# Proportionality Degree of Multiwinner Rules

We study multiwinner elections with approval-based preferences. An instance of a multiwinner election consists of a set of alternatives, a population of voters---each voter approves a subset of alternatives, and the desired committee size k; the goal is to select a committee (a subset) of k alternatives according to the preferences of the voters. We investigate a number of election rules and ask whether the committees that they return represent the voters proportionally. In contrast to the classic literature, we employ quantitative techniques that allow to measure the extent to which the considered rules are proportional. This allows us to arrange the rules in a clear hierarchy. For example, we find out that Proportional Approval Voting (PAV) has better proportionality guarantees than its sequential counterpart, and that Phragmén's Sequential Rule is worse than Sequential PAV. Yet, the loss of proportionality for the two sequential rules is moderate and in some contexts can be outweighed by their other appealing properties. Finally, we measure the tradeoff between proportionality and utilitarian efficiency for a broad subclass of committee election rules.

## Authors

• 24 publications
• ### Proportionality and Strategyproofness in Multiwinner Elections

Multiwinner voting rules can be used to select a fixed-size committee fr...
04/17/2021 ∙ by Dominik Peters, et al. ∙ 0

• ### A Quantitative Analysis of Multi-Winner Rules

To choose a multi-winner rule, i.e., a voting rule that selects a subset...
01/04/2018 ∙ by Martin Lackner, et al. ∙ 0

• ### Selecting Matchings via Multiwinner Voting: How Structure Defeats a Large Candidate Space

Given a set of agents with approval preferences over each other, we stud...
02/15/2021 ∙ by Niclas Boehmer, et al. ∙ 0

• ### Multiwinner Approval Rules as Apportionment Methods

We establish a link between multiwinner elections and apportionment prob...
11/26/2016 ∙ by Markus Brill, et al. ∙ 0

• ### Dynamic Proportional Rankings

Proportional ranking rules aggregate approval-style preferences of agent...
05/17/2021 ∙ by Jonas Israel, et al. ∙ 0

• ### Comparing Election Methods Where Each Voter Ranks Only Few Candidates

Election rules are formal processes that aggregate voters preferences, t...
01/30/2019 ∙ by Matthias Bentert, et al. ∙ 0

• ### Sequential Voting with Confirmation Network

We discuss voting scenarios in which the set of voters (agents) and the ...
07/11/2018 ∙ by Yakov Babichenko, et al. ∙ 0

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## 1 Introduction

An approval-based committee election rule (an ABC rule, in short) is a function that given a set of  candidates  (the candidates are also referred to as alternatives), a population of  voters—each approving a subset of —and an integer  representing the desired committee size, returns a -element subset of . Committee election rules are important tools that facilitate collective decision making in various contexts such as electing representative bodies (e.g., supervisory boards, trade unions, etc.), finding responses to database querying [11, 26], suggesting collective recommendations for groups [21, 22], and managing discussions on proposals within liquid democracy [4]. Further, since committee elections are a special case of participatory budgeting (PB) [8, 14], good understanding of ABC rules is a prerequisite for designing effective PB methods.

The applicability of various ABC rules often depends on the particular context, yet an important requirement one often imposes on a committee election rule is that it should be fair to (groups of) voters. While fairness is a broad concept putting various principles under the same umbrella, in certain types o collective decision-making—specifically when the goal is to elect a committee—it is often argued that a fair rule should be proportional111Other basic axioms describing fairness requirements are, for instance, anonymity or solid coalitions property [10]., as illustrated by the following example.222Indeed, proportional representation (PR) electoral systems are often argued to be more fair than non-proportional ones [15, 20]. Yet, the criticism of proportionality also appears in the literature, and it comes from the two main directions—one stems from the analysis of voting power in the elected committees [13, 25]; the other one is grounded in the arguments in favor of degressive proportionality [17]. Proportionality is also related to fairness in other domains, such as allocation of individual [27] and public goods [12].

###### Example 1.

Consider an election with 30 candidates, , and 100 voters having the following approval-based preferences. The first 60 voters approve the subset ; the next 30 voters approve , and the last 10 approve . When the size of the committee to be elected is 10, then a proportional rule should select a committee with members coming from , 3 members coming from , and 1 member from .

