Approval-Based Apportionment

The Thirty-Fourth AAAI Conference on Artiﬁcial Intelligence (AAAI-20)

Markus Brill,1 Paul Golz,¨ 2 Dominik Peters,2 Ulrike Schmidt-Kraepelin,1 Kai Wilker1 1Technische Universitat¨ Berlin, Chair of Efﬁcient Algorithms 2Carnegie Mellon University, Computer Science Department {brill, u.schmidt-kraepelin}@tu-berlin.de, {pgoelz, dominikp}@cs.cmu.edu, [email protected]

Abstract equally well represented by several political parties, there is no way to express this preference within the voting system. In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters In the context of single-winner elections, approval voting supporting each party. We study a generalization of this setting, has been put forward as a solution to this problem as it strikes in which voters cast approval ballots over parties, such that an attractive compromise between simplicity and expressiv- each voter can support multiple parties. This approval-based ity (Brams and Fishburn 2007; Laslier and Sanver 2010). apportionment setting generalizes traditional apportionment Under approval voting, each voter is asked to specify a set and is a natural restriction of approval-based multiwinner elec- of candidates she “approves of,” i.e., voters can arbitrarily tions, where approval ballots range over individual candidates. partition the set of candidates into approved candidates and Using techniques from both apportionment and multiwinner disapproved ones. Proponents of approval voting argue that elections, we are able to provide representation guarantees its introduction could increase voter turnout, “help elect the that are currently out of reach in the general setting of multiwinner elections: First, we show that core-stable committees strongest candidate,” and “add legitimacy to the outcome” of are guaranteed to exist and can be found in polynomial time. an election (Brams and Fishburn 2007, pp. 4–8). Second, we demonstrate that extended justified representation Due to the practical and theoretical appeal of approval vot- is compatible with committee monotonicity. ing in single-winner elections, a number of scholars have sug- gested to also use approval voting for multiwinner elections, in which a fixed number of candidates needs to be elected 1 Introduction (Kilgour and Marshall 2012). In contrast to the single-winner The fundamental fairness principle of proportional represen- setting, where the straightforward voting rule “choose the tation is relevant in a variety of applications ranging from candidate approved by the highest number of voters” enjoys recommender systems to digital democracy (Skowron et al. a strong axiomatic foundation (Fishburn 1978), several ways 2017). It features most explicitly in the context of political of aggregating approval ballots have been proposed in the elections, which is the language we adopt for this paper. In multiwinner setting (e.g., Aziz et al. 2017; Janson 2016). this context, proportional representation prescribes that the Most studies of approval-based multiwinner elections as- number of representatives championing a particular opinion sume that voters directly express their preference over individ- in a legislature be proportional to the number of voters who ual candidates; we refer to this setting as candidate-approval favor that opinion. elections. This assumption runs counter to widespread demo- In most democratic institutions, proportional represen- cratic practice, in which candidates belong to political parties tation is implemented via party-list elections: Candidates and voters indicate preferences over these parties (which in- are members of political parties and voters are asked to duce implicit preferences over candidates). In this paper, we indicate their favorite party; each party is then allocated therefore study party-approval elections, in which voters ex- a number of seats that is (approximately) proportional to press approval votes over parties and a given number of seats the number of votes it received. The problem of transform- must be distributed among the parties. We refer to the process ing a voting outcome into a distribution of seats is known of allocating these seats as approval-based apportionment. as apportionment. Analyzing the advantages and disadvan- We believe that party-approval elections are a promising tages of different apportionment methods has a long and framework for legislative elections in the real world. Allow- illustrious political history and has given rise to a deep ing voters to express approval votes over parties enables the and elegant mathematical theory (Balinski and Young 1982; aggregation mechanism to coordinate like-minded voters. For Pukelsheim 2014). example, two blocks of voters might currently vote for par- Unfortunately, forcing voters to choose a single party pre- ties that they mutually disapprove of. Using approval ballots vents them from communicating any preferences beyond could reveal that the blocks jointly approve a party of more their most preferred alternative. For example, if a voter feels general appeal; allocating more seats to this party leads to Copyright c 2020, Association for the Advancement of Artificial mutual gain. This cooperation is particularly necessary for Intelligence (www.aaai.org). All rights reserved. small minority opinions that are not centrally coordinated. In

