Approval-Based Apportionment

11/19/2019 ∙ by Markus Brill, et al. ∙ Berlin Institute of Technology (Technische Universität Berlin) Carnegie Mellon University 0

In the apportionment problem, a fixed number of seats must be distributed among parties in proportion to the number of voters supporting each party. We study a generalization of this setting, in which voters cast approval ballots over parties, such that each voter can support multiple parties. This approval-based apportionment setting generalizes traditional apportionment and is a natural restriction of approval-based multiwinner elections, where approval ballots range over individual candidates. Using techniques from both apportionment and multiwinner elections, we are able to provide representation guarantees that are currently out of reach in the general setting of multiwinner elections: First, we show that core-stable committees are guaranteed to exist and can be found in polynomial time. Second, we demonstrate that extended justified representation is compatible with committee monotonicity.

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1 Introduction

The fundamental fairness principle of proportional representation is relevant in a variety of applications ranging from recommender systems to digital democracy (SLB+17a SLB+17a). It features most explicitly in the context of political elections, which is the language we adopt for this paper. In this context, proportional representation prescribes that the number of representatives championing a particular opinion in a legislature be proportional to the number of voters who favor that opinion.

In most democratic institutions, proportional representation is implemented via party-list elections: Candidates are members of political parties and voters are asked to indicate their favorite party; each party is then allocated a number of seats that is (approximately) proportional to the number of votes it received. The problem of transforming a voting outcome into a distribution of seats is known as apportionment. Analyzing the advantages and disadvantages of different apportionment methods has a long and illustrious political history and has given rise to a deep and elegant mathematical theory [Balinski and Young1982, Pukelsheim2014].

Unfortunately, forcing voters to choose a single party prevents them from communicating any preferences beyond their most preferred alternative. For example, if a voter feels equally well represented by several political parties, there is no way to express this preference within the voting system.

In the context of single-winner elections, approval voting has been put forward as a solution to this problem as it strikes an attractive compromise between simplicity and expressivity [Brams and Fishburn2007, Laslier and Sanver2010]. Under approval voting, each voter is asked to specify a set of candidates she “approves of,” i.e., voters can arbitrarily partition the set of candidates into approved candidates and disapproved ones. Proponents of approval voting argue that its introduction could increase voter turnout, “help elect the strongest candidate,” and “add legitimacy to the outcome” of an election (BrFi07c BrFi07c, pp. 4–8).

Due to the practical and theoretical appeal of approval voting in single-winner elections, a number of scholars have suggested to also use approval voting for multiwinner elections, in which a fixed number of candidates needs to be elected [Kilgour and Marshall2012]. In contrast to the single-winner setting, where the straightforward voting rule “choose the candidate approved by the highest number of voters” enjoys a strong axiomatic foundation [Fishburn1978], several ways of aggregating approval ballots have been proposed in the multiwinner setting (e.g., ABC+16a ABC+16a; Jans16a Jans16a).

Most work on approval-based multiwinner elections assumes that voters directly express their preference over individual candidates; we refer to this setting as candidate-approval elections. This assumption runs counter to widespread democratic practice, in which candidates belong to political parties and voters indicate preferences over these parties (which induce an implicit preference over candidates). In this paper, we therefore study party-approval elections, in which voters express approval votes over parties and a given number of seats must be distributed among the parties. We refer to the process of allocating these seats as approval-based apportionment.

We believe that party-approval elections are a promising framework for legislative elections in the real world. Allowing voters to express approval votes over parties enables the aggregation mechanism to coordinate like-minded voters. For example, two blocks of voters might currently vote for parties that they mutually disapprove of. Using approval ballots could reveal that the blocks jointly approve a party of more general appeal; giving this party more seats leads to mutual gain. This cooperation is particularly necessary for small minority opinions that are not centrally coordinated. In such cases, finding a commonly approved party can make the difference between being represented or votes being wasted because the individual parties receive insufficient support.

In contrast to approval voting over individual candidates, party-approval voting does not require a break with the current role of political parties—it can be combined with both “open list” and “closed list” approaches to filling the seats allocated to a party.

1.1 Related Work

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 aggregating approval votes over parties has not been considered before. Indeed, several scholars have explored possible generalizations of existing aggregation procedures.

For instance, BKP19a (BKP19a) study multiwinner approval rules that are inspired by classical apportionment methods. Besides the setting of candidate approval, they explicitly consider the case where voters cast party-approval votes. They conclude that these rules could “encourage coalitions across party or factional lines, thereby diminishing gridlock and promoting consensus.”

Such desire of compromise is only one motivation for considering party-approval elections, as exemplified by recent work by SpGe19a (SpGe19a). To allow for more efficient governing, they aim to concentrate the power of a legislature in the hands of few big parties, while nonetheless preserving the principle of proportional representation. To this end, they let voters cast party-approval votes and transform these votes into a party-list election by assigning each voter to one of her approved parties. One method for doing this (referred to as majoritarian portioning later in this paper) assigns voters to parties in such a way that the strongest party has as many votes as possible.

Several other papers consider extensions of approval-based voting rules to accommodate party-approval elections [Brams and Kilgour2014, Mora and Oliver2015, Janson2016, Janson and Öberg2019]. All of these papers have in common that they study specific rules or classes of rules, rather than exploring the party-approval setting in its own right.

Candidate-approval elections [Kilgour and Marshall2012, Aziz et al.2017]

Party-approval elections [approval-based apportionment]

Party-list elections [apportionment] [Balinski and Young1982, Pukelsheim2014]

(ii)

(i)

(iii)
Figure 1: Relations between the different settings of multiwinner elections. An arrow from to signifies that is a generalization of . The relationship corresponding to arrow (iii) has been explored by BLS18a (BLS18a). We establish and explore the relationship (i) in Section 3 and the relationship (ii) in Section 4.

1.2 Relation to Other Settings

Party-approval elections can be positioned between two well-studied voting settings (see Figure 1).