Identifying a proportional committee in Example 1 was easy due to the voters’ preferences having a very specific structure: each two approval sets where either the same or disjoint. In the more general case, when the approval sets can arbitrarily overlap, the answer is no longer straightforward. Several approaches to formalizing proportionality have been proposed in the literature [1, 2, 7, 12, 19, 24]

. These approaches are all axiomatic—they formally define natural properties referring to proportionality, and classify known rules based on whether they satisfy these axioms or not. Thus, these approaches are qualitative, giving only a yes/no answer to the question of “Is a given rule proportional or not?”. Hence, in essence, they are not capable of measuring the extent to which proportionality is satisfied or violated. This is a serious drawback since there is no single rule satisfying all the desired properties, and the choice of the rule boils down to a judgment call. The mechanism designer must decide on which properties she finds most critical and which tradeoff she is willing to accept. However, in order to make such a decision one primarily needs to understand these tradeoffs; in particular the extent of violations of certain properties.

In our study we employ a quantitative approach. For several rules of our interest we will determine their proportionality degrees, i.e., we will assess the extent to which they satisfy proportionality. Informally speaking, the proportionality degree of a rule is a function specifying how the rule treats groups of voters with cohesive preferences, depending on the size of these groups. At the same time, our proportionality degree has the form of a guarantee; it gives the best possible bounds on the proportionality that the rule cannot violate, no matter what are the voters’ preferences.

Our definition of the proportionality degree is closely related to the concept proposed by Skowron et al. [26]

. In fact, their work establishes certain bounds on the proportionality degree for a number of ABC rules. Unfortunately, these bounds cannot be used for comparing the rules, since they are not tight; not even asymptotically. Our main contribution is that we derive new almost tight bounds that allow to arrange the rules that we study in a clear hierarchy, based on how proportional they are. A tight estimation of the proportionality degree is already known for Proportional Approval Voting (PAV)

[2]—we generalize this result and calculate the proportionality degree for convex Thiele methods, a broad class of rules which—in particular—includes PAV. Additionally, we find close estimations of the proportionality degree for two sequential methods often considered in the literature: Sequential Proportional Approval Voting and Phragmén’s Sequential Rule.

Our findings can be summarized as follows. As far as the proportionality guarantees are concerned, Sequential PAV is better than Phragmén’s Sequential Rule; further, PAV is better than both of the sequential rules. For reasonable committee sizes the proportionality degree of Sequential PAV is no worse than 70% of the proportionality degree of PAV. The proportionality degree of Phragmén’s Sequential Rule is roughly twice lower than that for PAV. On the one hand, our results suggest that PAV should be preferred whenever proportionality is the primary goal. On the other hand, they demonstrate that the loss of proportionality for Sequential PAV and Phragmén’s Sequential Rule is moderate. Thus, using these rules can be justified in cases when the decision maker considers their other distinctive properties equally important to proportionality (in Sections 4 and 3 we briefly recall a few arguments that sometimes can speak in favor of one of the two sequential rules).

Finally, in Section 6 we apply the same quantitative methodology to other criteria than proportionality. Specifically, we focus on the utilitarian efficiency, measured as the total number of approvals that the members of the elected committee obtain. We establish asymptotically tight bounds on the loss of the utilitarian efficiency for convex Thiele methods. For Sequential PAV and for Phragmén’s Sequential Rule such bounds are already known [18]. Together with the aforementioned proportionality guarantees, this allows to quantify the tradeoff between the level of proportionality and utilitarian efficiency, that is to estimate the price of fairness for particular (classes of) rules. Similar tradeoffs have been considered in the resource allocation domain [5, 9].

## 2 The Model

For each natural number , we set and . For each set  by we denote the set of all -element subsets of , and by we denote the powerset of , i.e., .

An approval-based election is a triple , where is the set of voters, is the set of candidates, and is an approval-based profile (or, in short, a profile), i.e., a function that maps each voter to a subset of ; intuitively, consists of candidates that voter finds acceptable and is referred to as the approval set of . Whenever we consider an election, we will implicitly assume that and refer to the sets of voters and candidates, respectively. Similarly, we will always assume that and denote the number of voters and candidates, respectively. For the sake of the simplicity of the notation we will also identify elections with approval-based profiles, assuming that the sets of voters and candidates are implicitly encoded as, respectively, the domain and the range of . We denote the set of all elections by (yet, following our convention, we will treat the elements of as if these were simply profiles).