1854 such cases, finding a commonly approved party can make the Candidate-approval elections difference between being represented or votes being wasted (Kilgour and Marshall 2012; Aziz et al. 2017) because the individual parties receive insufficient support. In contrast to approval voting over individual candidates, (ii) party-approval voting does not require a break with the cur- Party-approval elections rent role of political parties—it can be combined with both (iii) [approval-based apportionment] “open list” and “closed list” approaches to filling the seats allocated to a party. (i) 1.1 Related Work Party-list elections [apportionment] (Balinski and Young 1982; Pukelsheim 2014) To the best of our knowledge, this paper is the first to formally develop and systematically study approval-based apportionment. That is not to say that the idea of expressing and Figure 1: Relations between the different settings of multi- aggregating approval votes over parties has not been consid- winner elections. An arrow from X to Y signifies that X is a ered before. Indeed, several scholars have explored possible generalization of Y . The relationship corresponding to arrow generalizations of existing aggregation procedures. (iii) has been explored by Brill, Laslier, and Skowron (2018). For instance, Brams, Kilgour, and Potthoff (2019) study We establish and explore the relationship (i) in Section 3 and multiwinner approval rules that are inspired by classical ap- the relationship (ii) in Section 4. portionment methods. Besides the setting of candidate approval, they explicitly consider the case where voters cast party-approval votes. They conclude that these rules could embedded in this setting by replacing each party by multi- “encourage coalitions across party or factional lines, thereby ple candidates belonging to this party, and by interpreting diminishing gridlock and promoting consensus.” a voter’s approval of a party as approval of all of its candi- Such desire for compromise is only one motivation for con- dates. This embedding establishes party-approval elections as sidering party-approval elections, as exemplified by recent a subdomain of candidate-approval elections (see arrow (ii) work by Speroni di Fenizio and Gewurz (2019). To allow for in Figure 1). In Section 4, we explore the axiomatic and more efficient governing, they aim to concentrate the power computational ramifications of this domain restriction. of a legislature in the hands of few big parties, while nonetheless preserving the principle of proportional representation. 1.3 Contributions To this end, they let voters cast party-approval votes and trans- In this paper, we formally introduce the setting of approval- form these votes into a party-list election by assigning each based apportionment and explore different possibilities of voter to one of her approved parties. One method for doing constructing axiomatically desirable aggregation methods for this (referred to as majoritarian portioning later in this paper) this setting. Besides its conceptual appeal, this setting is also assigns voters to parties in such a way that the strongest party interesting from a technical perspective. has as many votes as possible. Exploiting the relations described in Section 1.2, we re- Several other papers consider extensions of approval-based solve problems that remain open in the more general setting voting rules to accommodate party-approval elections (Brams of approval-based multiwinner voting. First, we prove that and Kilgour 2014; Mora and Oliver 2015; Janson 2016; committee monotonicity is compatible with extended justi- Janson and Oberg¨ 2019). All of these papers have in common fied representation (a representation axiom proposed by Aziz that they study specific rules or classes of rules, rather than et al. 2017) by providing a rule that satisfies both properties. exploring the party-approval setting in its own right. Second, we show that the core of an approval-based apportionment problem is always nonempty and that core-stable 1.2 Relation to Other Settings committees can be found in polynomial time. Party-approval elections can be positioned between two well- Besides these positive results, we verify for a wide range of studied voting settings (see Figure 1). multiwinner voting rules that their axiomatic guarantees do First, approval-based apportionment generalizes standard not improve in the party-approval setting, and that some rules apportionment, which corresponds to party-approval elec- remain NP-hard to evaluate. On the other hand, we show that tions in which all approval sets are singletons. This relation it becomes tractable to check whether a committee provides (depicted as