First, approval-based apportionment generalizes standard apportionment, which corresponds to party-approval elections in which all votes are singletons. This relation (depicted as arrow (i) in Figure 1) provides a generic two-step approach to define aggregation rules for approval-based apportionment problems: transform a party-approval instance to an apportionment instance, and then apply an apportionment method. In Section 3, we employ this approach to construct approval-based apportionment methods satisfying desirable properties.

Second, our setting can be viewed as a special case of approval-based multiwinner voting, in which voters cast candidate-approval votes. A party-approval election can be embedded in this setting by replacing each party by multiple candidates belonging to this party, and by interpreting a voter’s approval of a party as approval of all of its candidates. This embedding establishes party-approval elections as a subdomain of candidate-approval elections (see arrow (ii) in Figure 1). In Section 4, we explore the axiomatic and computational ramifications of this domain restriction.

1.3 Contributions

In this paper, we formally introduce the setting of approval-based apportionment and explore different possibilities of constructing axiomatically desirable aggregation methods for this setting. Besides its conceptual appeal, this setting is also interesting from a technical perspective.

Exploiting the relations described in Section 1.2, we resolve problems that remain open in the more general setting of approval-based multiwinner voting. First, we prove that committee monotonicity is compatible with extended justified representation (a representation axiom proposed by ABC+16a ABC+16a) by providing a rulemethod that satisfies both properties. Second, we show that the core of an approval-based apportionment problem is always nonempty and that core-stable committees can be found in polynomial time.

Besides these positive results, we verify for a wide range of multiwinner voting rules that their axiomatic guarantees do not improve in the party-approval setting, and that some rules remain NP-hard to evaluate. On the other hand, we show that it becomes tractable to check whether a committee provides extended justified representation or the weaker axiom of proportional justified representation.

Omitted proofs as well as further definitions and results can be found in Appendices B and A.

2 The Model

A party-approval election is a tuple consisting of a set of voters , a finite set of parties , a ballot profile where each ballot is the set of parties approved by voter , and the committee size . We assume that for all . When considering computational problems, we assume that is encoded in unary (see Footnote 5).

A committee in this setting is a multiset over parties, which determines the number of seats assigned to each party . The size of a committee is given by , and we denote multiset addition and subtraction by and , respectively. A party-approval rule is a function that takes a party-approval election as input and returns a committee of valid size .111This definition implies that rules are resolute, that is, only a single committee is returned. In the case of a tie between multiple committees, a tiebreaking mechanism is necessary. Our results hold independently of the choice of a specific tiebreaking mechanism.

In our axiomatic study of party-approval rules, we focus on two axioms capturing proportional representation: extended justified representation and core stability [Aziz et al.2017].222Some results in the appendix refer to the weaker representation axioms of justified representation (JR) [Aziz et al.2017] and proportional justified representation (PJR) [Sánchez-Fernández et al.2017]; see Section A.1. It is well known that EJR implies PJR and that PJR implies JR. Both axioms are derived from their analogs in multiwinner elections (see Section 4.2) and can be defined in terms of quota requirements.

For a party-approval election and a subset of voters, define the quota of as . Intuitively, corresponds to the number of seats that the group “deserves” to be represented by (rounded down).

Definition 1.

A committee provides extended justified representation (EJR) for a party-approval election if there is no subset of voters such that and for all .

In words, EJR requires that for every voter group with a commonly approved party, at least one voter of the group should be represented by many candidates. A party-approval rule is said to satisfy EJR if it only produces committees providing EJR.

We can obtain a stronger representation axiom by removing the requirement of a commonly approved party.

Definition 2.

A committee is core stable for a party-approval election if there is no nonempty subset and committee of size such that for all . The core of a party-approval election is defined as the set of all core-stable committees.

Core stability requires adequate representation even for voter groups that cannot agree on a common party, by ruling out the possibility that the group can deviate to a smaller committee that represents all voters in the group strictly better. It follows from the definitions that core stability is a stronger requirement than EJR: If a committee violates EJR, there is a group that would prefer any committee of size that assigns all seats to the commonly approved party.

A final, non-representational axiom that we will discuss is committee monotonicity. A party-approval rule satisfies this axiom if, for all party-approval elections , it holds that . Committee monotonic rules avoid the so-called Alabama paradox, in which a party loses a seat when the committee size increases. Besides, committee monotonic rules can be used to construct proportional rankings [Skowron et al.2017].

3 Constructing Party-Approval Rules via Portioning and Apportionment

Party-approval elections are a generalization of party-list elections, which can be thought of as party-approval elections in which all approval sets are singletons. Since there is a rich body of research on apportionment methods, it is natural to examine whether we can employ these methods for our setting as well. To use them, we will need to translate party-approval elections into the party-list domain on which apportionment methods operate. This translation thus needs to transform approval votes over parties into a target percentage for each party. Motivated by time sharing, BMS05 (BMS05) have developed a theory of such transformation rules, further studied by Dudd15a (Dudd15a) and ABM19 (ABM19). We will refer to this framework as portioning.

The approach explored in this section, then, divides the construction of a party-approval rule into two independent steps: (1) portioning, which maps a party-approval election to a vector of parties’ shares; followed by (2) apportionment, which transforms the shares into a seat distribution.

Both the portioning and the apportionment literature have discussed representation axioms similar in spirit to EJR and core stability. For both settings, several rules have been found to satisfy these properties. One might hope that by composing two rules that are each representative, we obtain a party-approval rule that is also representative (and satisfies, say, EJR). If we succeed in finding such a combination, it is likely that the resulting voting rule will automatically satisfy committee monotonicity since most apportionment methods satisfy this property. In the general candidate-approval setting (considered in Section 4), the existence of a rule satisfying both EJR and committee monotonicity is an open problem.

3.1 Preliminaries

We start by introducing relevant notions from the literature of portioning [Bogomolnaia, Moulin, and Stong2005, Aziz, Bogomolnaia, and Moulin2019] and apportionment [Balinski and Young1982, Pukelsheim2014], with notations and interpretations suitably adjusted to our setting.

