leximin ordering (see, e.g., Moulin [30],Definition 1.1). In the literature, the leximax ordering is referred to as “lexicographic minimax” by Ogryczak[31] and as “lexicographical” by Schmeidler [47]. bca b c
14 12 34 A = { a } A = { b } A = { b, c } A = { a, b, c } A = { d } abba d A = { a } A = { b } A = { b, c } A = { a, b, c } A = { d } Figure 1: Illustration of Example 2. The diagram on the left illustrates a load distribution mini-mizing the maximal voter load max i ∈ N ¯ x i , and the diagram on the right illustrates theunique load distribution minimizing (cid:80) i ∈ N ¯ x i . var-Phragm´en: The rule var-Phragm´en selects all committees corresponding to load distribu-tions minimizing (cid:80) i ∈ N ¯ x i .Minimizing (cid:80) i ∈ N ¯ x i indeed minimizes the variance of (¯ x , . . . , ¯ x n ) , as is well-known: Since n (cid:80) i ∈ N ¯ x i = kn , it holds that the variance of (¯ x , . . . , ¯ x n ) equals n (cid:88) i ∈ N (cid:18) ¯ x i − kn (cid:19) = 1 n (cid:88) i ∈ N (cid:18) ¯ x i − x i · kn + k n (cid:19) = 1 n (cid:88) i ∈ N ¯ x i − n · k · kn + 1 n · n · k n = 1 n (cid:88) i ∈ N ¯ x i − k n . When minimizing this expression, we can ignore multiplicative or additive constants ( n and k )and thus equivalently minimize (cid:80) i ∈ N ¯ x i .The following example demonstrates that the maximal voter load under var-Phragm´en mayindeed be greater than under leximax-Phragm´en. Example 2.
Let C = { a, b, c, d } , k = 3 , and consider the preference profile A = ( { a } , { b } , { b, c } , { a, b, c } , { d } ) . For this instance, leximax-Phragm´en selects the committee { a, b, c } andvar-Phragm´en selects the committee { a, b, d } . Optimal load distributions corresponding to thesecommittees are illustrated in Figure 1. Load distributions minimizing the maximal voter load(like the one illustrated by the first diagram in Figure 1) satisfy max i ∈ N ¯ x i = and (cid:80) i ∈ N ¯ x i =4( ) = , and the load distribution minimizing the variance of voter loads (illustrated by thesecond diagram in Figure 1) satisfies max i ∈ N ¯ x i = 1 and (cid:80) i ∈ N ¯ x i = 4( ) + 1 = 2 . Rather than minimizing the maximum load, one could also aim to maximize the minimal voterload . This variant would select committees minimizing the number of unrepresented voters, evenin the face of large cohesive groups of voters. Therefore, this method will not do well in terms ofthe representation axioms considered in Section 6. For this reason, we do not consider it furtherin this paper. 7 .2. Sequential Method
We now introduce the sequential method, which can be seen as a greedy algorithm for minimiz-ing the maximal voter load. seq-Phragm´en:
The rule seq-Phragm´en starts with an empty committee and iteratively addscandidates, always choosing the candidate that minimizes the (new) maximal voter load. Let ¯ x ( j ) i denote the voter loads after round j . At first, all voters have a load of , i.e., ¯ x (0) i = 0 forall i ∈ N . As a first candidate we select one that is supported by most voters as it is the onethat increases the maximal load the least. In the next round, we again choose a candidate thatinduces a (new) maximal voter load that is as small as possible, but now we have to take intoaccount that some voters already have a non-zero load. The new maximal load if c is chosen asthe ( j + 1) -st committee member is calculated as s ( j +1) c = 1 + (cid:80) i ∈ N c ¯ x ( j ) i | N c | . (5)This is because we distribute the load of among all voters in N c in such a way that all thesevoters have the same voter load afterwards. Let c be the candidate that minimizes s ( j +1) c amongthose that are not yet in the committee. (If there are several candidates minimizing s ( j +1) c , weuse a fixed tie-breaking rule to decide which candidate to add.) Then we add c to the committeeand set ¯ x ( j +1) i = (cid:40) s ( j +1) c if i ∈ N c ¯ x ( j ) i otherwise. (6)It follows that (cid:80) i ∈ N ¯ x ( j +1) i = j + 1 . After k iterations, we have obtained a load distributionand a committee.The definitions ensure that voter loads never decrease, i.e., ¯ x ( j +1) i ≥ ¯ x ( j ) i for all i ∈ N andall j < k . This is because a candidate minimizing the new maximal load is selected in eachround. If the selection of candidate c in round j + 1 led to a load distribution x ( j +1) with s ( j +1) c = ¯ x ( j +1) i < ¯ x ( j ) i for some i ∈ N c , then candidate c would have been selected in an earlierround, a contradiction. (See also Lemma 5 (i) .)Phragm´en [37] illustrates his sequential method by imagining the different ballots as repre-sented by cylindrical vessels, with base area proportional to the number of voters casting thatballot. The already elected candidates are represented by a liquid that is fixed in the vessels, andthe additional unit of load incurred by adding another candidate to the committee is representedby pouring 1 unit of a liquid into the vessels representing voters approving this candidate. Theliquid then distributes among these vessels such that the height of the liquid is the same in allvessels. This is to be tried for each candidate; the candidate that requires the smallest height iselected, and the corresponding amounts of liquid are added to the vessels and fixed there.Phragm´en’s sequential method is committee monotonic by definition. As mentioned above,seq-Phragm´en can be seen as a (polynomial-time computable) heuristic to approximate the op-timization method leximax-Phragm´en. Unsurprisingly, the load distribution constructed by seq-8 bb cb a c
13 23 A = { a } A = { b } A = { b, c } A = { a, b, c } A = { d } abbb ad
13 23 A = { a } A = { b } A = { b, c } A = { a, b, c } A = { d } Figure 2: Illustration of Example 3. The diagram on the left (respectively, right) illustrates theload distributions obtained by seq-Phragm´en with ties broken in favor of candidate c (respectively, d ).Phragm´en might not be optimally balanced. Example 3.
Consider again the instance from Example 2. We have s (1) b = , s (1) a = s (1) c = ,and s (1) d = 1 . Therefore, candidate b is chosen in the first round. In the second round, wehave s (2) a = , s (2) c = , and s (2) d = 1 , so candidate a is chosen. In the third round, there isa tie between c and d because s (3) c = s (3) d = 1 . Thus, the final committee is either { a, b, c } or { a, b, d } , depending on which tie-breaking method is used. Figure 2 illustrates the resultingload distributions, both of which are suboptimal for the optimization problems corresponding toleximax-Phragm´en and var-Phragm´en. One can also define a sequential version of var-Phragm´en, by in each iteration selecting acandidate minimizing the variance of the resulting load distribution [28]. This variant does notfare well in terms of the representation axioms considered in Section 6, and we therefore do notconsider it any further.