arrow (i) in Figure 1) provides a generic two-step extended justified representation or the weaker axiom of approach to define aggregation rules for approval-based ap- proportional justified representation. portionment problems: transform a party-approval instance to Omitted proofs as well as further definitions and results an apportionment instance, and then apply an apportionment can be found in the full version of the paper (Brill et al. 2019). method. In Section 3, we employ this approach to construct approval-based apportionment methods satisfying desirable 2 The Model properties. A party-approval election is a tuple (N, P, A, k) consisting Second, our setting can be viewed as a special case of of a set of voters N = {1,...,n}, a finite set of parties P ,a approval-based multiwinner voting, in which voters cast ballot profile A =(A1,...,An) where each ballot Ai ⊆ P candidate-approval votes. A party-approval election can be is the set of parties approved by voter i, and the committee

1855 size k ∈ N. We assume that Ai = ∅ for all i ∈ N. When it holds that f(N, P, A, k) ⊆ f(N, P, A, k +1). Committee considering computational problems, we assume that k is monotonic rules avoid the so-called Alabama paradox,in encoded in unary (see Footnote 5). which a party loses a seat when the committee size increases. A committee in this setting is a multiset W : P → N over Besides, committee monotonic rules can be used to construct parties, which determines the number of seats W (p) assigned proportional rankings (Skowron et al. 2017). to each party p ∈ P . The size of a committee W is given by |W | = W (p) p∈P , and we denote multiset addition and 3 Constructing Party-Approval Rules via subtraction by + and −, respectively. A party-approval rule Portioning and Apportionment is a function that takes a party-approval election (N, P, A, k) as input and returns a committee W of valid size |W | = k.1 Party-approval elections are a generalization of party-list In our axiomatic study of party-approval rules, we focus on elections, which can be thought of as party-approval elec- two axioms capturing proportional representation: extended tions in which all approval sets are singletons. Since there justified representation and core stability (Aziz et al. 2017).2 is a rich body of research on apportionment methods, it is Both axioms are derived from their analogs in multiwinner natural to examine whether we can employ these methods elections (see Section 4.2) and can be defined in terms of for our setting as well. To use them, we will need to translate quota requirements. party-approval elections into the party-list domain on which For a party-approval election (N, P, A, k) and a subset apportionment methods operate. This translation thus needs S ⊆ N of voters, define the quota of S as q(S)=k·|S|/n. to transform a collection of approval votes over parties into Intuitively, q(S) corresponds to the number of seats that the vote shares for each party. Motivated by time sharing, Bogo- group S “deserves” to be represented by (rounded down). molnaia, Moulin, and Stong (2005) have developed a theory of such transformation rules, further studied by Duddy (2015) W : P → N Definition 1. A committee provides extended and Aziz, Bogomolnaia, and Moulin (2019). We will refer to justified representation (EJR) for a party-approval election this framework as portioning. (N, P, A, k) S ⊆ N if there is no subset of voters such that The approach explored in this section, then, divides the Ai = ∅ and W (p) W (p) for all i ∈ S. The p∈Ai p∈Ai committee monotonicity since most apportionment methods core of a party-approval election is defined as the set of all satisfy this property. In the general candidate-approval setting core-stable committees. (considered in Section 4), the existence of a rule satisfying Core stability requires adequate representation even for both EJR and committee monotonicity is an open problem. voter groups that cannot agree on a common party, by ruling out the possibility that the group can deviate to a smaller 3.1 Preliminaries committee that represents all voters in the group strictly better. We start by introducing relevant notions from the litera- It follows from the definitions that core stability is a stronger ture of portioning (Bogomolnaia, Moulin, and Stong 2005; requirement than EJR: If a committee violates EJR, there is a Aziz, Bogomolnaia, and Moulin 2019) and apportionment S q(S) group that would prefer any committee of size that (Balinski and Young 1982; Pukelsheim 2014), with notations assigns all seats to the commonly approved party. and interpretations suitably adjusted to our setting. A final, non-representational axiom that we will discuss is committee monotonicity. A party-approval rule f satisfies Portioning A portioning problem is a triple (N,P,A), just this axiom if, for all party-approval elections (N, P, A, k), as in party-approval voting but without a committee size. A portioning is a function r : P → [0, 1] with p∈P r(p)=1. 