### 2.1 Approval-Based Committee Election Rules

An approval-based committee election rule (an ABC rule, in short) is a function that for each approval-based profile and a positive integer returns a nonempty set of size- subsets of candidates, i.e., . We will refer to the elements of as to size- committees, or simply as to committees when the committee size  is known from the context. We will call the elements of winning committees.

Below, we recall definitions of several prominent (classes of) approval-based committee election rules, often studied in the literature in the context of proportional representation.

Thiele Methods.

For a function the -Thiele rule is defined as follows. The -score that a committee gets from a voter is equal to . The total -score of is the sum of the scores that it garners from all the voters: . The -Thiele rule returns the committees with the highest -score.

Proportional Approval Voting (PAV).

PAV is the -Thiele rule for . Using the harmonic sequence to compute the score ensures that PAV satisfies particularly appealing properties pertaining to proportionality [1, 7, 19].

Sequential Proportional Approval Voting (Seq-PAV).

This is an iterative rule that involves steps. It starts with an empty committee and in each step it adds to the candidate that increases the PAV score of most. Thus, intuitively, each voter starts with the same voting power (equal to 1), which can decrease over time. When the voter gets her -th representative in the committee, then her voting power decreases from to .

Phragmén’s Sequential Rule.

This is also an iterative rule that is usually defined as a load balancing procedure. Each candidate is associate with one unit of load; if is selected this unit of load needs to be distributed among the voters who approve . In each step the rule selects the candidate that minimizes the load assigned to the maximally loaded voter. Formally, let be the load assigned to voter after the -th iteration. In each step the rule selects a candidate and finds a load distribution such that: (i) , (ii) for each , (iii) for each , and (iv) is minimized. Finally, for each the rule updates the load, . For more discussion on the rule and its properties we refer the reader to the work of Brill et al. [6] and to the survey by Janson [16].

### 2.2 Proportionality Degree of ABC Rules

This section defines and explains the notion of proportionality used in our comparative study.

Given an approval-based profile and a desired committee size , we say that a group of voters is -large if . The satisfaction of the group from a committee is defined as the average number of representatives that a voter from has in committee :

 satV(A,W)=1|V|∑i∈V|W∩A(i)|%.
###### Definition 1.

We sat that a function is a -proportionality guarantee for an ABC rule if for each each approval-based profile , each -large group of voters , and for each winning committee , the following implication holds:

 |⋂i∈VA(i)|≥g(ℓ,k)⟹satV(A,W)≥g(ℓ,k).

Let us give an informal explanation of Definition 1; intuitively, this definition says that an -large group of voters is guaranteed representatives in the size- committee elected by a rule , no matter what the voters’ preferences are. The average satisfaction of at least is guaranteed only to -large groups that have coherent enough preferences, that is for groups that agree on some common candidates. Indeed, it is clear that if each member of a group approves different candidates, then cannot be satisfied by any rule; in fact we will call such simply a set instead of a group to indicate that there is no agreement between the voters in , so they cannot be related or grouped based of their preferences. Thus, summarizing, Definition 1 specifies how the rule treats cohesive groups of certain size.

One intuitively expects that a proportional rule should have a guarantee of . Indeed, a group such that (i) , and (ii) , is large enough to deserve representatives in the elected committee, and choosing candidates that all members of the group agree on is feasible. Unfortunately, such a guarantee is not possible to achieve. From the recent results of Aziz et al. [2] it follows that there exists no rule with the proportionality guarantee satisfying . The same work proves that PAV has the proportionality guarantee of .

In order to facilitate our further discussion we define two related concepts.

###### Definition 2.

Let be the set of all -proportionality guarantees for a rule . The -proportionality degree of an ABC rule is the function:

 d(k)R(ℓ,k)=sup\mathclapg∈k-Guar(R)g(ℓ,k).

In other words, the -proportionality degree is the best possible -proportionality guarantee one can find for the rule. The proportionality degree of is the function , i.e., it defines the best possible guarantee that holds irrespectively of the size of the committee.

In the remainder of this section we compare our notion of the proportionality guarantee/degree with other concepts pertaining to proportionality, studied in the literature.

Extended Justified Representation (EJR).