In addition to the methods described in the previous sections, there is another rule that is at-tributed, at least partially, to Phragm´en. Following Camps et al. [12], we refer to this methodas
Enestr¨om-Phragm´en .The method predates the load balancing methods and is similar in spirit to single transferablevote (STV) methods [54]. It uses a quota q , which is defined either as the Hare quota q H = nk or as the Droop quota q D = nk +1 . The choice between q H and q D does not affect the axiomaticperformance of the rule with respect to the properties studied in this paper (see Table 1). Whileseq-Phragm´en and Enestr¨om-Phragm´en do not differ with respect to the representation prop-erties studied in Section 6, a crucial difference is that Enestr¨om-Phragm´en is not committeemonotonic [12]. The approximability of leximax-Phragm´en has recently been studied by Cevallos and Stewart [14], who showed,in particular, that seq-Phragm´en does not offer a constant-factor approximation guarantee. For details on the origin of this rule, see footnote 38 in the paper by Janson [20], who refers to this method as
Phragm´en’s first method . S´anchez-Fern´andez et al. [44] refer to the method as
Phragm´en-STV . nestr¨om-Phragm´en: Initially, all voters have a voting weight of . Each ballot is countedfully, with its present voting weight, for each unelected candidate on the ballot. In each round,a candidate with maximal weighted approval score is chosen and the voting weight of votersapproving this candidate are reduced: If the maximal weighted approval score v is strictly greaterthan the quota (i.e., v > q ), then each of these ballots has its voting power multiplied by v − qv ;if v ≤ q , then these ballots all get voting power (and are thus ignored in the sequel). This isrepeated until the desired number of candidates are elected.Note that the total voting weight of all voters is decreased by ( q/v ) · v = q each time, as longas some candidate reaches the quota. This rule has been extensively analyzed by Camps et al.[12] (mostly using q D ). Independently, it has been studied by S´anchez-Fern´andez et al. [44](using q H ). In the following example, we use q H . Example 4.
Consider again the instance from Example 2. We have q H = nk = . In the firstround, candidate b is chosen with a (weighted) approval score of . Since > q H , the votingpower of the three voters approving b is multiplied by − q H = . In the second round, theweighted approval scores of the remaining candidates are = for a , + = for c ,and for d . Therefore, candidate a is chosen. Since ≤ q H , both voters approving a havetheir voting power reduced to . In the third and final round, the weighted approval score of c is and candidate d is chosen with a weighted approval score of .
5. Computational Aspects
In this section, we study the computational complexity of Phragm´en’s methods, and we providealgorithms for finding winning committees. S´anchez-Fern´andez et al. [45] have shown that everyrule satisfying perfect representation (see Section 6) is NP-hard; this essentially follows fromearlier work by Procaccia et al. [41]. Since we show that leximax-Phragm´en and var-Phragm´enboth satisfy this condition (Theorems 8 and 11), it follows that there do not exist polynomial-time algorithms for computing a committee for either of these rules, unless P = NP.We complement these hardness results by considering two basic decision problems.
LEXIMAX -P HRAGM ´ EN asks whether an instance allows a load distribution x such that (¯ x , . . . , ¯ x n ) ˙ < ( y , . . . , y n ) for some given n -tuple ( y , . . . , y n ) . And VAR -P HRAGM ´ EN asks whether an in-stance allows a load distribution x such that (cid:80) i ∈ N ¯ x i < α for some given threshold value α > . Both problems can be interpreted as asking whether a given load distribution is optimal.We show that both problems are NP-complete even for rather restricted instances. For a pref-erence profile A , let s ( A ) denote the maximum number of candidates a voter approves, and let d ( A ) denote the maximum number of voters that approve a candidate. Theorem 1.
LEXIMAX -P HRAGM ´ EN , and VAR -P HRAGM ´ EN are NP-complete, even restricted toinstances with s ( A ) = 2 and d ( A ) = 3 .Proof. To show hardness for both problems, we reduce from the NP-complete Independent Setproblem on cubic graphs [18, 17], which is defined as follows: given a cubic graph ( V, E ) (i.e.,a graph such that every vertex has degree 3) and a positive integer k , is there a set of vertices S ⊆ V with | S | = k such that | e ∩ S | ≤ for all edges e ∈ E ? Let E = ( e , . . . , e n ) . We10onstruct an instance of LEXIMAX -P HRAGM ´ EN and VAR -P HRAGM ´ EN by identifying candidateswith vertices ( C = V ) and voters with edges, i.e., A = ( e , . . . , e n ) . It is easy to see that s ( A ) = 2 and d ( A ) = 3 . Without loss of generality we assume that n ≥ k because cubicgraphs with fewer than k edges cannot have an independent set of size k (two vertices in everyset of size k share an edge).To prove that LEXIMAX -P HRAGM ´ EN is NP-hard, we claim that ( V, E ) has an independentset of size k if and only if there exists a load distribution x with (¯ x , . . . , ¯ x n ) ˙ < ( y , . . . , y n ) ,where ( y , . . . , y n ) is the sequence containing k entries of + k followed by zeros. If S is an independent set, then S , viewed as a committee, contains candidates that are approvedby disjoint sets of (three) voters. Hence, there are exactly k voters that bear a load of ; allothers have load . Conversely, let S be a committee so that (¯ x , . . . , ¯ x n ) ˙ < ( y , . . . , y n ) . Sincecandidates are approved by three voters, if there exists a voter with more than one approvedcandidate in S , then the average load (and thus the maximal load) is at least k k − > + k ,which contradicts our assumption that (¯ x , . . . , ¯ x n ) ˙ < ( y , . . . , y n ) . Hence S is an independentset.To prove that VAR -P HRAGM ´ EN is NP-hard, we claim that ( V, E ) has an independent set ofsize k if and only if there exists a load distribution x with (cid:80) i ∈ N ¯ x i < k + . It is straightforwardto see that an independent set S corresponds to a committee with (cid:80) i ∈ N ¯ x i = 3 k · = k . Forthe other direction, let S be a committee with (cid:80) i ∈ N ¯ x i < k + . Note that at most k votershave approved candidates in the committee. Let N (cid:48) ⊆ N be such that it contains all voters i with ¯ x i > . Hence (cid:80) i ∈ N ¯ x i = (cid:80) i ∈ N (cid:48) ¯ x i . The value of (cid:80) i ∈ N (cid:48) ¯ x i is minimal only if all x i , i ∈ N (cid:48) , are equal and we then have (cid:80) i ∈ N (cid:48) ¯ x i = | N (cid:48) | · k | N (cid:48) | = k | N (cid:48) | . If | N (cid:48) | < k , we thus seethat (cid:80) i ∈ N (cid:48) ¯ x i ≥ k k − > k + . Hence | N (cid:48) | = 3 k and we can conclude that S corresponds toan independent set.It remains to be shown that LEXIMAX -P HRAGM ´ EN and VAR -P HRAGM ´ EN are contained inNP. This is achieved for LEXIMAX -P HRAGM ´ EN by encoding it as a mixed-integer linear pro-gram (see the discussion following this proof). Solving a mixed-integer linear program (i.e.,its corresponding decision problem) is known to be NP-complete [see, e.g., 48]. For showingNP-membership of VAR -P HRAGM ´ EN , we proceed in a similar fashion: we encode it as a mixed-integer quadratic program (see Theorem 3). NP-membership then follows from a result by Piaet al. [39].We now turn to algorithms for computing Phragm´en’s methods. First, we show how theoutcome of leximax-Phragm´en can be computed