Portioning

A portioning problem is a triple , just as in party-approval voting but without a committee size. A portioning is a function with . We interpret as the “vote share” of party . A portioning method maps each triple to a portioning.

Our minimum requirement on portioning methods will be that they uphold proportionality if all approval sets are singletons, i.e., if we are already in the party-list domain. Formally, we say that a portioning method is faithful if for all with for all , the resulting portioning satisfies for all . Among the portioning methods considered by ABM19 (ABM19), only three are faithful. They are defined as follows.

Conditional utilitarian portioning

selects, for each voter , as a party in approved by the highest number of voters. Then, for all .

Random priority

computes portionings, one for each permutation of , and returns their average. The portioning for maximizes , breaking ties by maximizing , and so forth.

Nash portioning

selects the portioning maximizing the Nash welfare .

The last method seems particularly promising because it satisfies portioning versions of core stability and EJR [Aziz, Bogomolnaia, and Moulin2019].

We will also make use of a more recent portioning approach, which was proposed by SpGe19a (SpGe19a) in the context of party-approval voting.

Majoritarian portioning

proceeds in rounds . Initially, all parties and voters are active. In iteration , we select the active party that is approved by the highest number of active voters. Let be the set of active voters who approve . Then, set to , and mark and all voters in as inactive. If active voters remain, the next iteration is started; else, is returned.

Under majoritarian portioning, the approval preferences of voters who have been assigned to a party are ignored in further iterations. Note that conditional utilitarian portioning can similarly be seen as a sequential method, in which the preferences of inactive voters are not ignored.

Apportionment

An apportionment problem is a tuple , which consists of a finite set of parties , a portioning specifying the vote shares of parties, and a committee size . Committees are defined as for party-approval elections, and an apportionment method maps apportionment problems to committees of size .

An apportionment method satisfies lower quota if each party is always allocated at least seats in the committee. Furthermore, an apportionment method is committee monotonic if for every apportionment problem .

Among the standard apportionment methods, only one satisfies both lower quota and committee monotonicity: the D’Hondt method (aka Jefferson method).333All other divisor methods fail lower quota, and the Hamilton method is not committee monotonic [Balinski and Young1982]. The method assigns the seats iteratively, each time giving the next seat to the party with the largest quotient , where denotes the number of seats already assigned to . Another apportionment method satisfying lower quota and committee monotonicity is the quota method, due to BaYo75a (BaYo75a). It is identical to the D’Hondt method, except that, in the th iteration, only parties satisfying are eligible for the allocation of the next seat.

Composition

If we take any portioning method and any apportionment method, we can compose them to obtain a party-approval rule. Note that if the apportionment method is committee monotonic then so is the composed rule, since the portioning is independent of .

3.2 Composed Rules That Fail EJR

Perhaps surprisingly, many pairs of portioning and apportionment methods fail EJR. This is certainly true if the individual parts are not representative themselves. For example, if an apportionment method properly fails lower quota (in the sense that there is a rational-valued input on which lower quota is violated), then one can construct an example profile on which any composed rule using fails EJR: Construct a party-approval election with singleton approval sets in which the voter counts are proportional to the shares in the counter-example . Then any faithful portioning method, applied to this election, must return . Since fails lower quota on , the resulting committee will violate EJR. By a similar argument, an apportionment method that violates committee monotonicity on some rational portioning will, when composed with a faithful portioning method, give rise to a party-approval rule that fails committee monotonicity.

To our knowledge, among the named and studied apportionment methods, only two satisfy both lower quota and committee monotonicity: D’Hondt and the quota method. However, it turns out that the composition of either option with the conditional-utilitarian, random-priority, or Nash portioning methods fails EJR, as the following examples show.

Example 1.

Let , , .

Then, the conditional utilitarian solution sets , , and . Any apportionment method satisfying lower quota allocates four seats to , one each to and , and none to . The resulting committee does not provide EJR since the last two voters, who jointly approve , have a quota of that is not met.

Example 2.

Let , , and .

Random priority chooses the portioning , , and . Both D’Hondt and the quota method allocate four seats to , two seats to , and none to the other two parties. This clearly violates the claim to representation of the sixth voter (with ).

Nash portioning produces a fairly similar portioning, with , , and . D’Hondt and the quota method produce the same committee as above, leading to the same EJR violation.

At first glance, it might be surprising that Nash portioning combined with a lower-quota apportionment method violates EJR (and even the weaker axiom JR). Indeed, Nash portioning satisfies core stability in the portioning setting, which is a strong notion of proportionality, and the lower-quota property limits the rounding losses when moving from the portioning to a committee. As expected, in the election of Example 2, the Nash solution itself gives sufficient representation to the sixth voter since . However, since both and are below on their own, lower quota does not apply to either of the two parties, and the sixth voter loses all representation in the apportionment step.

3.3 Composed Rules That Satisfy EJR

As we have seen, several initially promising portioning methods fail to compose to a rule that satisfies EJR. One reason is that these portioning methods are happy to assign small shares to several parties. The apportionment method may round several of those small shares down to zero seats. This can lead to a failure of EJR when not enough parties obtain a seat. It is difficult for an apportionment method to avoid this behavior since the portioning step obscures the relationships between different parties that are apparent from the approval ballots of the voters.

Majoritarian portioning is designed to maximize the size of parties. Thus, it tends to avoid the problem we have identified. While it fails the strong representation axioms that Nash portioning satisfies, this turns out not be crucial: Composing majoritarian portioning with any apportionment method satisfying lower quota yields an EJR rule. If we use an apportionment method that is also committee monotonic, such as D’Hondt or the quota method, we obtain a party-approval rule that satisfies both EJR and committee monotonicity.444As long as the apportionment method is computable in polynomial time (which is the case for D’Hondt and the quota method), the same holds for the resulting party-approval rule.

Theorem 1.