1 This definition implies that rules are resolute, that is, only a We interpret r(p) as the vote share of party p.Aportioning single committee is returned. In the case of a tie between multiple method maps each triple (N,P,A) to a portioning. committees, a tiebreaking mechanism is necessary. Our results hold independently of the choice of a specific tiebreaking mechanism. Our minimum requirement on portioning methods will 2Some results in the full version refer to the weaker represen- be that they uphold proportionality if all approval sets are tation axioms of justified representation (JR) (Aziz et al. 2017) singletons, i.e., if we are already in the party-list domain. and proportional justified representation (PJR) (Sanchez-Fern´ andez´ Formally, we say that a portioning method is faithful if for et al. 2017). It is well known that EJR implies PJR and that PJR all (N,P,A) with |Ai| =1for all i ∈ N, the resulting implies JR. portioning r satisfies r(p)=|{i ∈ N | Ai = {p}}|/n for all

1856 p ∈ P . Among the portioning methods considered by Aziz, Composition If we take any portioning method and any Bogomolnaia, and Moulin (2019), only three are faithful. apportionment method, we can compose them to obtain a They are defined as follows. party-approval rule. Note that if the apportionment method is committee monotonic then so is the composed rule, since the Conditional utilitarian portioning selects, for each voter portioning is independent of k. i, pi as a party in Ai approved by the highest number of r(p)=|{i ∈ N | p = p}|/n p ∈ P voters. Then, i for all . 3.2 Composed Rules That Fail EJR Random priority computes n! portionings, one for each per- Perhaps surprisingly, many pairs of portioning and apportion- mutation σ of N, and returns their average. The portioning ment methods fail EJR. This is certainly true if the individual for σ =(i1,...,in) maximizes p∈A r(p), breaking i1 parts are not representative themselves. For example, if an r(p) apportionment method M properly fails lower quota (in the ties by maximizing p∈Ai , and so forth. 2 sense that there is a rational-valued input r on which lower Nash portioningselects the portioning r maximizing the quota is violated), then one can construct an example profile r(p) Nash welfare i∈N p∈Ai . on which any composed rule using M fails EJR: Construct a The last method seems particularly promising because it party-approval election with singleton approval sets in which satisfies portioning versions of core stability and EJR (Aziz, the voter counts are proportional to the shares in the counter- r Bogomolnaia, and Moulin 2019). example . Then any faithful portioning method, applied to this election, must return r. Since M fails lower quota on We will also make use of a more recent portioning ap- r proach, which was proposed by Speroni di Fenizio and , the resulting committee will violate EJR. By a similar Gewurz (2019) in the context of party-approval voting. argument, an apportionment method that violates committee monotonicity on some rational portioning will, when Majoritarian portioning proceeds in rounds j =1, 2,.... composed with a faithful portioning method, give rise to a Initially, all parties and voters are active. In iteration j,we party-approval rule that fails committee monotonicity. select the active party pj that is approved by the highest To our knowledge, among the named and studied appor- number of active voters. Let Nj be the set of active voters tionment methods, only two satisfy both lower quota and who approve pj. Then, set r(pj) to |Nj|/n, and mark pj committee monotonicity: D’Hondt and the quota method. and all voters in Nj as inactive. If active voters remain, the However, it turns out that the composition of either option next iteration is started; else, r is returned. with the conditional-utilitarian, random-priority, or Nash por- Under majoritarian portioning, the approval preferences of tioning methods fails EJR, as the following examples show. voters who have been assigned to a party are ignored in Example 1. Let n = k =6, P = {p0,p1,p2,p3}, A = further iterations. Note that conditional utilitarian portioning ({p0}, {p0}, {p0,p1,p2}, {p0,p1,p2}, {p1,p3}, {p2,p3}). can similarly be seen as a sequential method, in which the Then, the conditional utilitarian solution sets r(p0)=4/6, preferences of inactive voters are not ignored. r(p1)=r(p2)=1/6, and r(p3)=0. Any apportionment method satisfying lower quota allocates four seats to p0, one Apportionment An apportionment problem is a tuple each to p1 and p2, and none to p3. The resulting committee (P, r, k) P , which consists of a finite set of parties , a por- does not provide EJR since the last two voters, who jointly r : P → [0, 1] tioning specifying the vote shares of parties, approve p3, have a quota of q({5, 6})=2that is not met. and a committee size k ∈ N. Committees are defined as for n = k =6 P = {p ,p ,p ,p } A = party-approval elections, and an apportionment method maps Example 2. Let , 0 1 2 3 , and ({p }, {p }, {p ,p ,p }, {p ,p ,p }, {p }, {p ,p }) apportionment