with the help of mixed-integer linear pro-grams (MILPs). We start by formulating a MILP that solves the decision problem LEXIMAX -P HRAGM ´ EN . We use variables x i,c (for i ∈ N , c ∈ C ), e i,j (for i, j ∈ N ), s i (for i ∈ N ), t j (for j ∈ N ), and (cid:15) . For a given n -tuple y = ( y , . . . , y n ) of real numbers, let P ( y ) be the MILP that11aximizes (cid:15) under the constraints (1)–(4) and (7)–(14). e i,j ∈ { , } for all i, j ∈ N (7) s i ∈ { , } for all i ∈ N (8) t j ∈ { , } for all j ∈ N (9) s i + (cid:88) j ∈ N e i,j = 1 for all i ∈ N (10) t j + (cid:88) i ∈ N e i,j ≤ for all j ∈ N (11) (cid:88) j ∈ N t j = 1 (12) ¯ x i − k (1 − e i,j ) ≤ y j for all i, j ∈ N (13) ¯ x i − k (2 − s i − t j ) ≤ y j − (cid:15) for all i, j ∈ N (14)The main idea of this MILP is as follows: The variables e i,j encode a partial bijection π froma subset of N to a subset of N ; the variables s i encode the subset S ⊆ N where π is not defined;and the variables t j encode t ∈ N , an index of an element in { y j : j / ∈ range ( π ) } . Constraint(10) encodes the relation between π and S : for every i ∈ N , either s i = 1 or e i,j = 1 for some j ∈ N . In a similar fashion, constraint (11) encodes the relation between π and t : for every i ∈ N , t i = 1 only if e i,j = 0 for all j ∈ N . Together with constraint (12), we enforce that thereexists exactly one j ∈ N such that t j = 1 . Hence at least one voter has a load strictly smallerthan y t and (¯ x , . . . , ¯ x n ) ˙ < ( y , . . . , y n ) .The final two constraints ensure that indeed (¯ x , . . . , ¯ x n ) ˙ < ( y , . . . , y n ) . From constraint (13)it follows that ¯ x i ≤ y j whenever π ( i ) = j . This is because if e i,j = 0 (i.e., π ( i ) (cid:54) = j ),constraint (13) reduces to ¯ x i − k ≤ y j , which is trivially satisfied because every load distribution x satisfies ¯ x i ≤ k for all i ∈ N . If e i,j = 1 (i.e., π ( i ) = j ), however, constraint (13) reads ¯ x i ≤ y j . Similarly, constraint (14) enforces that x i ≤ y t − (cid:15) ≤ max j ∈ N \ range ( π ) y j − (cid:15) for i ∈ S . As wemaximize (cid:15) , we look for a solution where x i < max j ∈ N \ range ( π ) y j . We conclude that a feasiblesolution with objective function value (cid:15) > encodes a load distribution x with (¯ x , . . . , ¯ x n ) ˙ < ( y , . . . , y n ) . Observe that P ( y ) solves the LEXIMAX -P HRAGM ´ EN decision problem: givenvoter loads y , P ( y ) returns (cid:15) > if and only if LEXIMAX -P HRAGM ´ EN with input y is a Yes-instance.We now present a MILP-based algorithm that computes the outcome of leximax-Phragm´en.Our algorithm solves a sequence of at most n instantiations of the MILP P , using the optimalsolutions of previously solved instances as constraints for subsequent calls. We assume that P returns the load distribution x and the objective function value (cid:15) . For an overview of theprocedure, see Algorithm 1.We start with y = ( k, , . . . , , an n -tuple consisting of one k and n − zeroes. We employ P to find a strictly better solution. The only entry of y that can be improved is y (1) = k and hencethe solution x returned by P minimizes the largest load; let ¯ x (1) be the largest load and ¯ x (2) thesecond-largest. We repeat this procedure with y = (¯ x (1) , ¯ x (2) , , . . . , . We already know that ¯ x (1) is optimal and cannot be further decreased (and 0 cannot be improved), hence the next P lgorithm 1: Computing leximax-Phragm´en y ← ( k, , . . . , for (cid:96) = 1 . . . n do x, (cid:15) ← P ( y )¯ x ← (¯ x , . . . , ¯ x n ) // ¯ x (1) , . . . , ¯ x ( (cid:96) ) optimal if (cid:15) = 0 then // no improvement x (cid:48) , (cid:15) (cid:48) ← P (¯ x ) if (cid:15) (cid:48) = 0 then // ¯ x optimal return { c ∈ C : (cid:80) i ∈ N x i,c = 1 } y ← (¯ x (1) , . . . , ¯ x ( (cid:96) +1) , , . . . , return { c ∈ C : (cid:80) i ∈ N x i,c = 1 } instance minimizes the second-largest load. We iterate this process and in step (cid:96) guarantee thatthe (cid:96) -th largest load is optimal. If at some point P returns (cid:15) = 0 , we verify whether the currentsolution is optimal: if P (¯ x ) also returns (cid:15) = 0 , the load distribution x is indeed optimal and thealgorithm terminates. In any case Algorithm 1 returns { c ∈ C : (cid:80) i ∈ N x i,c = 1 } , the committeecorresponding to the load distribution x .We have therefore proven the following result. Theorem 2. leximax-Phragm´en can be computed by solving at most n mixed-integer linearprograms with O ( nm + n ) variables. To compute var-Phragm´en, we solve a mixed-integer quadratic program, i.e., a program con-sisting of linear constraints and a quadratic optimization statement.
Theorem 3. var-Phragm´en can be computed by solving one mixed-integer quadratic programwith O ( nm ) variables.Proof. Our MIQP uses the variables x i,c (for i ∈ N , c ∈ C ) and the constraints (1)–(4). Thequadratic optimization statement is min (cid:88) i ∈ N (cid:32)(cid:88) c ∈ C x i,c (cid:33) .Since minimizing (cid:80) i ∈ N ¯ x i minimizes the variance (see Section 4.1), this MIQP computes loaddistributions corresponding to var-Phragm´en committees.Finally, we study the runtime for computing seq-Phragm´en. A naive estimate is that seq-Phragm´en can be computed in O ( kmn ) time. This estimate ignores the cost of computingthe quantities s ( j ) c , i.e., numerical operations are assumed to require constant time. While thisis a sensible assumption in many cases, here it is questionable since computing s ( j ) c exactly13equires fractions with large numerators and denominators. Indeed, the denominator of s ( j ) c cangrow exponentially with j . Hence, the following theorem also takes the complexity of theseoperations into account. Theorem 4.
The output of seq-Phragm´en can be computed in O ( k mn (log n ) ) time.Proof. In the following analysis we also consider the complexity of arithmetic operations in thealgorithms, as exact numerical computation of the involved quantities may require numbers ofsubstantial size. Let us consider the procedure described in Section 4.2. In each of the k rounds,one candidate is chosen. For this, the quantity s ( j ) c is computed for every c not yet placed in thecommittee. To ensure correct results, we represent s ( j ) c as fractions, i.e., pairs of integers. Let { c , . . . , c j − } be the first j − chosen candidates. It is easy to see that the denominator of s ( j ) c can be bounded by | N c | · . . . · | N c j − | · | N c | ≤ n j ≤ n k , assuming we reduce fractions.Furthermore, since s ( j ) c ≤ k , the numerator of s ( j ) c is at most kn k . Hence, the space required tostore s ( j ) c is bounded by O ( k log n ) . The necessary computations for calculating s ( j ) c (addition,division, reducing fractions) can all be performed in O ( b ) time, where b is the number of bitsrequired to store any of s ( j − c , and O ( n ) such operations are required. Since b = O ( k log n ) ,we conclude that s ( j ) c can be computed in O ( nk (log n ) ) time. This has to be done in each ofthe k rounds for at most | C | = m many candidates c ∈ C . The consequent update of x ( j ) i doesnot increase the runtime bound further.
6. Phragm ´en’s Methods and Representation
In this section, we study which representation axioms are satisfied by Phragm´en’s methods. Ourresults are summarized in Table 1. Particularly noteworthy are the results that seq-Phragm´ensatisfies PJR and that leximax-Phragm´en and var-Phragm´en satisfy PR. For completeness, thetable also contains results obtained by S´anchez-Fern´andez et al. [44] and Camps et al. [12]regarding Enestr¨om-Phragm´en.