EJR [1] requires that each -large group of voters with must contain a voter who approves at least members of the winning committee(s). This property is very natural and interesting, yet it also has certain drawbacks. On the one hand, it seems quite weak—it aims at analyzing how voting rules treat groups of voters, yet for each group it only enforces the requirement that there must exist some voter in the group who is well represented. On the other hand, it appears very strong—if each voter from approves members of the winning committee, then already witnesses that the rule violates the property. Indeed, PAV is almost the only natural known rule satisfying EJR.333Other rules satisfying EJR are either very similar to PAV (e.g., the local search algorithm for PAV) or very technical and specifically tailored to satisfy the particular property [2] Our approach, on the other hand, provides a fine-grained guarantee on how well a given rule represents certain groups of voters.

Our bounds are also stronger. Clearly, if the proportionality guarantee of a rule satisfies , then—by the pigeonhole principle— also satisfies EJR. For the reverse direction a weaker implication has been shown by Sánchez-Fernández et al. [24]: if a rule satisfies EJR, then it has the proportionality guarantee of .

Uniform Proportionality.

Definition 1 requires that the average satisfaction of the voters from large, cohesive groups must be high enough. Instead, it seems natural to enforce this requirement for each member of these groups rather than on average. Such a definition is more natural when we assume that the utilities of the voters are not transferable between the group members.444We thank Jérôme Lang for suggesting this approach and for fruitful discussions on different notions of proportionality. Unfortunately, such a definition would be much too strong to lead to any meaningful results. Indeed, consider an election where is even and . Further, for each let . For each , the group consists of half of the population of the voters and agrees on a candidate. Thus, each such a group should be well represented, even in a reasonably small committee. Yet, when we set , then for each possible size- committee there exist a large group containing a voter who does not approve anyone from , hence any committee would violate the property.

Further, let us informally explain that Definition 1 is, to a certain degree, resilient to the aforementioned problem of the transferability of utilities. For example, for it is not possible that the whole utility of an -large group is held by a small subgroup of the voters—say, for the sake of concreteness, by less then of them. Indeed, if this were the case, then a large subset of (with ) would have no (or very little) representatives. Yet, such a subset would form an -large group, and so, by Definition 1, it had to be well represented.

Lower Quota.

Another approach to investigating proportionality of ABC rules is to analyze how they behave for certain structured preferences; for example, when the voters and the candidates can be divided into disjoint groups so that each group of voters approves exactly one group of candidates. Such profiles can be represented as party-list elections, and so the classic notions of proportionality for apportionment methods [3, 23] apply. For instance, it can be shown that EJR generalizes the definition of lower-quota. Brill et al [7] discussed how different ABC rules behave for such restricted preferences, and Lackner and Skowron [19] proved that under certain assumptions the behavior of ABC rules uniquely extends from party-list profiles to general preferences. Our study, on the other hand, answers whether the considered rules still behave proportionally for general preferences.

Finally, let us note that Skowron et al. [26] studied the proportionality degree of a number of ABC rules, focusing on those that satisfy the committee enlargement monotonicity. However, their estimations are not asymptotically tight, and are not sufficient to compare the studied rules. E.g., for the Phragmén’s rule, they showed that ; we will strengthen this result by showing that (note that , and that for “small” groups is much greater than ).

## 3 Proportionality Degree of Phragmén’s Sequential Rule

In this section we establish the proportionality degree of the Phragmén’s Sequential Rule. However, we first provide an alternative definition of the rule that will be more convenient to work with. According to our new definition the voters gradually earn virtual money (credits) which they then use to buy committee members. Specifically, we assume that each voter earns money with the constant speed of one credit per one time unit. Buying a candidate costs credits, and a voter pays only for a candidate that she approves of. We say that a candidate is electable if is approved by the voters who altogether have at least

credits. In the first time moment when there exists an electable candidate

the rule adds this candidate to the committee (ties are broken arbitrarily) and resets to zero the credits of all voters who approve —intuitively, these voters pay the total amount of credits for adding to the committee. The rule stops when candidates are selected.

Let us now argue that the so-described process is equivalent to the Phragmén’s Sequential Rule. First, it is apparent that in the original definition of the Phragmén’s rule we can assume that each candidate is associated with units of load instead of one. When a candidate is selected then its load is distributed so that all the voters who approve have the same total load, which is the maximum load among all the voters. The same candidate would be selected by the above described process, and each voter approving would pay for the number of credits which is equal to the difference between its current total load and the previous one (just before was selected).