problems to committees W of size k. 0 0 0 1 2 0 1 3 1 2 3 . r(p )=23/45 An apportionment method satisfies lower quota if each Random priority chooses the portioning 0 , p k · r(p) r(p1)=23/90, and r(p2)=r(p3)=7/60. Both D’Hondt party is always allocated at least seats in the p committee. Furthermore, an apportionment method f is com- and the quota method allocate four seats to 0, two seats to f(P, r, k) ⊆ f(P, r, k +1) p1, and none to the other two parties. This clearly violates the mittee monotonic if for every q({6})=1 apportionment problem (P, r, k). claim to representation of the sixth voter (with ). Among the standard apportionment methods, only one Nash portioning produces a fairly similar portioning, with r(p0) ≈ 0.5302, r(p1) ≈ 0.2651, and r(p2)=r(p3) ≈ satisfies both lower quota and committee monotonicity: the 0.1023 D’Hondt method (aka Jefferson method).3 The method as- . D’Hondt and the quota method produce the same signs the k seats iteratively, each time giving the next seat to committee as above, leading to the same EJR violation. the party p with the largest quotient r(p)/(s(p)+1), where At first glance, it might be surprising that Nash portioning s(p) denotes the number of seats already assigned to p. An- combined with a lower-quota apportionment method violates other apportionment method satisfying lower quota and com- EJR (and even the weaker axiom JR). Indeed, Nash portion- mittee monotonicity is the quota method, due to Balinski ing satisfies core stability in the portioning setting, which is a and Young (1975). It is identical to the D’Hondt method, strong notion of proportionality, and the lower-quota property except that, in the jth iteration, only parties p satisfying limits the rounding losses when moving from the portioning s(p)/j < r(p) are eligible for the allocation of the next seat. to a committee. As expected, in the election of Example 2, the Nash solution itself gives sufficient representation to the 3All other divisor methods fail lower quota, and the Hamilton sixth voter since r(p2)+r(p3) ≈ 0.2047 > 1/6. However, method is not committee monotonic (Balinski and Young 1982). since both r(p2) and r(p3) are below 1/6 on their own, lower

1857 quota does not apply to either of the two parties, and the sixth While the party-approval rules identified by Theorem 1 voter loses all representation in the apportionment step. satisfy EJR and committee monotonicity, they do not quite reach our gold standard of representation, i.e., core stability. 3.3 Composed Rules That Satisfy EJR Example 3. Let n = k =16, let P = {p0,...,p4}, with the {p ,p } {p ,p } As we have seen, several initially promising portioning meth- following approval sets: 4 times 0 1 , 3 times 1 2 , once {p2}, 4 times {p0,p3}, 3 times {p3,p4}, and once {p4}. ods fail to compose to a rule that satisfies EJR. One reason p p p p is that these portioning methods are happy to assign small Note the symmetry between 1 and 3, and between 2 and 4. Majoritarian portioning allocates 1/2 to p0 and 1/4 each shares to several parties. The apportionment method may p p round several of those small shares down to zero seats. This to 2 and 4. Any lower-quota apportionment method must translate this into 8 seats for p0 and 4 seats each for p2 and can lead to a failure of EJR when not enough parties obtain a p S seat. It is difficult for an apportionment method to avoid this 4. This committee is not in the core: Let be the coalition of all 14 voters who approve multiple parties, and let T allocate behavior since the portioning step obscures the relationships p p p between different parties that are apparent from the approval 4 seats to 0 and 5 seats each to 1 and 3. This gives strictly ballots of the voters. higher representation to all members of the coalition. Majoritarian portioning is designed to maximize the seat The example makes it obvious why majoritarian portion- allocations to the largest parties. Thus, it tends to avoid the ing cannot satisfy the core: All voters approving of p0 get problem we have identified. While it fails the strong repre- deactivated after the first round, which makes p2 seem uni- sentation axioms that Nash portioning satisfies, this turns out versally preferable to p1. However, p1 is a useful vehicle for not to be crucial: Composing majoritarian portioning with cooperation between the group approving {p0,p1} and the any apportionment method satisfying lower quota yields an group approving {p1,p2}. Since majoritarian portioning is EJR rule. If we use an apportionment method that is also blind to this opportunity, it cannot guarantee core stability. committee monotonic, such as D’Hondt or the quota method, The example also illustrates the power of core stability: we obtain a party-approval rule that satisfies both EJR and The deviating coalition does not agree on any single party committee monotonicity.4 they support, but would nonetheless benefit from the devi- ation. There is room for collaboration, and core stability is Theorem 1. Let M be a committee monotonic