We start by stating the definitions of Aziz et al. [1] and S´anchez-Fern´andez et al. [45].
Definition 2.
A committee S ⊆ C with | S | = k provides• justified representation (JR) if there does not exist a set N ∗ ⊆ N of voters with | N ∗ | ≥ nk , | (cid:84) i ∈ N ∗ A i | ≥ and | S ∩ A i | = 0 for all i ∈ N ∗ .• proportional justified representation (PJR) if there does not exist an integer (cid:96) > and aset N ∗ ⊆ N of voters with | N ∗ | ≥ (cid:96) nk , | (cid:84) i ∈ N ∗ A i | ≥ (cid:96) and | S ∩ ( (cid:83) i ∈ N ∗ A i ) | < (cid:96) .• extended justified representation (EJR) if there does not exist an integer (cid:96) > and a set N ∗ ⊆ N of voters with | N ∗ | ≥ (cid:96) nk , | (cid:84) i ∈ N ∗ A i | ≥ (cid:96) and | S ∩ A i | < (cid:96) for all i ∈ N ∗ . This quadratic bound is a very rough estimate and does not use any of the more sophisticated methods for multi-plication such as the Sch¨onhage–Strassen algorithm [51] or computing greatest common divisors [26, 7].
14R PJR EJR PRseq-Phragm´en (cid:88) (Corollary 7) (cid:88) (Theorem 6) – (Example 6) – (Example 5)leximax-Phragm´en (cid:88) (Corollary 10) (cid:88) (Theorem 9) – (Example 5) (cid:88) (Theorem 8)var-Phragm´en (cid:88) (Theorem 12) – (Example 7) – (Example 5) (cid:88) (Theorem 11)Enestr¨om-Phragm´en (cid:88) [44, 12] (cid:88) [44, 12] – [44, 12] – (Example 5)Table 1: Phragm´en’s methods and representation axioms
A rule f satisfies JR (respectively, PJR or EJR) if, for every instance ( A, k ) , every committee S ∈ f ( A, k ) provides JR (respectively, PJR or EJR). It follows immediately from the definitions that a rule satisfying EJR also satisfies PJR, andthat a rule satisfying PJR also satisfies JR.The following definition is due to S´anchez-Fern´andez et al. [45].
Definition 3.
Consider an instance ( A, k ) so that k divides n = | N | . A committee S = { c , . . . , c k } ⊆ C provides perfect representation if there exists a partition of the set N of votersinto k pairwise disjoint subsets N , . . . , N k such that, for all j ∈ { , . . . , k } , | N j | = nk and c j ∈ (cid:84) i ∈ N j A i . Let PR ( A, k ) denote the set of all committees providing perfect representationfor the instance ( A, k ) . A rule f satisfies perfect representation (PR) if, for every instance ( A, k ) where k divides n and PR ( A, k ) (cid:54) = ∅ , we have f ( A, k ) ⊆ PR ( A, k ) . The following example, which also appears in the papers by Aziz et al. [1] and S´anchez-Fern´andez et al. [45], illustrates the requirements of the different axioms.
Example 5.
Let C = { a, b, c, d, e, f } and consider the -voter preference profile given by A = { a } , A = { b } , A = { c } , A = { d } , A = { a, e, f } , A = { b, e, f } , A = { c, e, f } , A = { d, e, f } . Let k = 4 and assume that ties are broken alphabetically. Then, seq-Phragm´enchooses e , f , a , and b (in this order). The final loads are (¯ x , . . . , ¯ x ) = ( , , , , , , , ) .This is indeed not optimal as there is a perfect load distribution y with ¯ y i = for all i ∈ N . Thecorresponding committee { a, b, c, d } is selected by both leximax-Phragm´en and var-Phragm´en.Consider the group of voters N ∗ = { , , , } , of size (cid:96) nk = 2 = 4 , where (cid:96) = 2 . Sincethe voters all approve candidates e and f , a set of size (cid:96) = 2 , the conditions for JR, PJR, andEJR all bind. JR requires that at least one candidate approved by at least one voter in N ∗ ischosen. PJR requires that at least candidates are chosen that are each supported by at leastone voter from N ∗ , while EJR requires that some voter from N ∗ is represented twice. Thus,EJR dictates that either e or f is chosen. On the other hand, the only committee providingPR is { a, b, c, d } . As a consequence, no rule can satisfy both PR and EJR. Note that leximax-Phragm´en and var-Phragm´en both violate EJR in this example, and that seq-Phragm´en violatesPR. Enestr¨om-Phragm´en also yields { e, f, a, b } , and thus violates PR. The incompatibility of PR and EJR was first observed by S´anchez-Fern´andez et al. [45].15 .2. Results for seq-Phragm ´en
In this section we establish our main result: seq-Phragm´en satisfies proportional justified repre-sentation.We need the following notation. For the committee S that is selected by seq-Phragm´en (usinga fixed tie-breaking rule), we can relabel the candidates such that S = { c . . . , c k } and candidate c j was chosen in round j . Then, we have c j = arg min c ∈ C \{ c ,...,c j − } s ( j ) c . Using this conven-tion, we define s ( j ) = s ( j ) c j . That is, s ( j ) is the new load of all voters in N c j after candidate c j isadded to the committee in round j . We call ( s (1) , . . . , s ( k ) ) the max-load sequence . (Note thatdifferent tie-breaking rules can lead to different max-load sequences.)The following lemma has two parts. The first part states that the max-load sequence is mono-tonically increasing. The second part states that, when computing the optimal distribution of theload of a candidate c among its voters, it never helps to restrict attention to a subset N (cid:48) ⊂ N c . Lemma 5.