Using the new definition allows us to obtain a new accurate estimation of the proportionality degree for the Phragmén’s Sequential Rule. Informally speaking, our guarantee says that the Phragmén’s rule can be at at most twice less proportional than PAV. The crucial element of the proof is the analysis of a certain potential function. Intuitively, this function measures how unfair the committee iteratively built by the Phragmén’s Sequential Rule can be; we will show that (i) this unfairness cumulates, and that (ii) eventually the accumulated unfairness must be used to compensate the voters who got less representatives than they deserved. (The most difficult part of the proof was coming up with its high-level idea and finding the right potential function.)

###### Theorem 1.

The proportionality degree of Phragmén’s Sequential Rule satisfies .

###### Proof.

In the proof we will be using the alternative definition of the Phragmén’s Sequential Rule, using the concept of virtual money (credits).

Let be a committee returned by Phragmén’s Sequential Rule. For the sake of contradiction, let us assume that there exists an -large group of voters with and with the average satisfaction lower than . We set .

Observe that the rule stops after at least time units. Indeed, the total amount of credits earned by all the voters in the first time units is equal to . This allows to buy at most candidates.

The total amount of credits collected by the voters from in the first time units is equal to . The number of credits left after the whole committee is elected is at most equal to (otherwise, these credits would have been spent earlier for buying an additional candidate approved by all the voters from ; such a candidate would exist as and ). Thus, the voters from spent at least credits for buying candidates they approve.

We now move to the central argument of the proof—we will estimate how much on average a voter from pays for buying a committee member. Our arguments are based on the analysis of a potential function, defined as follows.

Let denote the number of credits held by voter at time ; further, let . For a time we define the potential value as:

 ϕt=∑i∈V(pt(i)−pt(av))2.

We will now analyze how the potential value changes over time. First, observe that the potential value remains unchanged when the number of credits of each voter is incremented (earning credits does not change the potential value). Next, we analyze what happens when the voters use their credits to pay for the committee members that they approve. Consider the time moment when a new committee member is selected. The voters who approve pay for her with their credits; furthermore, a voter who pays uses all her available credits. We consider these voters separately, one by one. Consider a voter and assume that her number of credits decreases to 0 (since we consider the voters separately, the number of credits held by each other voter remains unchanged). In such a case the average decreases by . We assess the change in the potential value:

 Δϕ =∑i∈V,i≠j(pt(i)−(pt(av)−pt(j)nV))2+(0−(pt(av)−pt(j)nV))2 −∑i∈V(pt(i)−pt(av))2 =∑i∈V(pt(i)−(pt(av)−pt(j)nV))2−∑i∈V(pt(i)−pt(av))2 +(pt(av)−pt(j)nV)2−(pt(j)−(pt(av)−pt(j)nV))2 =∑i∈Vpt(j)nV(2pt(i)−2pt(av)+pt(j)nV)−pt(j)2+2pt(j)(pt(av)−pt(j)nV)

Now, observe that , thus:

 Δϕ =pt(j)2nV+pt(j)(2pt(av)−2pt(j)nV−pt(j)) =pt(j)(2pt(av)−pt(j)(nV+1)nV).

We further observe that in each time we have as otherwise the sum of credits within the group would be greater than , and such credits would be earlier spent for buying a committee member who is approved by all the voters within the group.

Let us now interpret the above calculations. Intuitively, we will argue that a voter from will on average pay at most for a committee member. Let us set . Then,

 Δϕ =pt(j)(2pt(av)−2kℓ−xt,j(nV+1)nV)≤−xt,j(nV+1)nV⋅pt(j).

If , then decreases by at least . Similarly, if , then increases by at most . Since, the potential value is always non-negative, we infer that the values of for and are on average lower than or equal to . Recall that in the definition of is equal to how much voter pays for the selected candidate. Thus, the voters from on average pay at most for a committee member. Consequently, the average number of committee members that the voters from approve is at least equal to:

 nVk−n2kℓ⋅nV=k2kℓ−n2kℓ⋅nV≥ℓ2−n2kℓ⋅nℓk=ℓ−12.

Proposition 1 below upper-bounds the proportionality degree of Phragmén’s Sequential Rule. For the sake of simplicity we present the proof for the case when is divisible by ; an analogous construction holds also in the general case (with slightly worse, but asymptotically the same, bounds), but the analysis is more complex.