apportion- sensitive to this demand for better representation. ment method satisfying lower quota. Then, the party-approval rule composing majoritarian portioning and M satisfies EJR 4 Constructing Party-Approval Rules via and committee monotonicity. Multiwinner Voting Rules (N, P, A, k) In the previous section, we applied tools from apportionment, Proof. Consider a party-approval election and a more restrictive setting, to our party-approval setting. Now, let r be the outcome of majoritarian portioning applied (N,P,A) N ,N ,... p ,p ,... we go in the other direction, and apply tools from a more gen- to . Let 1 2 and 1 2 be the voter eral setting: As mentioned in Section 1.2, party-approval elec- groups and parties in the construction of majoritarian portion- r(p )=|N |/n j tions can be viewed as a special case of candidate-approval ing, so that j j for all . elections, i.e., multiwinner elections in which approvals are W = M(P, r, k) Consider the committee and suppose expressed over individual candidates rather than parties. Af- S ⊆ N that EJR is violated, i.e., that there exists a group ter introducing relevant candidate-approval notions, we show with Ai = ∅ and W (p)

1858 Janson 2016; Aziz et al. 2017), out of which we will only in- approval setting will satisfy the corresponding axiom in the troduce the one which we use for our main positive result. Let party-approval setting as well. Note that, by restricting our j Hj denote the jth harmonic number, i.e., Hj = t=1 1/t. view to party approval, the cohesiveness requirement of EJR Given (N, C, A, k), the candidate-approval rule propor- is reduced to requiring a single commonly approved party. tional approval voting (PAV), introduced by Thiele (1895), chooses a candidate committee W maximizing the PAV score 4.3 PAV Guarantees Core Stability PAV(W )= H i∈N |W ∩Ai|. A powerful stability concept in economics, core stability is a We now describe EJR and core stability in the candidate- natural extension of EJR. It is particularly attractive because approval setting, from which our versions of these axioms blocking coalitions are not required to be coherent at all, are derived. Recall that q(S)=k |S|/n. A candidate com- just to be able to coordinate for mutual gain. Our earlier mittee W provides EJR if there is no subset S ⊆ N and Example 3 illustrates how a coalition might deviate in spite >0 q(S) | A | no integer such that , i∈S i , and of not agreeing on any approved party. |Ai ∩ W | |Ai ∩ W | for all i ∈ S. be counterexamples to their notion of core stability. They pro- The core consists of all core-stable candidate committees. vide a nonconstant approximation to the core in our sense, but nonemptyness remains open. Recently, Cheng et al. (2019) 4.2 Embedding Party-Approval Elections showed that there always exist randomized committees providing core stability (over expected representation), but it is We have informally argued in Section 1.2 that party-approval not clear how their approach based on two-player zero-sum elections constitute a subdomain of candidate-approval elec- game duality would extend to deterministic committees. tions. We formalize this notion by providing an embedding of All standard candidate-approval rules either already fail party-approval elections into the candidate-approval domain. (N, P, A, k) weaker representation axioms such as EJR or fail core sta- For a given party-approval election , we define bility. In particular, Aziz et al. (2017) have shown that PAV a corresponding candidate-approval election with the same satisfies EJR, but may produce non-core-stable candidate set of voters N and the same committee size k. The set of k p(1),...,p(k) committees even for candidate-approval elections for which candidates contains many “clone” candidates core-stable candidate committees are known to exist. p ∈ P p(j) for each party , and a voter approves a candidate By contrast, we show that PAV guarantees core stability in the candidate-approval election iff she approves the cor- in the party-approval setting. We follow the structure of the p responding party in the party-approval election. This em- aforementioned proof showing that PAV satisfies EJR for bedding establishes party-approval elections as a subdomain candidate-approval elections (Aziz et al. 2017). of candidate-approval elections. As a consequence, we can apply rules and axioms from the more general candidate- Theorem 2. For every party-approval election, PAV chooses approval setting also in the party-approval setting. a core-stable committee. In particular, the generic way to apply a candidate-approval Proof. Consider a party-approval election (N, P, A, k) and rule for a party-approval election consists in (1) translating let W : P → N be the committee selected by PAV. Assume the party-approval election into a candidate-approval election, for contradiction that W is not core stable. Then, there is a (2) applying the candidate-approval rule, and (3) counting nonempty coalition S and a committeeT : P → N such that the number