Fix an instance ( A, k ) .(i) The max-load sequence satisfies s (1) ≤ . . . ≤ s ( k ) .(ii) For a candidate c ∈ C , a subset N (cid:48) ⊆ N c , and j ≤ k , let s ( j ) c [ N (cid:48) ] denote the maximal voterload after optimally distributing the load of c among all voters in N (cid:48) . Then, s ( j ) c [ N c ] ≤ s ( j ) c [ N (cid:48) ] for all N (cid:48) ⊆ N c .Proof. Part (i) is clear since if we increase j to j + 1 , we minimize over a smaller set of distri-butions of the additional load, and for each of these, the maximum is at least as large as as inround j ; for details see Proposition 7.2 by Mora and Oliver [29].We now prove part (ii). Since j is fixed, we omit all superscripts ( j ) and ( j − for improvedreadability. Part (i) and (6) imply that ¯ x i ≤ s c [ N c ] . (15)We can therefore, using (5), deduce that s c [ N c ] = 1 + (cid:80) i ∈ N c ¯ x i | N c | = 1 + (cid:80) i ∈ N (cid:48) ¯ x i + (cid:80) i ∈ N c \ N (cid:48) ¯ x i | N c |≤ (cid:80) i ∈ N (cid:48) ¯ x i + | N c \ N (cid:48) | · s c [ N c ] | N c | = (cid:18) (cid:80) i ∈ N (cid:48) ¯ x i | N (cid:48) | (cid:19) | N (cid:48) || N c | + s c [ N c ] − | N (cid:48) || N c | s c [ N c ]= s c [ N c ] + | N (cid:48) || N c | ( s c [ N (cid:48) ] − s c [ N c ]) ,where the inequality follows from (15) and the last equality is due to s c [ N (cid:48) ] = | N (cid:48) | (1 + (cid:80) i ∈ N (cid:48) ¯ x i ) . Since | N (cid:48) | / | N c | > , we have s c [ N (cid:48) ] − s c [ N c ] ≥ and thus s c [ N c ] ≤ s c [ N (cid:48) ] ,as desired.We are now ready to prove our main theorem.16 heorem 6. seq-Phragm´en satisfies PJR.Proof. PJR requires that | S ∩ ( (cid:83) i ∈ N ∗ A i ) | ≥ (cid:96) for all groups N ∗ ⊆ N of voters satisfying | N ∗ | ≥ (cid:96) nk and | (cid:84) i ∈ N ∗ A i | ≥ (cid:96) for some integer (cid:96) > . We show that seq-Phragm´en satisfies astrictly stronger property by weakening the constraint | N ∗ | ≥ (cid:96) nk to | N ∗ | > (cid:96) nk +1 .Consider an instance ( A, k ) and let S be the committee selected by seq-Phragm´en. Assumefor contradiction that there exists a group N ∗ ⊆ N of voters and an integer (cid:96) > with | N ∗ | >(cid:96) nk +1 such that | (cid:84) i ∈ N ∗ A i | ≥ (cid:96) and | S ∩ ( (cid:83) i ∈ N ∗ A i ) | ≤ (cid:96) − .Let c ∈ ( (cid:84) i ∈ N ∗ A i ) \ S and consider round k (the last round) of the seq-Phragm´en procedure.Adding candidate c to the committee would have caused a maximal voter load of s ( k ) c = 1 + (cid:80) i ∈ N c ¯ x ( k − i | N c | ≤ (cid:80) i ∈ N ∗ ¯ x ( k − i | N ∗ |≤ (cid:96) − | N ∗ | = (cid:96) | N ∗ | < k + 1 n . Here, the first inequality follows from part (ii) of Lemma 5 (observe that N ∗ ⊆ N c ), the secondinequality follows from | S ∩ ( (cid:83) i ∈ N ∗ A i ) | ≤ (cid:96) − , and the strict inequality follows from | N ∗ | >(cid:96) nk +1 .Let c k be the candidate that was chosen in round k . Since candidate c was not chosen, wehave c (cid:54) = c k and s ( k ) c k ≤ s ( k ) c . Using part (i) of Lemma 5, we have s (1) ≤ . . . ≤ s ( k ) = s ( k ) c k ≤ s ( k ) c < k +1 n . In particular, this implies that at the end of round k , every voter i ∈ N has a load ¯ x ( k ) i that is strictly less than k +1 n . Summing the loads over all voters, we get (cid:88) i ∈ N ¯ x ( k ) i = (cid:88) i ∈ N ∗ ¯ x ( k ) i + (cid:88) i ∈ N \ N ∗ ¯ x ( k ) i ≤ ( (cid:96) −
1) + | N \ N ∗ | · s ( k ) < (cid:96) − nk + 1 ( k + 1 − (cid:96) ) k + 1 n = k ,where we have used the fact that | N \ N ∗ | ≤ nk +1 ( k + 1 − (cid:96) ) . But (cid:80) i ∈ N ¯ x ( k ) i < k is acontradiction, because the sum of all voter loads (at the end of the seq-Phragm´en procedure)must equal k . This completes the proof.We note that the proof of Theorem 6 shows that seq-Phragm´en satisfies a property that isstrictly stronger than PJR, because the constraint on the size of group N ∗ has been relaxed. Asan immediate corollary of Theorem 6, we obtain that seq-Phragm´en satisfies JR.
Corollary 7. seq-Phragm´en satisfies JR.
However, seq-Phragm´en violates EJR, as the following example demonstrates. Replacing the constraint | N ∗ | ≥ (cid:96) nk with | N ∗ | > (cid:96) nk +1 is similar to replacing the Hare quota with the Droopquota in the context of single transferable vote elections (see Section 4.3). The condition | N ∗ | > (cid:96) nk +1 is thebest possible here; see also Janson [21]. xample 6. Let C = { a, b, c , c , . . . , c } , k = 12 , and consider the following profile with n = 24 voters: × { a, b, c } × { c , c , . . . , c } × { a, b, c } × { c , c , . . . , c } × { c , c , . . . , c } seq-Phragm´en selects S = { c , c , . . . , c } . (For details of the calculation, see the table in theappendix.) To see that S does not provide EJR, consider the group N ∗ consisting of the fourvoters on the left. We have | N ∗ | = 4 = 2 nk and | (cid:84) i ∈ N ∗ A i | = |{ a, b }| = 2 . Therefore, EJRrequires that at least one voter in N ∗ approves at least candidates in S , which is not the case. Note that seq-Phragm´en also fails PR (see Example 5). This is not surprising, consideringthat PR is computationally intractable [45].
In Example 5, leximax-Phragm´en selects the committee providing perfect representation. Wenow show that leximax-Phragm´en satisfies PR in general.
Theorem 8. leximax-Phragm´en satisfies PR.Proof.
Consider an instance ( A, k ) and assume that PR ( A, k ) (cid:54) = ∅ (otherwise, there is nothingto show). Recall that a load distribution x = ( x i,c ) i ∈ N,c ∈ C is perfect if ¯ x i = kn for all i ∈ N .We first show that there is a perfect load distribution. Let { c , . . . , c k } ⊆ C be a committeeproviding perfect representation and let N , . . . , N k be a corresponding partition of N . Defineload distribution x ∗ by x ∗ i,c j = (cid:40) kn if i ∈ N j , otherwise.It is straightforward to check x ∗ is a valid load distribution and that x ∗ is perfect.Clearly, a perfect load distribution is an optimal solution for the minimization problem inleximax-Phragm´en. It follows that every optimal load distribution is perfect. We now show thatevery perfect load distribution corresponds to a committee providing perfect representation. Itthen follows that every committee S output by leximax-Phragm´en provides perfect representa-tion for ( A, k ) .Let x = ( x i,c ) i ∈ N,c ∈ C be a perfect load distribution and let S = { c ∈ C : (cid:80) i ∈ N x i,c = 1 } bethe corresponding committee. Define M = ( m i,c ) i ∈ N,c ∈ S as the n × k -matrix whose entries aregiven by m i,c = x i,c nk .Every row of M sums to (cid:80) c ∈ S x i,c nk = nk ¯ x i = 1 , and every column of M sums to (cid:88) i ∈ N x i,c nk = nk . M (cid:48) be an n × n -matrix with non-negative entries that results from M by replacing,for each c ∈ S , the column corresponding to c with nk equal columns c , c , . . . c nk with entries kn m i,c . Then all column sums of M (cid:48) are (i.e., (cid:80) i ∈ N m (cid:48) i,c j = 1 for all c ∈ S and j = 1 , . . . , nk ),as are all row sums, so M (cid:48) is doubly stochastic. We can now apply the Birkhoff–von Neumanntheorem and get that M (cid:48) is a convex combination of permutation matrices. Choose a permu-tation matrix P in this convex combination. P encodes a bijection between the sets N and (cid:83) c ∈ S (cid:83) n/kj =1 c j . From this bijection, we can extract a partition { N c : c ∈ S } of N by defining N c as the set of voters that are mapped to an element of the set c , c , . . . c nk , for each c ∈ S . Itis easily verified that this partition satisfies the conditions in Definition 3. Therefore, S providesperfect representation for ( A, k ) .Since EJR is incompatible with PR (see Example 5), leximax-Phragm´en fails EJR. However,it satisfies PJR. Theorem 9. leximax-Phragm´en satisfies PJR.Proof.