Let be a committee monotonic apportionment method satisfying lower quota. Then, the party-approval rule composing majoritarian portioning and satisfies EJR and committee monotonicity.

Proof.

Consider a party-approval election and let be the outcome of majoritarian portioning applied to . Let and be the voter groups and parties in the construction of majoritarian portioning, so that for all .

Consider the committee and suppose that EJR is violated, i.e., that there exists a group with and for all .

Let be minimal such that . We now show that . By the definition of , no voter in approves of any of the parties ; thus, all those voters remain active in round . Consider a party . In the th iteration of majoritarian portioning, this party had an approval score of at least . Therefore, the party that is chosen in the th iteration has an approval score that is at least (of course, is possible). The approval score of party equals . Therefore, .

Since , we have . Since satsifies lower quota, it assigns at least seats to party . Now consider a voter . Since this voter approves party , we have , a contradiction.

This shows that EJR is indeed satisfied; committee monotonicity follows from the committee monotonicity of . ∎

While the party-approval rules identified by Theorem 1 satisfy EJR and committee monotonicity, they do not quite reach our gold standard of representation, i.e., core stability.

Example 3.

Let , let , with the following approval sets: 4 times , 3 times , once , 4 times , 3 times , and once . Note the symmetry between and , and between and . Majoritarian portioning allocates to and each to and . Any lower-quota apportionment method must translate this into 8 seats for and 4 seats each for and . This committee is not in the core: Let be the coalition of all 14 voters who approve multiple parties, and let allocate 4 seats to and 5 seats each to and . This gives strictly higher representation to all members of the coalition.

The example makes it obvious why majoritarian portioning cannot satisfy the core: All voters approving of get deactivated after the first round, which makes seem universally preferable to . However, is a useful vehicle for cooperation between the group approving and the group approving . Since majoritarian portioning is blind to this opportunity, it cannot guarantee core stability.

The example also illustrates the power of core stability: The deviating coalition does not agree on any single party they support, but would nonetheless benefit from the deviation. There is room for collaboration, and core stability is sensitive to this demand for better representation.

4 Constructing Party-Approval Rules via Multiwinner Voting Rules

In the previous section, we applied tools from apportionment, a more restrictive setting, to our party-approval setting. Now, we go in the other direction, and apply tools from a more general setting: As mentioned in Section 1.2, party-approval elections can be viewed as a special case of candidate-approval elections, i.e., multiwinner elections in which approvals are expressed over individual candidates rather than parties. After introducing relevant candidate-approval notions, we show how party-approval elections can be translated into candidate-approval elections. This embedding allows us to apply established candidate-approval rules to our setting. Exploiting this fact, we will prove the existence of core-stable committees for party-approval elections.

4.1 Preliminaries

A candidate-approval election is a tuple . Just as for party-approval elections, is a set of voters, is a finite set, is an -tuple of nonempty subsets of , and is the committee size. The conceptual difference is that is a set of individual candidates rather than parties. This difference manifests itself in the definition of a committee because a single candidate cannot receive multiple seats. That is, a candidate committee is now simply a subset of with cardinality . (Therefore, it is usually assumed that .) A candidate-approval rule is a function that maps each candidate-approval election to a candidate committee.

A diverse set of such voting rules has been proposed since the late 19th century [Kilgour and Marshall2012, Aziz et al.2017, Janson2016], out of which we will only introduce the one which we use for our main positive result. Let denote the th harmonic number, i.e., . Given , the candidate-approval rule proportional approval voting (PAV), introduced by Thie95a (Thie95a), chooses a candidate committee maximizing the PAV score .

We now describe EJR and core stability in the candidate-approval setting, from which our versions of these axioms are derived. Recall that . A candidate committee provides EJR if there is no subset and no integer such that , , and for all . (The requirement is often referred to as cohesiveness.) A candidate-approval rule satisfies EJR if it always produces EJR committees.

The definition of core stability is even closer to the version in party-approval: A candidate committee is core stable if there is no nonempty group and no set of size such that for all . The core consists of all core-stable candidate committees.

4.2 Embedding Party-Approval Elections

We have informally argued in Section 1.2 that party-approval elections constitute a subdomain of candidate-approval elections. We formalize this notion by providing an embedding of party-approval elections into the candidate-approval domain. For a given party-approval election , we define a corresponding candidate-approval election with the same set of voters and the same committee size . The set of candidates contains many “clone” candidates for each party , and a voter approves a candidate  in the candidate-approval election iff she approves the corresponding party  in the party-approval election. This embedding establishes party-approval elections as a subdomain of candidate-approval elections. As a consequence, we can apply rules and axioms from the more general candidate-approval setting also in the party-approval setting.

In particular, the generic way to apply a candidate-approval rule for a party-approval election consists in (1) translating the party-approval election into a candidate-approval election, (2) applying the candidate-approval rule, and (3) counting the number of chosen clones per party to construct a committee over parties. Note that, since is encoded in unary, the running time is blown up by at most a polynomial factor.555For candidate-approval elections, it does not make sense to have more seats than candidates, whereas for party-approval elections it is natural to have more seats than parties. If was encoded in binary, even greedy candidate-approval algorithms would suddenly have exponential running time; this would make running times between candidate approval and party-approval hard to compare and would blur the intuitive distinction between simple and complex algorithms. Encoding in unary sidesteps this technical complication.

Having established party-approval elections as a subdomain of candidate-approval elections, our variants of EJR and core stability (Definitions 2 and 1) are immediately induced by their candidate-approval counterparts. In particular, any candidate-approval rule satisfying an axiom in the candidate-approval setting will satisfy the corresponding axiom in the party-approval setting as well. Note that, by restricting our view to party approval, the cohesiveness requirement of EJR is reduced to requiring a single commonly approved party.

4.3 PAV Guarantees Core Stability

A powerful stability concept in economics, core stability is a natural extension of EJR. It is particularly attractive because blocking coalitions are not required to be coherent at all, just to be able to coordinate for mutual gain. Our earlier Example 3 illustrates how a coalition might deviate in spite of not agreeing on any approved party.