of chosen clones per party to construct a commit- |T | k |S|/n and T (p) > W (p) for every k p∈Ai p∈Ai tee over parties. Note that, since is encoded in unary, the voter i ∈ S. running time is blown up by at most a polynomial factor.5 Let ui(W ) denote the number of seats in W that are Having established party-approval elections as a subdo- allocated to parties approved by voter i, i.e., ui(W )= main of candidate-approval elections, our variants of EJR and p∈A W (p). Furthermore, for a party p with W (p) > 0, core stability (Definitions 1 and 2) are immediately induced i we let MC (p, W ) denote the marginal contribution to the by their candidate-approval counterparts. In particular, any PAV score of allocating a seat to p, i.e., MC (p, W )= candidate-approval rule satisfying an axiom in the candidate- PAV( W ) − PAV(W −{p}). Observe that MC (p, W )= 1/ui(W ), where Np = {i ∈ N | p ∈ Ai}. The sum 5For candidate-approval elections, it does not make sense to have i∈Np more seats than candidates, whereas for party-approval elections of all marginal contributions satisfies k it is natural to have more seats than parties. If was encoded in W (p) binary, even greedy candidate-approval algorithms would suddenly W (p) MC (p, W )= have exponential running time. This would complicate running-time ui(W ) p∈P p∈P i∈Np comparisons between the candidate-approval and party-approval setting and would blur the intuitive distinction between simple and W (p) k = = |{i ∈ N | ui(W ) > 0}| n. complex algorithms. Encoding in unary sidesteps this technical ui(W ) complication. i∈N p∈Ai

1859 Note that terms MC (p, W ) for W (p)=0and quotients An immediate follow-up question is whether core-stable 1/ui(W ) for ui(W )=0are undefined in the calculation committees can be computed efficiently. PAV committees are above, but that they only appear with factor 0. known to be NP-hard to compute in the candidate-approval It follows that the average marginal contribution of all k setting, and we confirm in the full version of the paper that seats in W is at most n/k, and consequently, that there has to hardness still holds in the party-approval subdomain. be a party p1 with a seat in W such that MC (p1,W) n/k. Equally confronted with the computational complexity of Using a similar argument, we show that there is also a party p2 PAV, Aziz et al. (2018) proposed a local-search variant of with T (p2) > 0 which would increase the PAV score by at PAV, which runs in polynomial time and guarantees EJR in least n/k if it received an additional seat in W : the candidate-approval setting. Using the same approach, we can find a core-stable committee in the party-approval setting. T (p) T (p) MC (p, W + {p})= We defer the proof to the full version of the paper. ui(W + {p}) p∈P i∈N p∈Ai Theorem 4. Given a party-approval election, a core-stable T (p) T (p) committee can be computed in polynomial time. = ui(W + {p}) ui(W )+1 Theorem 2 motivates the question of whether other i∈S p∈Ai i∈S p∈Ai candidate-approval rules satisfy stronger representation ax- T (p) ioms when restricted to the party-approval subdomain. We = |{i ∈ S | ui(T ) > 0}| = |S|. ui(T ) have studied this question for various rules besides PAV, and i∈S p∈Ai the answer was always negative.6 The second inequality holds because every voter in S strictly While the party-approval setting does not reduce the com- increases their utility when deviating from W to T ; the last plexity of computing PAV, it allows us to efficiently check equality holds because every voter in S must get some rep- whether a given committee provides EJR or PJR; both prob- resentation in T to deviate. As desired, it follows that there lems are coNP-hard in the candidate-approval setting (Aziz et al. 2017; 2018). For EJR, this follows from coherence be- has to be a party p2 in the support of T with MC (p2,W + coming simpler for party-approval elections. Our algorithm {p2}) |S|/|T | n/k. for checking PJR employs submodular minimization. Again, If any of these inequalities would be strict, that is, if details can be found in the full version. MC (p1,W) n/k, then W −{p } + {p } the committee 1 2 would have a PAV-score of 5 Discussion PAV(W ) − MC (p1,W)+MC (p2,W−{p1}+{p2}) In this paper, we have initiated the axiomatic analysis of PAV(W ) − MC (p1,W)+MC (p2,W + {p2}) (1) approval-based apportionment. On a technical level, it would > PAV(W ), be interesting to see whether the party-approval domain allows us to satisfy other combinations of axioms that are not which would contradict the choice of W . known to be attainable in candidate-approval elections. For Else, suppose that MC (p, W )=n/k for all parties p in instance, the compatibility between strong representation ax- the support of W and MC (p, W +{p})=n/k for all parties ioms and certain notions of support monotonicity is an open p in the support of T . If there is a party p1 in W that is ap- problem (Sanchez-Fern´ andez´ and Fisteus 2019). proved by some voter i ∈ S, we can choose an arbitrary party We have presented our setting