We introduce one new piece of notation for this proof. For a committee S ⊆ C , let x S bea leximax-optimal load distribution, given that S is selected. As usual, we let ¯ x Si = (cid:80) c ∈ S x Si,c .Consider an instance ( A, k ) and a committee S output by leximax-Phragm´en. Assume that S does not satisfy PJR. That is, there exists a group N ∗ ⊆ N of voters and an integer (cid:96) > with | N ∗ | ≥ (cid:96) nk such that | (cid:84) i ∈ N ∗ A i | ≥ (cid:96) and | S ∩ ( (cid:83) i ∈ N ∗ A i ) | ≤ (cid:96) − .Let S (cid:48) = { c ∈ S : c (cid:54)∈ (cid:83) i ∈ N ∗ A i } ⊂ S be the chosen candidates that are not approved by any i ∈ N ∗ . Since S does not satisfy PJR, we know that | S (cid:48) | ≥ k − (cid:96) + 1 . Since no voters in N ∗ approve any candidate in S (cid:48) , the entire load from S (cid:48) is allocated to the (at most) ( k − (cid:96) ) nk votersin N \ N ∗ . The average voter load among these voters is therefore at least k − (cid:96) + 1( k − (cid:96) ) nk = k ( k − (cid:96) + 1) n ( k − (cid:96) ) > kn . In particular, consider the voter i (cid:48) with maximal load among all voters in N \ N ∗ . It must be thecase that this voter has load ¯ x Si (cid:48) > kn .We next show that ¯ x Si < kn for all i ∈ N ∗ . For contradiction, suppose that there exists some i ∈ N ∗ with ¯ x i ≥ nk . Then we can replace the subset of chosen candidates that are approved byat least one voter in N ∗ (of which there are at most (cid:96) − , since S violates PJR), by candidatesthat are approved by all of N ∗ (of which there are at least (cid:96) , by the definition of N ∗ ). Further,we can distribute the load of these candidates equally among N ∗ , so that each i ∈ N ∗ has loadat most (cid:96) − (cid:96) nk < kn . This leads to a leximax-smaller load distribution and we have contradictedour assumption that there exists some i ∈ N ∗ with ¯ x i ≥ nk .We can now complete the proof by constructing a committee T which has a leximax-smallervector of voter loads than S , contradicting the optimality of S . Let T (cid:48) ⊂ S (cid:48) consist of any k − (cid:96) candidates from S (cid:48) so that there exists a candidate c ∈ S (cid:48) \ T (cid:48) with x Si (cid:48) ,c > . Furtherlet T (cid:48)(cid:48) consist of (cid:96) candidates from (cid:84) i ∈ N ∗ A i . Let T = T (cid:48) ∪ T (cid:48)(cid:48) . We show that there exists aload distribution x Ti,c for T such that (¯ x T , . . . , ¯ x Tn ) ˙ < (¯ x S , . . . , ¯ x Sn ) . For candidates c ∈ T (cid:48)(cid:48) , wespread the load equally among voters in N ∗ , i.e., x Ti,c = | N ∗ | ≤ k(cid:96)n . For candidates c ∈ T (cid:48) , let19 Ti,c = x Si,c . Since no voter i ∈ N ∗ approves any candidate c ∈ T (cid:48) ⊂ S (cid:48) , it is necessarily thecase that x Ti,c = 0 for all i ∈ N ∗ and c ∈ T (cid:48) . Consequently, for i ∈ N ∗ , ¯ x Ti = (cid:88) c ∈ T x Ti,c = (cid:88) c ∈ T (cid:48)(cid:48) x Ti,c ≤ (cid:96) k(cid:96)n = kn < ¯ x Si (cid:48) . For i ∈ N \ N ∗ , we have ¯ x Ti = (cid:88) c ∈ T (cid:48) x Ti,c = (cid:88) c ∈ T (cid:48) x Si,c ≤ (cid:88) c ∈ S (cid:48) x Si,c = ¯ x Si . It remains to note that there exists at least one c ∈ S (cid:48) \ T (cid:48) with x Si (cid:48) ,c > . Thus ¯ x Ti (cid:48) < ¯ x Si (cid:48) , i.e.,there are fewer voters with the maximal load of ¯ x Si (cid:48) , and hence (¯ x T , . . . , ¯ x Tn ) ˙ < (¯ x S , . . . , ¯ x Sn ) . Corollary 10. leximax-Phragm´en satisfies JR.
We note that Example 1 shows that simply minimizing the maximal voter load (without lexi-max tie-breaking) does not even yield committees satisfying JR.
The proof of Theorem 8 directly applies to var-Phragm´en.
Theorem 11. var-Phragm´en satisfies PR.
Unlike leximax-Phragm´en, var-Phragm´en fails PJR.
Example 7. C = { a, b, c, d, e, f, g } , k = 6 , and consider the following profile with 100 voters:67 voters approve { a, b, c, d } , 12 voters approve { e } , 11 voters approve { f } , and 10 votersapprove { g } . Let N ∗ be the set of voters approving { a, b, c, d } . We have | N ∗ | = 67 ≥ nk and | (cid:84) i ∈ N ∗ A i | = 4 . Thus, PJR requires that all four candidates in (cid:84) i ∈ N ∗ A i = { a, b, c, d } areselected. However, var-Phragm´en selects { a, b, c, e, f, g } . Example 7 also shows that the sequential version of var-Phragm´en violates PJR. Example 7is an instance of the party-list setting (with four disjoint parties). An alternative proof that var-Phragm´en violates PJR consists in noting that in the party-list setting, var-Phragm´en reduces toSainte-Lagu¨e’s apportionment method [42], and PJR is equivalent to lower quota [9]. It is wellknown that Sainte-Lagu¨e’s method violates lower quota [4].Finally, we show that var-Phragm´en satisfies JR.
Theorem 12. var-Phragm´en satisfies JR.
The proof of Theorem 12 can be found in the appendix.20 . Conclusion
We have shown that Phragm´en’s load-balancing methods satisfy interesting representation ax-ioms. In particular, the polynomial-time computable variant seq-Phragm´en satisfies PJR. More-over, we have shown that both leximax-Phragm´en and var-Phragm´en satisfy PR and that leximax-Phragm´en additionally satisfies PJR. Arguably, leximax-Phragm´en is the first known example ofa “natural” rule satisfying both PR and PJR—the only other rule known to satisfy these twoproperties is an artificial construct that returns a PR committee if one exists and otherwise runsPAV [45]. The Monroe rule (i.e., the optimization variant of
Greedy Monroe ) satisfies PR bydefinition, but fails PJR if the committee size does not divide the number of voters [45].Since seq-Phragm´en violates EJR, it remains an open problem whether EJR is compatiblewith committee monotonicity. Further, the intricate nature of Example 6 seems to suggest thatinstances on which seq-Phragm´en violates EJR are rare. It would be interesting to see whetherseq-Phragm´en satisfies EJR for realistic distributions of preferences and/or for reasonable do-main restrictions. Finally, it would be of great interest to find axiomatic characterizationsof Phragm´en’s rules, i.e., to find sets of axiomatic properties that uniquely define leximax-Phragm´en, var-Phragm´en and seq-Phragm´en.
Acknowledgements
We would like to thank Xavier Mora for many fruitful discussions and for providing us withcopies of the original papers by Phragm´en. We also thank Marie-Louise Lackner for point-ing out essential literature and providing us with translations. Furthermore, we thank VincentConitzer, Edith Elkind, Dominik Peters, Luis S´anchez-Fern´andez, and Piotr Skowron for helpfulcomments. This material is based on work supported by ERC-StG 639945, NSF IIS-1527434and ARO W911NF-12-1-0550, by a Feodor Lynen return fellowship of the Alexander von Hum-boldt Foundation, by COST Action IC1205 on Computational Social Choice, by a grant fromthe Knut and Alice Wallenberg Foundation, by the Isaac Newton Institute for MathematicalSciences (EPSRC Grant Number EP/K032208/1), by a grant from the Simons foundation, bythe Deutsche Forschungsgemeinschaft (DFG) under grant BR 4744/2-1, and by the AustrianScience Foundation FWF, grant P31890.