###### Proposition 1.

The -proportionality degree of Phragmén’s Sequential Rule satisfies the following inequality: For each , and divisible by we have .

###### Proof.

Let us fix two natural numbers, ; and is divisible by . Let and let be an integer divisible by , , and by . We construct a profile with voters and candidates , as follows:

 N(b1)={1,2,…,\nicefracLx}∪{L+1,L+2,…,n−\nicefracLx}, N(b2)={\nicefracLx+1,…,\nicefrac2Lx}∪{L+1,L+2,…,n−\nicefrac2Lx}, … N(bx)={L−\nicefracLx+1,…,L}∪{L+1,…,n−L}, N(bx+1)={1,2,…,\nicefracLx}∪{L+1,…,n−L}, N(bx+2)={\nicefracLx+1,…,\nicefrac2Lx}∪{L+1,…,n−L}, … N(bk)=(a cyclic shift of~{}{1,2,…,\nicefracLx})∪{L+1,…,n−L} N(c1)=…=N(ck)={n−L,n−L+1,…,n} N(d1)=…=N(dℓ)={1,…,L}.

In particular, candidate is approved by voters in total, is approved by voters, and by voters. Each candidate from is approved by the same voters from and by some voters from , which cyclically shift. The voters from who approve , are those who are right after the voters who approve -th candidate, unless those who approve form the last segment, i.e., . In such a case, the voters from who approve are exactly .

Let ; this ensures that . In our profile Phragmén’s Sequential Rule selects first at time . Indeed, at this time the group of voters collects credits. At time each voter from has already credits, and each voter from has credits; altogether, they have credits, thus is selected second. By a similar reasoning, we infer that in the first steps candidates will be selected by Phragmén’s Sequential Rule, and the last one of them will be selected at time . At this time the rule would also select one candidate from . Indeed, at time the voters from have the following number of credits:

 Lx⋅t⋅(1+2+…+x)=Lx⋅kk−ℓ⋅x(x+1)2=nℓ(x+1)2(k−ℓ)=n.

Let us now analyze what happens in the next steps. First, let us consider how the Phragmén’s Sequential Rule would behave if there were no candidates from . At time voters from have credits each. Similarly, each voter from has credits, each voter from has credits, etc. The amount of credits held by the voters from {1, 2, …, L} altogether at time is:

 Lx⋅t1⋅(1+2+…+x)=n.

At the same time voters from have credits altogether, thus, can be selected next, before any candidate from is chosen555Here we assume adversarial tie breaking, yet the construction can be strengthen so that it does not depend on the particular tie-breaking mechanism.. Similarly, at time each voter from has credits, each from has credits, each from has credits, etc.; altogether they have at most credits, and the voters from have exactly ; thus can be selected next. Through a similar reasoning we conclude that in the first steps Phragmén’s Sequential Rule would select candidates .

Now, consider the candidates from . After candidates are selected, candidates from are approved only by voters who do not approve other remaining candidates. Thus, their selection does not interfere with the relative order of selecting the other candidates. Further, observe that over time the last voters collect the following number of credits:

 (x+1)tL=2(k−ℓ)ℓ⋅kk−ℓ⋅nℓk=2n.

Thus, every time moments the rule will select one candidate from . Consequently, in the first time moments, the rule will select at least candidates from . Indeed, the first candidate will be selected after the first time moments. After the remaining time moments the number of candidates from that will be selected is equal to:

 t(k−x−⌊2k−2xx+3⌋)(x+1)t2≥2(k−x−2k−2xx+3)x+1=2k−2xx+3≥⌊2k−2xx+3⌋.

Next, observe that:

Consequently, the winning committee has at most candidates from and the other committee members are from .

The group of voters is -cohesive. Let us assess their average number of representatives. Observe that, except for candidates from , each candidate from the selected committee is approved by exactly voters of . Thus:

 1|V|∑i∈V|W∩A(i)|=1L⋅Lx⋅(k−ℓ+1)=ℓ2⋅2k−2ℓ+22k−3ℓ.

Next, observe that for large enough the proportionality guarantee from Proposition 1 can be arbitrarily close to . Thus, we obtain the following corollary, which shows that the guarantee from Theorem 1 is almost tight (up to the constant of ).

###### Corollary 1.

The proportionality degree of Phragmén’s Sequential Rule satisfies .