Unfortunately, it is still unknown whether core-stable candidate committees exist for all candidate-approval elections. FMS18 (FMS18) give positive approximate results for a variant of core stability in which blocking coalitions get to provide sets of candidates  of size  but have to increase their utilities by at least a factor of to be counterexamples to their notion of core stability. They provide a nonconstant approximation to the core in our sense, but nonemptyness remains open. Recently, CJM+19 (CJM+19) showed that there always exist randomized committees providing core stability (over expected representation), but it is not clear how their approach based on two-player zero-sum game duality would extend to deterministic committees.

All standard candidate-approval rules either already fail weaker representation axioms such as EJR or fail the core. In particular, ABC+16a (ABC+16a) have shown that PAV satisfies EJR, but may produce non-core-stable candidate committees even for candidate-approval elections for which core-stable candidate committees are known to exist.

By contrast, we show that PAV guarantees core stability in the party-approval setting. We follow the structure of the aforementioned proof showing that PAV satisfies EJR for candidate-approval elections [Aziz et al.2017].

Theorem 2.

For every party-approval election, PAV chooses a core-stable committee.

Proof.

Consider a party-approval election and let be the committee selected by PAV. Assume for contradiction that is not core stable. Then, there is a nonempty coalition and a committee such that and for every voter .

Let denote the number of seats in that are allocated to parties approved by voter , i.e., . Furthermore, for a party with , we let denote the marginal contribution to the PAV score of allocating a seat to , i.e., . Observe that , where . The sum of all marginal contributions satisfies

Note that terms for and quotients for are undefined in the calculation above, but that they only appear with factor .

It follows that the average marginal contribution of all  seats in is at most , and consequently, that there has to be a party  with a seat in such that . Using a similar argument, we show that there is also a party  with which would increase the PAV score by at least if it received an additional seat in :

The second inequality holds because every voter in strictly increases their utility when deviating from to ; the last equality holds because every voter in must get some representation in to deviate. As desired, it follows that there has to be a party in the support of with .

If any of these inequalities would be strict, that is, if or , then the committee would have a PAV-score of

which would contradict the choice of .

Else, suppose that for all parties in the support of and for all parties in the support of . If there is a party in that is approved by some voter , we can choose an arbitrary party from the support of that approves as well. Then, for voter , the marginal contribution of in is , but the marginal contribution of in for is only . This implies , which makes inequality (1) strict and again contradicts the optimality of .

Thus, one has to assume that no voter in approves any party in the support of . Pick an arbitrary in the support of , and recall that for all in the support of . Thus, all inequalities in the derivation of above must be equalities, which implies that this increase in PAV score must solely come from voters in . Thus, there are at least voters in who are not represented at all in , but commonly approve . This would be a violation of EJR, contradicting the fact that PAV satisfies this axiom. ∎

Corollary 3.

The core of a party-approval election is nonempty.

An immediate follow-up question is whether core-stable committees can be computed efficiently. PAV committees are known to be NP-hard to compute in the candidate-approval setting, and we confirm in Section B.1 that hardness still holds in the party-approval subdomain.

Equally confronted with the computational complexity of PAV, AEH+18a (AEH+18a) proposed a local-search variant of PAV, which runs in polynomial time and guarantees EJR in the candidate-approval setting. Using the same approach, we can find a core-stable committee in the party-approval setting. We defer the proof to Section B.2.

Theorem 4.

Given a party-approval election, a core-stable committee can be computed in polynomial time.

Theorem 2 motivates the question of whether other candidate-approval rules satisfy stronger representation axioms when restricted to the party-approval subdomain. We have studied this question for various rules besides PAV, and the answer was always negative.666We consider the candidate-approval rules SeqPAV, RevSeqPAV, Approval Voting (AV), SatisfactionAV, MinimaxAV, SeqPhragmén, MaxPhragmén, VarPhragmén, Phragmén-STV, MonroeAV, GreedyMonroeAV, GreedyAV, HareAV, and Chamberlin–CourantAV. Besides EJR and core stability, we consider JR and PJR (see Footnote 2). Definitions and results can be found in Appendix A.

While the party-approval setting does not reduce the complexity of computing PAV, it allows us to check efficiently check whether a given committee provides EJR or PJR; both problems are coNP-hard in the candidate-approval setting [Aziz et al.2017, Aziz et al.2018]. For EJR, this follows from coherence becoming simpler for party-approval elections. Our algorithm for checking PJR employs submodular minimization. For details, we refer to Section B.3.

5 Discussion

In this paper, we have initiated the axiomatic analysis of approval-based apportionment. On a technical level, it would be interesting to investigate whether the party-approval domain allows to satisfy other combinations of axioms that are (currently) not attainable in candidate-approval elections. For instance, it is not known whether strong representation axioms are compatible with certain notions of support monotonicity [Sánchez-Fernández and Fisteus2019].

We have presented our setting guided by the application of apportioning parliamentary seats to political parties. We believe that this is an attractive application worthy of practical experimentation. Our formal setting has other interesting applications. An example would be participatory budgeting settings in which the provision of items of equal cost is decided, where the items come in different types. For instance, a university department could decide how to allocate Ph.D. scholarships across different research projects, in a way that respects the preferences of funding organizations.

As another example, the literature on multiwinner elections suggests many applications to recommendation problems [Skowron, Faliszewski, and Lang2016]. For instance, one might want to display a limited number of news articles, movies, or advertisements in a way that fairly represents the preferences of the audience. These preferences might be expressed not over individual pieces of content, but over content producers (such as newspapers, studios, or advertising companies), in which case our setting provides rules that decide how many items should be contributed by each source. Expressing preferences on the level of content producers is natural in repeated settings, where the relevant pieces of content change too frequently to elicit voter preferences on each occasion. Besides, content producers might reserve the right to choose which of their content should be displayed.