References [1] H. Aziz, M. Brill, V. Conitzer, E. Elkind, R. Freeman, and T. Walsh. Justified representa-tion in approval-based committee voting. In
Proceedings of the 29th AAAI Conference onArtificial Intelligence (AAAI) , pages 784–790. AAAI Press, 2015.[2] H. Aziz, S. Gaspers, J. Gudmundsson, S. Mackenzie, N. Mattei, and T. Walsh. Computa-tional aspects of multi-winner approval voting. In
Proceedings of the 14th International In the approval-based apportionment setting, where candidates can obtain multiple seats in the committee, EJRand committee monotonicity can be achieved simultaneously [10]. Recent experimental work by Bredereck et al. [6] showed that committees satisfying JR very often satisfy EJR aswell, supporting the hypothesis that instances for which seq-Phragm´en fails EJR are rare. onference on Autonomous Agents and Multi-Agent Systems (AAMAS) , pages 107–115.IFAAMAS, 2015.[3] H. Aziz, E. Elkind, S. Huang, M. Lackner, L. S´anchez-Fern´andez, and P. Skowron. Onthe complexity of extended and proportional justified representation. In Proceedings ofthe 32nd AAAI Conference on Artificial Intelligence (AAAI 2018) , pages 902–909. AAAIPress, 2018.[4] M. Balinski and H. P. Young.
Fair Representation: Meeting the Ideal of One Man, OneVote . Yale University Press, 1982. (2nd Edition [with identical pagination], BrookingsInstitution Press, 2001).[5] N. Betzler, A. Slinko, and J. Uhlmann. On the computation of fully proportional represen-tation.
Journal of Artificial Intelligence Research , 47:475–519, 2013.[6] R. Bredereck, P. Faliszewski, A. Kaczmarczyk, and R. Niedermeier. An experimental viewon committees providing justified representation. In
Proceedings of the 28th InternationalJoint Conference on Artificial Intelligence (IJCAI) , pages 109–115. IJCAI, 2019.[7] R. P. Brent and P. Zimmermann.
Modern computer arithmetic , volume 18. CambridgeUniversity Press, 2010.[8] M. Brill, R. Freeman, S. Janson, and M. Lackner. Phragm´en’s voting methods and justifiedrepresentation. In
Proceedings of the 31st AAAI Conference on Artificial Intelligence (AAAI2017) , pages 406–413. AAAI Press, 2017.[9] M. Brill, J.-F. Laslier, and P. Skowron. Multiwinner approval rules as apportionment meth-ods.
Journal of Theoretical Politics , 30(3):358–382, 2018.[10] M. Brill, P. G¨olz, D. Peters, U. Schmidt-Kraepelin, and K. Wilker. Approval-based appor-tionment. In
Proceedings of the 34th AAAI Conference on Artificial Intelligence (AAAI) ,pages 1854–1861. AAAI Press, 2020.[11] W. D. Cairns. The International Mathematical Congress at Toronto.
The American Math-ematical Monthly , 31(9):411–417, 1924.[12] R. Camps, X. Mora, and L. Saumell. The method of Enestr¨om and Phragm´en for par-liamentary elections by means of approval voting. Technical report, arXiv:1907.10590[econ.TH], 2019.[13] I. Caragiannis, S. Nath, A. D. Procaccia, and N. Shah. Subset selection via implicit utili-tarian voting.
Journal of Artificial Intelligence Research , 58:123–152, 2017.[14] A. Cevallos and A. Stewart. A verifiably secure and proportional committee election rule.Technical report, arXiv:2004.12990v2 [cs.DS], 2020.[15] E. Elkind, P. Faliszewski, P. Skowron, and A. Slinko. Properties of multiwinner votingrules. In
Proceedings of the 13th International Conference on Autonomous Agents andMulti-Agent Systems (AAMAS) , pages 53–60. IFAAMAS, 2014.2216] P. Faliszewski, P. Skowron, A. Slinko, and N. Talmon. Multiwinner voting: A new chal-lenge for social choice theory. In U. Endriss, editor,
Trends in Computational SocialChoice , chapter 2. AI Access, 2017.[17] M. R. Garey and D. S. Johnson.
Computers and Intractability: A Guide to the Theory ofNP-Completeness . W. H. Freeman, 1979.[18] M. R. Garey, D. S. Johnson, and L. J. Stockmeyer. Some simplified NP-complete graphproblems.
Theor. Comput. Sci. ∼ svante/papers/sjV6.pdf, 2012.[20] S. Janson. Phragm´en’s and Thiele’s election methods. Technical report,arXiv:1611.08826v2 [math.HO], 2018.[21] S. Janson. Thresholds quantifying proportionality criteria for election methods. Technicalreport, arXiv:1810.06377 [cs.GT], 2018.[22] D. M. Kilgour. Approval balloting for multi-winner elections. In Handbook on ApprovalVoting , chapter 6. Springer, 2010.[23] M. Lackner and P. Skowron. Approval-based committee voting: Axioms, algorithms, andapplications. Technical report, arXiv:2007.01795 [cs.GT], 2020.[24] M. Lackner and P. Skowron. Consistent approval-based multi-winner rules.
Journal ofEconomic Theory , 192:105173, 2021.[25] T. Lu and C. Boutilier. Budgeted social choice: From consensus to personalized decisionmaking. In
Proceedings of the 22nd International Joint Conference on Artificial Intelli-gence (IJCAI) , pages 280–286. AAAI Press, 2011.[26] N. M¨oller. On Sch¨onhage’s algorithm and subquadratic integer GCD computation.
Math-ematics of Computation , 77(261):589–607, 2008.[27] B. L. Monroe. Fully proportional representation.
The American Political Science Review ,89(4):925–940, 1995.[28] X. Mora. Phragm´en’s sequential method with a variance criterion. Technical report,arXiv:1611.06833 [math.OC], 2016.[29] X. Mora and M. Oliver. Eleccions mitjanc¸ant el vot d’aprovaci´o. El m`etode de Phragm´eni algunes variants.
Butllet´ı de la Societat Catalana de Matem`atiques , 30(1):57–101, 2015.[30] H. Moulin.
Axioms of Cooperative Decision Making . Cambridge University Press, 1988.[31] W. Ogryczak. On the lexicographic minimax approach to location problems.
EuropeanJournal of Operational Research , 100(3):566–585, 1997.2332] D. Peters and P. Skowron. Proportionality and the limits of welfarism. In
Proceedings ofthe 21st ACM Conference on Economics and Computation , pages 793–794, 2020.[33] E. Phragm´en. Om proportionella val.
Stockholms Dagblad , 14 March 1893, 1893. Sum-mary of a public lecture published in a newspaper.[34] E. Phragm´en. Sur une m´ethode nouvelle pour r´ealiser, dans les ´elections, la repr´esentationproportionnelle des partis. ¨Ofversigt af Kongliga Vetenskaps-Akademiens F¨orhandlingar ,51(3):133–137, 1894.[35] E. Phragm´en.