guided by the application p2 from the support of T that i approves as well. Then, for of apportioning parliamentary seats to political parties. We voter i, the marginal contribution of p2 in W −{p1} + {p2} believe that this is an attractive application worthy of practi- 1 = 1 cal experimentation. Our formal setting has other interesting is ui(W −{p1}+{p2}) ui(W ) , but the marginal contribution 1 1 applications. An example would be participatory budgeting of p2 in W +{p2} for i is only = . This ui(W +{p2}) ui(W )+1 settings in which the provision of items of equal cost is de- implies MC (p2,W −{p1} + {p2}) > MC (p2,W + {p2}), cided, where the items come in different types. For instance, which makes inequality (1) strict and again contradicts the a university department could decide how to allocate Ph.D. optimality of W . scholarships across different research projects, in a way that Thus, one has to assume that no voter in S approves any respects the preferences of funding organizations. party in the support of W . Pick an arbitrary p in the support As another example, the literature on multiwinner elec- of T , and recall that MC (p,W + {p})=n/k for all p in tions suggests many applications to recommendation prob- the support of T . Thus, all inequalities in the derivation of lems (Skowron, Faliszewski, and Lang 2016). For instance, p∈P T (p) MC (p, W + {p}) |S| above must be equali- one might want to display a limited number of news arti- ties, which implies that this increase in PAV score must solely cles, movies, or advertisements in a way that fairly represents S n/k come from voters in . Thus, there are at least voters in 6 S who are not represented at all in W , but commonly approve We consider the candidate-approval rules SeqPAV, RevSeqPAV, p. This would be a violation of EJR, contradicting the fact Approval Voting (AV), SatisfactionAV, MinimaxAV, SeqPhragmen,´ MaxPhragmen,´ VarPhragmen,´ Phragmen-STV,´ MonroeAV, Greedy- that PAV satisfies this axiom. MonroeAV, GreedyAV, HareAV, and Chamberlin–CourantAV. Be- sides EJR and core stability, we consider JR and PJR (see Foot- Corollary 3. The core of a party-approval election is note 2). Definitions and results can be found in the full version of nonempty. the paper.

1860 the preferences of the audience. These preferences might be Brill, M.; Laslier, J.-F.; and Skowron, P. 2018. Multiwin- expressed not over individual pieces of content, but over con- ner approval rules as apportionment methods. Journal of tent producers (such as newspapers, studios, or advertising Theoretical Politics 30(3):358–382. companies), in which case our setting provides rules that Cheng, Y.; Jiang, Z.; Munagala, K.; and Wang, K. 2019. decide how many items should be contributed by each source. Group Fairness in Committee Selection. In Proceedings of Expressing preferences on the level of content producers is the 2019 ACM Conference on Economics and Computation natural in repeated settings, where the relevant pieces of con- (EC), 263–279. tent change too frequently to elicit voter preferences on each Duddy, C. 2015. Fair sharing under dichotomous preferences. occasion. Besides, content producers might reserve the right Mathematical Social Sciences 73:1–5. to choose which of their content should be displayed. In the general candidate-approval setting, the search con- Fain, B.; Munagala, K.; and Shah, N. 2018. Fair allocation tinues for rules that satisfy EJR and committee monotonicity, of indivisible public goods. In Proceedings of the 2018 ACM or core stability. But for the applications mentioned above, Conference on Economics and Computation (EC), 575–592. these guarantees are already achievable today. Fishburn, P. C. 1978. Axioms for approval voting: Direct proof. Journal of Economic Theory 19(1):180–185. Acknowledgements This work was partially supported Janson, S., and Oberg,¨ A. 2019. A piecewise contractive by the Deutsche Forschungsgemeinschaft under Grant BR dynamical system and Phragmen’s´ election method. Bulletin 4744/2-1. We thank Steven Brams and Piotr Skowron for de la Societ´ e´ Mathematique´ de France 147(3):395–441. suggesting the setting of party approval to us, and we thank Janson, S. 2016. Phragmen’s´ and Thiele’s election methods. Anne-Marie George, Ayumi Igarashi, Svante Janson, Jer´ omeˆ Technical report, arxiv:1611.08826. Lang, and Ariel Procaccia for helpful comments and discus- Kilgour, D. M., and Marshall, E. 2012. Approval balloting sions. for fixed-size committees. In Electoral Systems, Studies in Choice and Welfare. Springer. 305–326. References Laslier, J.-F., and Sanver, M. R., eds. 2010. Handbook on Aziz, H.; Brill, M.; Conitzer, V.; Elkind, E.; Freeman, R.; and Approval Voting. Studies in Choice and Welfare. Springer. Walsh, T. 2017. Justified representation in approval-based Mora, X., and Oliver, M. 2015. Eleccions mitjançant el committee voting. Social Choice and Welfare 48(2):461–485. vot d’aprovacio.´ El metode` de Phragmen´ i algunes variants. 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