In the general candidate-approval setting, the search continues for rules that satisfy EJR and committee monotonicity, or core stability. But for the applications mentioned above, these guarantees are already achievable today.

Acknowledgements

This work was partially supported by the Deutsche Forschungsgemeinschaft under Grant BR 4744/2-1. We thank Steven Brams and Piotr Skowron for suggesting the setting of party approval to us, and we thank Anne-Marie George, Ayumi Igarashi, Svante Janson, Jérôme Lang, and Ariel Procaccia for helpful comments and discussions.

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Appendix

The appendix is organized as follows.

In Appendix A we investigate for several candidate-approval rules whether they fulfill stronger axiomatic properties in the party-approval subdomain. To this end, we introduce all studied axioms and rules in Sections A.2 and A.1, respectively, and present the results in Section A.3. Table 1 summarizes the results of this section.

In Appendix B we analyze the introduced axioms and rules from a computational complexity perspective. First, we show that computing a committee with maximum PAV score (which is core stable due to Theorem 2) as well as computing a winning committee for MaxPhragmén is still NP-hard in the party-approval subdomain (Section B.1). Second, we show that a core-stable committee can be computed in polynomial time (Section B.2). Third, we provide polynomial-time algorithms for checking whether a given committee satisfies EJR or PJR for a party-approval election (Section B.3).

Appendix A Axiomatic Study

a.1 Weaker Representation Axioms

Let be a party-approval election and be a committee.

The committee provides justified representation (JR) if there is no coalition of voters of size at least , all of whose voters commonly approve of a party, but for all . A voting rule satisfies JR if it returns committees providing JR for every party-approval election.

The committee provides proportional justified representation (PJR), if there is no coalition of voters , all of whose voters commonly approve a party, but . A voting rule satisfies PJR if it returns committees providing PJR for every party-approval election.

a.2 Candidate-Approval Rules

In the following we introduce approval-based committee voting rules that were repeatedly studied in the literature. Note that we use the language of the candidate-approval setting and in particular, is a set (not a multiset) of candidates. In order to apply the described rules in our setting, we can transform any party-approval election to a candidate-approval election by introducing clones of each party. For a formal description of the transformation see Section 4.2 in the main part of the paper.

Sequential PAV (SeqPAV) and Reverse Sequential PAV (RevSeqPAV)

Additionally to the PAV rule, Thiele proposed two strongly related voting rules which run in polynomial time and greedily try to maximize the PAV score: SeqPAV and RevSeqPAV, both of which construct committees in a sequential manner [Thiele1895]. SeqPAV starts with the empty committee and iteratively adds the candidate that increases the PAV score the most, until candidates are selected. RevSeqPAV initially assumes every candidate to be winning; afterwards, the candidate whose removal from the committee decreases the committee’s PAV score the least is removed, until there are only candidates left.

Approval Voting (AV)

Approval voting is the straightforward generalization of Approval Voting for single-winner elections [Brams and Fishburn2007]. The committee consists of the candidates with the largest number of voters who approve the respective candidate. Thus, approval voting returns the set of size that maximizes the approval score .

Satisfaction Approval Voting (SAV)

SAV selects the candidates with the highest satisfaction score [Brams and Kilgour2014]. Thereby, SAV outputs the committee with that maximizes the score

Minimax Approval Voting (MAV)

MAV returns the committee that minimizes the maximum Hamming distance of the committee to any approval ballot [Brams, Kilgour, and Sanver2007]. The Hamming distance between a ballot and a committee is simply the symmetric difference between the two candidate sets, i.e. . MAV outputs the committee with minimal score .

Phragmén’s rules (MaxPhragmén, VarPhragmén, Seq-Phragmén, and Phragmén-STV)

Phragmén’s voting rules, which were like PAV developed in the late 19th century.777Phragmén’s original papers are written in French and Swedish [Phragmén1894, Phragmén1895, Phragmén1896, Phragmén1899]; an English account of this work was composed by Jans16a (Jans16a). The central concept behind his rules is the idea that every candidate in the committee carries some “load”, which has to be distributed among the voters who approve this candidate. Phragmén’s rules then aim to select a committee where this load can be distributed evenly among the voters.

Following BFJL16a (BFJL16a) for the formal definition, we call a real-valued vector a load distribution if the following properties hold:

(1)
(2)
(3)
(4)

In this definition, represents the load of candidate on voter . Thus, the load that voter carries is . Properties (3) and (4) ensure that each load distribution corresponds to a committee of size —namely, every candidate whose total load is one is part of the committee, while candidates whose total load is zero are not.

There are different ways of measuring how balanced a load distribution is. This corresponds to different Phragmén rules. The first rule MaxPhragmén minimizes the maximal total load of any voter. Formally, MaxPhragmén returns the committee that corresponds to the load distribution where is minimal.888For technical reasons, the usual lexicographic tie-breaking does not suffice here and has to be replaced with a more involved tie-breaking mechanism; for details, see the work of BFJL16a (BFJL16a). The second rule VarPhragmén

minimizes the variance of the voter loads by computing the load distribution that minimizes

and returning the corresponding committee. The third rule SeqPhragmén constructs it’s committee sequentially by starting with an empty committee and then iteratively adding the candidate that increases the maximum voter load the least. See Algorithm 1 for a formal definition.

function SeqPhragmén()
     
      for every
     while  do
          for every
         
         
          for every
     end while
     return W
end function
Algorithm 1 SeqPhragmén (Candidate-Approval Rule)

If we want to make use of MaxPhragmén or VarPhragmén in the party-approval setting, we can do so without utilizing the previously described embedding of party-approval elections into the space of candidate-approval elections. We do so by simply treating the parties as candidates and slightly adjusting the definition of a load distribution such that constraint (1) becomes and constraint (4) becomes .