Proportionella val. En valteknisk studie . Svenska sp¨orsm˚al 25. LarsH¨okersbergs f¨orlag, Stockholm, 1895.[36] E. Phragm´en. Sur la th´eorie des ´elections multiples. ¨Ofversigt af Kongliga Vetenskaps-Akademiens F¨orhandlingar , 53:181–191, 1896.[37] E. Phragm´en. Till fr˚agan om en proportionell valmetod.
Statsvetenskaplig Tidskrift , 2(2):297–305, 1899.[38] E. Phragm´en and E. Lindel¨of. Sur une extension d’un principe classique de l’analyse et surquelques propri´et´es des fonctions monog`enes dans le voisinage d’un point singulier.
ActaMathematica , 31(1):381–406, 1908.[39] A. D. Pia, S. S. Dey, and M. Molinaro. Mixed-integer quadratic programming is in NP.
Mathematical Programming , 162:225–240, 2017.[40] R. F. Potthof and S. J. Brams. Proportional representation: Broadening the options.
Journalof Theoretical Politics , 10(2):147–178, 1998.[41] A. D. Procaccia, J. S. Rosenschein, and A. Zohar. On the complexity of achieving propor-tional representation.
Social Choice and Welfare , 30:353–362, 2008.[42] A. Sainte-Lagu¨e. La repr´esentation proportionnelle et la m´ethode des moindres carr´es. In
Annales scientifiques de l’ ´Ecole Normale Sup´erieure , volume 27, pages 529–542, 1910.[43] L. S´anchez-Fern´andez and J. A. Fisteus. Monotonicity axioms in approval-based multi-winner voting rules. In
Proceedings of the 18th International Conference on AutonomousAgents and Multiagent Systems , pages 485–493. International Foundation for AutonomousAgents and Multiagent Systems, 2019.[44] L. S´anchez-Fern´andez, E. Elkind, and M. Lackner. Committees providing EJR can becomputed efficiently. Technical report, arXiv:1704.00356v3 [cs.GT], 2017.[45] L. S´anchez-Fern´andez, E. Elkind, M. Lackner, N. Fern´andez, J. A. Fisteus, P. Basanta Val,and P. Skowron. Proportional justified representation. In
Proceedings of the 31st AAAIConference on Artificial Intelligence (AAAI) , pages 670–676. AAAI Press, 2017.2446] L. S´anchez-Fern´andez, N. Fern´andez, J. A. Fisteus, and M. Brill. The maximin supportmethod: An extension of the D’Hondt method to approval-based multiwinner elections. In
Proceedings of the 35th AAAI Conference on Artificial Intelligence (AAAI) . AAAI Press,2021. Forthcoming.[47] D. Schmeidler. The nucleolus of a characteristic function game.
SIAM Journal on AppliedMathematics , 17(6):1163–1170, 1969.[48] A. Schrijver.
Theory of Linear and Integer Programming . John Wiley & Sons, 1986.[49] P. Skowron, P. Faliszewski, and J. Lang. Finding a collective set of items: From pro-portional multirepresentation to group recommendation. In
Proceedings of the 29th AAAIConference on Artificial Intelligence (AAAI) , pages 2131–2137. AAAI Press, 2015.[50] P. Skowron, M. Lackner, M. Brill, D. Peters, and E. Elkind. Proportional rankings. In
Proceedings of the 26th International Joint Conference on Artificial Intelligence (IJCAI) ,pages 409–415. IJCAI, 2017.[51] A. S. V. Strassen. Schnelle Multiplikation großer Zahlen.
Computing , 7(3-4):281–292,1971.[52] A. Stubhaug.
G¨osta Mittag-Leffler: A man of conviction . Springer Science & BusinessMedia, 2010.[53] T. N. Thiele. Om flerfoldsvalg.
Oversigt over det Kongelige Danske Videnskabernes Sel-skabs Forhandlinger , pages 415–441, 1895.[54] N. Tideman. The single transferable vote.
Journal of Economic Perspectives , 9(1):27–38,1995.
A. Appendix
A.1. Proof of Theorem 12
We first prove a lemma.
Lemma 13.
Let < c < and ( x i ) ≤ i ≤ n be a sequence with ≤ x i ≤ c for all i ∈ { , . . . , n } and (cid:80) ni =1 x i = 1 . Then, (cid:80) ni =1 x i ≤ c .Proof. (cid:80) ni =1 x i ≤ (cid:80) ni =1 cx i = c .We can now prove Theorem 12. Theorem 12.
Consider an instance ( A, k ) and a committee S output by var-Phragm´en. Assumethat S does not satisfy JR. That is, there exists a set of voters N ∗ , with | N ∗ | ≥ nk , such that (cid:84) i ∈ N ∗ A i (cid:54) = ∅ and | S ∩ ( (cid:83) i ∈ N ∗ A i ) | = ∅ . Clearly, | N ∗ | < n .25et i ∗ be a voter with maximal load (that is, ¯ x i ∗ ≥ ¯ x i for all i (cid:54) = i ∗ ), and let c be a candidatewith x i ∗ ,c > . Such a c must exist because the total load on i ∗ is non-zero.First note that the average load on voters in N \ N ∗ is k | N \ N ∗ | ≥ kn − nk = k kn − n . Therefore, since i ∗ is a voter with maximal load, it must be the case that ¯ x i ∗ ≥ k kn − n . Further, ¯ x i = ¯ x i ∗ for all voters i ∈ N c . If this were not the case for some voter i , it would be possibleto decrease the variance of the load distribution by reducing x i ∗ ,c by some small amount andincreasing x i,c accordingly, thus reducing the difference between the loads on i ∗ and i whileleaving all other loads unchanged, which reduces the variance.Let d ∈ (cid:84) i ∈ N ∗ A i , and let T = S ∪ { d } \ { c } . That is, T is the committee obtained by startingwith S and replacing c with a candidate approved by all voters in N ∗ . To complete the proof,we consider the effect that this replacement has on the quanity (cid:80) i ∈ N ¯ x i , which is the objectiveminimized by var-Phragm´en.It is possible to distribute the load of candidate d evenly across all (previously unrepresented)voters in N ∗ . Therefore, the addition of d contributes at most (cid:80) i ∈ N ∗ | N ∗ | = | N ∗ | ≤ kn to theobjective. On the other hand, removing c from the committee decreases the objective by (cid:88) i ∈ N c (¯ x i − (¯ x i − x i,c ) ) = (cid:88) i ∈ N c (¯ x i − ¯ x i + 2¯ x i x i,c − x i,c ) = (cid:88) i ∈ N c (2¯ x i x i,c − x i,c )= (cid:88) i ∈ N c (2¯ x i ∗ x i,c − x i,c ) = 2¯ x i ∗ − (cid:88) i ∈ N c x i,c ≥ x i ∗ − ¯ x i ∗ = ¯ x i ∗ ≥ k kn − n > kn ,where the first inequality follows from Lemma 13. Therefore, replacing c by d causes a netdecrease to the objective, contradicting minimality of the variance of committee S , and thus weobtain a contradiction to our assumption that S does not provide JR. A.2. seq-Phragm ´en violates EJR
Table 2 shows the necessary calculations for computing seq-Phragm´en in Example 6.26 s (1) c s (2) c s (3) c s (4) c s (5) c s (6) c s (7) c s (8) c s (9) c s (10) c s (11) c s (12) c c – – – – c – – – – – – – c – – – – – – – – – – – c – – – – – – – – – – c – – – – – – – – – c – – – – – – – – c – – – – – – c – – – – – c – – – c – – c – c a b Table 2: Table showing the values s ( j ) c (rounded to three decimal places) for each candidate c ∈ C and each round j = 1 , . . . , in Example 6. Entries in bold distinguish thecandidate with lowest s ( j ) cc