Let us now discuss the consequences of our results from the perspective of a decision maker facing the problem of choosing the right rule. First, Phragmén’s Sequential Rule offers a considerably lower proportionality guarantee than PAV, hence the latter should be recommended whenever proportionality is the primary concern. Second, the worst-case loss of proportionality for the Phragmén’s Sequential Rule is moderate. Thus, in some cases this loss can be compensated by other appealing properties of the rules. For example, Phragmén’s Sequential Rule satisfies committee enlargement monotonicity, which makes it applicable when the goal is to compute a representative ranking of alternatives (the recent work of Skowron et al. [26] discusses several domains where finding proportional rankings is critical). Further, as discussed by Janson [16, Examples 13.5], Phragmén’s Sequential Rule has the following appealing property:666We thank Dominik Peters for suggesting the property and for fruitful discussions on its consequences.

###### Definition 3 (Strong Unanimity).

An ABC rule satisfies strong unanimity if for each approval-based profile such that there exists a candidate who is approved by all the voters, it holds that , where denotes the profile obtained from by removing from the approval sets of all the voters.

It easily follows from the definition that Phragmén’s Sequential Rule satisfies strong unanimity. Further, this property is so natural, that it is quite surprising that PAV does not satisfy it (this is an argument often raised by the critics of PAV). Theorem 1 quantifies the loss of proportionality being the result of imposing strong unanimity and committee enlargement monotonicity, and as such allows to view Phragmén’s Sequential Rule as an appealing alternative for PAV (depending on how important the decision maker considers particular axioms).

## 4 Proportionality Degree of Sequential PAV

In this section we will assess the proportionality degree of Sequential PAV, focusing on comparing Seq-PAV with Phragmén’s Sequential Rule. Our method here is quite different from the one we used in the previous section. Specifically, instead of proving a bound on the proportionality degree of Seq-PAV directly, we will show how to construct an algorithm that given a desired committee size finds in polynomial time an upper bound on the -proportionality degree of the rule. We will compute these bounds for and will argue that they are fairly accurate.

Designing such an algorithm is not straightforward—the main challenge lays in reducing the size of the space that needs to be searched. Indeed, even for a fixed committee size there is an infinite number of possible profiles. Thus, we will use several observations that will allow us to reformulate the problem and to compactly represent it as an instance of Linear Programming (LP).

Let us introduce some additional notation. Let and be an approval-based profile and a desired committee size, respectively. Let be a set of tied winning committees returned by Sequential Proportional Approval Voting for and . For each winning committee let denote the candidate that Sequential Proportional Approval Voting has added to as the last committee member. Let denote the average, per voter, marginal increase of the PAV score due to adding to :

 Δseq-PAV(A,k,W)=1n(scorePAV(A,W)−scorePAV(A,W∖{clast(A,W)})).

We denote the maximal possible such an average marginal increase as :

 Δseq-PAV(A,k)=max\mathclapW∈Rseq-PAV(A,k)Δseq-PAV(A,k,W).

Further, we define as:

 Δseq-PAV(k)=supA∈AΔseq-PAV(A,k).

Our first lemma shows a close relation between the proportionality guarantee of Sequential Proportional Approval Voting and value . This is very useful, as the definition of is not based on -large cohesive groups, and thus it is much easier to handle.

###### Lemma 1.

The -proportionality degree of Sequential PAV satisfies:

 d(k)seq-PAV(ℓ,k)≥ℓ⋅1k⋅Δseq-PAV(k)−1.

The proportionality degree of Sequential PAV satisfies:

 dseq-PAV(ℓ)≤ℓ⋅1k⋅Δseq-PAV(k).
###### Proof.

First, we prove that the -proportionality guarantee of Sequential Proportional Approval Voting satisfies . For the sake of contradiction let us assume that this is not the case, and that there exists an approval-based profile with voters, a committee size , an -large group of voters with , and a winning committee such that:

 1|V|∑i∈V|W∩A(i)|<ℓ⋅1k⋅Δseq-PAV(k)−1.

By the pigeonhole principle, there exist a candidate who is approved by all voters from . Let us now estimate by how much adding to increases the PAV score. Fixing the average number of representatives that the voters from have in , it is straightforward to check that the increase would be the smallest when each voter in has roughly the same number of representatives. Thus, adding to would increase the PAV score by more than

 |V|ℓ⋅1k⋅Δseq-PAV(k)−1+1=|V|⋅kℓ⋅Δseq-PAV(k)≥nΔseq-PAV(k).