Additionally, Phragmén developed a voting rule which is closely related to the well-known Single Transferable Vote system [Janson2016, Sánchez-Fernández, Elkind, and Lackner2018, Camps, Mora, and Saumell2019]. We refer to this rule as Phragmén-STV; CMS19a (CMS19a) call it the Eneström-Phragmén method. In contrast to the previous three algorithms, it does not make use of load distributions. The core idea of the sequential procedure is the following: Initially, every voter has weight 1 and every candidate’s score is the sum of all approving voters’ weights. In every round, the candidate with the highest score is added to the committee. If a voter with weight approves the candidate who is added to the committee, then their weight will be updated to if , and to otherwise. This process is repeated until all seats are assigned.

Monroe’s rule (MonroeAV and GreedyMonroeAV)

MonroeAV was originally intended for linear preferences of voters [Monroe1995]. Every voter is represented by exactly one winning candidate, but with the restriction that every candidate can only represent or many voters. Formally, a mapping for a given committee describes that voter is represented by . Such a mapping is a valid mapping if for all . The score of a valid mapping is the number of voters who are represented in by a candidate they approve. The Monroe score of a committee is the maximum score of any valid mapping for , and MonroeAV returns the committee with the highest Monroe score.

SFS15a (SFS15a) proposed a greedy variant of MonroeAV, which was later adopted for the approval setting by SFF+17a (SFF+17a). We refer to this algorithm as GreedyMonroeAV. GreedyMonroeAV constructs its committee in rounds. It is initialized with and . In round , the algorithm selects and such that the number of voters in who approve is maximal. Meanwhile, is restricted to be of size if and to be of size otherwise. At the end of each round, is added to and the voters in are removed from .

GreedyAV and HareAV

In GreedyAV [Aziz et al.2017], the committee is constructed iteratively by adding the candidate that satisfies the most voters, who do not approve any of the previously added candidates. More formally, the algorithm initially starts with and . While and is nonempty, the candidate with highest approval score999As usual, ties are broken lexicographically. with regard to is chosen and added to . Then, the ballots of all voters in are removed from and the next candidate is picked analogously. If at some point and is empty, then is filled up with arbitrary candidates until .

HareAV is a variant of GreedyAV where—after is added to the committee—not the ballots of all voters in are removed from , but instead only many ballots are removed.

Chamberlin-Courant’s rule (CCAV)

ChCo83a (ChCo83a) developed a multiwinner voting rule which was originally defined for voters with linear preferences. We define an adapted version for the setting of approval votes. The utility of a voter for a committee is defined to be if and otherwise. CCAV selects a committee that maximizes the sum of voters’ utilities.

a.3 Results

Rule JR PJR EJR Core Stability
PAV
SeqPAV - - - -
RevSeqPAV - - - -
AV - - - -
SAV - - - -
MAV - - - -
SeqPhragmén - -
MaxPhragmén - -
VarPhragmén - - -
Phragmén-STV - -
MonroeAV - - -
GreedyMonroeAV - - -
GreedyAV - - -
HareAV - - -
CCAV - - -
Table 1: The table contains a summary of the axiomatic properties of candidate-approval rules within the subdomain of party-approval elections.

We summarize the axiomatic properties of all above introduced voting rules when applied in the party-approval subdomain in Table 1. In the following we present the corresponding proofs following the table from top to bottom.

The proof that PAV satisfies core stability can be found in Section 4.3 of the main part of the paper.

We start by showing that, equivalently to the candidate-approval setting, the two greedy variants of PAV, SeqPAV and RevSeqPAV do not fulfill JR in the party-approval setting. Indeed, the counterexamples presented by ABC+16a (ABC+16a) and Aziz17a-corrected (Aziz17a-corrected) also show that both rules violate JR for party-elections.

Proposition 5.

In party-approval elections, SeqPAV violates JR for and RevSeqPAV violates JR for .

Proof.

The adapted counterexample by ABC+16a (ABC+16a) has voters and parties . First, assume . The ballot profile is as follows:

In the beginning of the SeqPAV computation, both and have a score of . As they are approved by disjoint set of voters, in the first two iterations both parties receive each one seat. In the third iteration, has the highest score of and receives a seat. In the remaining iterations, have a score of , and have a score of , while has only score points. The parties already present in the committee all have a score smaller than . Furthermore, giving each of the parties with score one seat does not affect the score of the other parties—therefore all of them receive one seat. Thus, receive each one seat in the committee, making the voters approving only unrepresented. This violates JR.

For , one can add additional parties and additional voters to the ballot profile, such that for each new party , 120 new voters approve only . This does not affect the SeqPAV computation described above. After the first ten iterations, the remaining seats are given to the parties . Because there are more parties than seats left, one of these parties does not receive a seat, resulting again in 120 cohesive voters who are unrepresented. This violates JR, as .

In an analogous way, we can adapt the counterexample by Aziz17a-corrected (Aziz17a-corrected), showing that RevSeqPAV violates JR for . ∎

When applied in the party-approval setting, the voting rules AV and SAV select the party with the highest approval respectively satisfaction score and allocate all available seats to this party. Evidently, both rules do not satisfy JR. The same holds for MAV.

Proposition 6.

AV, SAV, and MAV violate JR in the party-approval setting for .

Proof.

For AV and SAV, consider a party-approval election with two parties and , and voters with approval ballots . In AV’s and SAV’s committee, receives all seats, resulting in uninamous unrepresented voters. This violates JR.

For MAV, consider an election with four parties and voters with ballots . JR would require to receive at least one seat. However, MAV does not select such a committee: First, consider the committee where receives all seats. In the induced candidate-approval committee, the Hamming distance between any approval set and the committee is .101010Keep in mind that in the induced candidate-approval election, the approval sets contain copies of every party as candidates. Secondly, consider any committee where has at least one seat. Then, we have in the induced candidate-approval election. Thus, has a smaller maximal distance than any . As a result, MAV does not return a committee that allocates at least one seat to , a contradiction to JR. ∎

Proposition 7.

SeqPhragmén fails EJR for on party-approval elections.

Proof.