Clearly, adding to any subset of would result in at least the same increase of the PAV score. Since Sequential Proportional Approval Voting always selects a candidate that increases the PAV score of the committee most, and since was not selected, we infer that , a contradiction.

Second, let us fix and . We will construct an approval-based profile that witnesses that . Let be an approval-based profile and be a size- committee winning in such that , for some ; the value of will become clear later on. Let us fix an integer —intuitively, is a large number; the exact value of will also become clear from the further part of the proof. Let . We construct by appending independent copies of (we clone both the voters and the candidates) and voters who all approve some candidates, not approved by any voter from any copy of —let us denote this set of voters by . Let ; clearly . We set the required committee size to . Observe that is -cohesive. Indeed, all voters from approve common candidates; further the relative size of , , is equal to

 yn′=LnℓLk−ℓ⋅Lk−ℓL2nk=ℓLk=ℓk′.

Now, we show that the voters from have on average less than representatives in some winning committee for . Towards a contradiction, let us assume that this is not the case. Since the voters in are identical, this means that each such a voter has more than representatives in each winning committee. Thus, for any winning committee, when the last representative of the voters from was added, the PAV-score of the committee increased by at most:

 Δ=y⋅k⋅Δseq-PAV(k)ℓ+ϵk⋅Δseq-PAV(k)

The above expression can be written as

 Δ=y⋅(k⋅Δseq-PAV(k)ℓ−ϵ2),

where is some parameter dependent on and . Further, observe that:

 limL→∞Δ =limL→∞y⋅(k⋅Δseq-PAV(k)ℓ−ϵ2)=limL→∞LnℓLk−ℓ⋅(k⋅Δseq-PAV(k)ℓ−ϵ2) =nℓk⋅(k⋅Δseq-PAV(k)ℓ−ϵ2)=nΔseq-PAV(k)−ϵ2⋅nℓk.

Thus, there exists large enough and small enough such that:

 Δ

These are exactly the values of and that we use in our construction. In other words, the increase of the PAV score due to adding the last representative of is lower than the increase of the PAV score due to adding the last committee member to . Thus, clearly there exists a winning committee that consists of some copies of the candidates from and less than candidates from those approved by the voters from , a contradiction. This completes the proof. ∎

As an immediate corollary of Lemma 1 we obtain an almost tight estimation of the proportionality guarantee of Sequential Proportional Approval Voting.

###### Corollary 2.

The proportionality guarantee of Sequential Proportional Approval Voting is for some .

In the next part of this section we will focus on assessing the expression , for different values of the parameter . According to Lemma 1 this expression will give us the lower bound on the -proportionality degree and an upper bound on the proportionality degree of Sequential PAV.

For each committee size we can compute by solving an appropriately constructed linear program; such an LP for each committee size finds a profile for which is maximal. Designing such an LP however requires some care, since ideally its size should not depend on the number of voters nor the number of candidates. Our first observation is that we can consider only the profiles with candidates (all of which form a winning committee). Indeed, if we take a profile that minimizes , then we can remove from all candidates which are not members of the winning committee. After such a removal we obtain a profile that still witnesses the minimality of . Thus, we will assume that in the profile that we look for there are candidates, and we will represent them as integers from . Further, without loss of generality we will assume that the candidates are added to the winning committee in the order of there corresponding numbers: candidate is added first, candidate is added second, etc.

Second, we observe that we do not need represent each voter, we can rather cluster the voters into the groups having the same approval sets and for each possible approval set we are only interested in the proportion of the voters who have this approval set. Such a proportion will be denoted by the variable . In Figure 1 we give the LP; for a logical expression , we set to be 1 if is true, and 0 otherwise.

The expression that we maximize is exactly . Indeed, is the marginal increase of the PAV score coming from a voter who approves , as a result of adding candidate to committee . The constraint (a) ensures that the proportions of clustered voters sum up to 1. The constraint (b) ensures that in the -th step of Sequential PAV, the marginal increase of the PAV score due to adding to committee is at least as large as due to adding to . These constraints ensure that the candidates are indeed added in the order .

We computed the above program for ; the resulting lower bounds for the -proportionality degree of Sequential PAV are given in Table 1.