Fix a natural number . We construct a party-approval election with parties . The ballot profile of the many voters is as follows:

We first ignore the voters approving and focus on the remaining ones. Initially, adding a seat to or would increase the maximal voter load to , while giving one seat would increase it to . Without loss of generality, assume receives this seat. Then, giving the next seat to or would increase the maximal load to ; giving it to would increase it to , and giving the seat to would increase it to . Thus, we can assume receives the seat. Analogously, the next two seats are allocated to and , respectively—the exact computations can be found in Table 2. The fifth seat would then be allocated again to , increasing the maximal voter load to .

Party Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5
A 0.125 0.25 0.34570 0.42944 0.50272
B 0.125 0.23438 0.35938 0.44312 0.51639
C 0.125 0.23438 0.33008 0.45508 0.52835
D 0.125 0.23438 0.33008 0.41382 0.53882
X 0.25 0.28125 0.33984 0.42236 0.52582
Table 2: The SeqPhragmén computation for the profile in Proposition 7, when the party is ignored. The table shows the values of for a party in the first five iterations, rounded to five significant digits. The bold entries denote which party receives a seat (with lexicographic tie-breaking).

Taking the voters for into account would not affect the computation above, because all voters who approve do not approve any other party. Thus, in every SeqPhragmén iteration, either receives a seat or one of receives a seat, until all have one seat. Adding a seat to increases the load of a voter approving by . Thus, if seats are allocated to , each -voter would have a load of . Observe that and indeed, for all . Therefore, SeqPhragmén returns a committee where each receive one seat and receives the remaining seats.

This is a contradiction to EJR: Since , EJR demands one of the four voters approving to be represented at least twice in the committee. This is not the case in the committee selected by SeqPhragmén. ∎

In order to prove that MaxPhragmén and VarPhragmén violate EJR, we take a detour of arguing that another proportional representation axiom (introduced by SFF+17a SFF+17a) which is incompatible to EJR in the candidate-approval setting remains incompatible to EJR in the party-approval setting. Since MaxPhragmén and VarPhragmén fulfill this property, this suffices to prove our claim.

The axiom reflects the idea that every candidate represents voters who are all approve of this representation. More precisely, given an election where divides , a committee satisfies perfect representation (PR) if it is possible to partition the voters into distinct sets all of size and assign to each a different winning candidate such that all voters in approve . A voting rule satisfies PR if it always returns a committee that satisfies PR, whenever such a committee exists.

The proof by SFF+17a (SFF+17a) showing that EJR and PR are incompatible for candidate-approval elections can be adapted straightforwardly to the party-approval subdomain by interpreting candidates as parties.

Proposition 8.

There exists no party-approval voting rule which satisfies EJR and PR.

Phragmén’s optimization rules, MaxPhragmén and VarPhragmén, both satisfy PR in the candidate-approval setting [Brill et al.2017] and hence also in the party-approval setting.

Corollary 9.

MaxPhragmén and VarPhragmén do not satisfy EJR for party-approval elections.

In the following we show that also Phragmén-STV does not show improved axiomatic properties for party-approval elections.

Proposition 10.

Phragmén-STV fails EJR for on party-approval elections.

Proof.

For , consider an election with voters and parties . The ballot profile is as follows:

If , we also add 120 voters approving for every .

At first, let us only consider the parties . In Table 3, the Phragmén-STV calculation for an election restricted to these parties is described. Note that in the first 6 iterations, the parties each receive one seat and all have, when selected as winners, a score that exceeds 120. Afterwards, every party has a score strictly smaller than 109.

Furthermore, observe that the parties are all approved by voters who only approve this one particular party. As a result, their scores are not affected when other parties receive a seat. The parties have a score of 109, a score of 110, and (if existing) a score of 120. When any of these parties receive a seat, their score is decreased to 0, as they are all approved by at most voters.

Together, this shows that Phragmén-STV firstly allots each one seat to . Then, the score of the parties is always smaller than 109, and therefore, all receive a seat, which fills the committee. Thus, in the committee selected by Phragmén-STV, each receive one seat. However, the voters who approve form a cohesive group, where at least one voter should be represented by at least two seats according to EJR. This is not the case in the committee generated by Phragmén-STV. ∎

Party Round 1 Round 2 Round 3 Round 4 Round 5 Round 6 Round 7
240.0 179.86 179.86 103.26 103.26 103.26 103.26
242.0 120.71 120.71 120.71 120.71 120.71 60.14
190.0 190.0 190.0 68.71 45.96 45.96 45.96
183.0 121.86 121.86 121.86 121.86 121.86 0.57
240.0 240.0 179.61 134.92 13.64 13.64 13.64
241.0 241.0 119.71 119.71 66.13 7.86 7.86
186.0 186.0 125.11 125.11 125.11 3.82 3.82
Table 3: The Phragmén-STV computation for the restricted profile in Proposition 10. The table shows the scores of the parties in the first seven iterations. The bold entries denote the party with the highest score.
Proposition 11.

GreedyAV and CCAV do not satisfy PJR for on party-approval elections.

Proof.

Regarding GreedyAV with , consider an election with many voters and a parties . The ballots are for and for . GreedyAV first gives one seat to party and afterwards many seats, each to a different party from . Then, is a set of many voters with equal ballot who are only represented by one approved candidate. However, for we have and therefore PJR claims that the voters in must be represented by at least two seats.

It is not hard to see that, for the above constructed election, CCAV selects the same committee like GreedyAV. To see this note that the CCAV-utility of this committee is . Now, consider a committee which gives two seats to one of the parties. Such a committee has a CCAV-utility of at most . Hence, both GreedyAV and CCAV fail PJR. ∎

Proposition 12.

MonroeAV, GreedyMonroeAV and HareAV violate PJR for on party-approval elections.

Proof.

For , consider an election with voters and parties . In the election, 6 voters approve only ; furthermore for , there is one voter who approves only . For this election, MonroeAV, GreedyMonroeAV and HareAV all return a committee where receives 3 seats, and the remaining seats are distributed among