1 Introduction
The study of preference aggregation mechanisms—in particular, voting rules—is an important part of multiagent systems research (e.g., Conitzer, 2010). Recent years have witnessed an increasing interest in multiwinner elections. In this setting, there is a set of agents who entertain preferences over a set of alternatives. Based on these preferences, the goal is to select a committee, i.e., a (fixedsize) subset of the alternatives. Preferences are usually specified either as rankings, i.e., complete linear orders over the set of all alternatives (e.g., Elkind et al., 2014), or as approval votes, i.e., yes/no assessments of all the alternatives (e.g., Kilgour et al., 2006). We are particularly interested in the latter variant, in which each agent can be thought of as specifying a subset of alternatives that are “acceptable” for that agent.
The decision scenario modeled by multiwinner elections—selecting a subset of objects from a potentially much larger pool of available objects—is ubiquitous: picking players to form a sports team, selecting items to display in an online shop, choosing the board of directors of a company, etc. Many of these scenarios are reminiscent of parliamentary elections, a topic that has been studied in great detail by political scientists. In a parliamentary election, the candidates are traditionally organized in political parties and the election determines how many parliamentary seats a party is allocated.
Under socalled proportional representation systems with “closed party lists,” a voter is allowed to give her vote to one and only one party. In a sense, this forces the voter to approve all candidates from one party and no candidates from any other party. Counting such ballots, and deciding how many candidates are elected from each list, is an apportionment problem. Any apportionment problem can be seen as a very simple approval voting instance: all voters approve all the candidates from their chosen party, and only those.
The present paper formally establishes and explores this analogy between multiwinner elections and apportionment problems. We show how an apportionment problem can be phrased as an instance of an approvalbased multiwinner election, thereby rendering multiwinner rules applicable to the apportionment setting. As a result, every approvalbased multiwinner rule induces a method of apportionment. Exploring this link between multiwinner rules and apportionment methods is interesting for at least two reasons. First, observing what kind of apportionment method a given multiwinner rule induces yields new insights into the nature of the rule. Second, every multiwinner rule inducing a given apportionment method can be seen as an extension of the apportionment method to a more general setting where candidates have no party affiliations (or party affiliations are ignored in the election process).
The relevance of the latter perspective is due to the realization that closedlist systems have a number of drawbacks. For instance, it is a known feature of closedlist systems that candidates tend to campaign within their parties (for being placed on a good position on the party list), rather than to campaign for the citizens’ votes. Closedlist systems thus favor party discipline, at the potential expense of alienating the political elites from the citizens (e.g., see Colomer, 2011; André et al., 2015; Ames, 1995; Chang, 2005).
In an attempt to overcome these drawbacks, many countries use “openlist” systems, leaving some flexibility to the voters by allowing them to vote for specific candidates inside the chosen party list. In some (rare) cases, voters are given even more freedom. Under socalled “panachage” systems, sometimes used in Luxembourg and in France, voters can vote for candidates from different parties. And sometimes voters vote directly for the candidates and the outcome of elections does not depend on how candidates are grouped into parties. Such is the case for some elections in Switzerland, were variants of multiwinner approval voting are used in several cantons (see Laslier and Van der Straeten, 2016). In a recent book, Renwick and Pilet (2016) extensively examine this “trend towards greater personalization,” which they see as “one of the key shifts in contemporary politics.” They find that this trend “is indeed changing core democratic institutions.”
After formally establishing the link between approvalbased multiwinner rules and apportionment methods, we consider several multiwinner rules and observe that they induce (and extend) apportionment methods that are wellestablished in the apportionment literature. For instance, Proportional Approval Voting (PAV) induces the D’Hondt method (aka Jefferson method) and Monroe’s rule induces the largest remainder method (aka Hamilton method). We also consider properties of apportionment methods (such as lower quota or the Penrose condition) and exhibit multiwinner rules that induce apportionment methods satisfying these properties.
The paper is organized as follows. Section 2 formally introduces both the apportionment problem and the multiwinner election setting. Section 3 shows how approvalbased multiwinner rules can be employed as apportionment methods, and contains several results related to proportional representation. Section 4 is devoted to nonproportional representation in the form of degressive proportionality and thresholds, and Section 5 concludes.
2 The Apportionment Problem and ApprovalBased Multiwinner Elections
In this section we provide the formal setting for the apportionment problem and for approvalbased multiwinner elections. For a natural number , let denote the set .
2.1 Apportionment Methods
In the apportionment setting, there is a finite set of voters and a finite set of parties . Every voter votes for exactly one party, and for each , we let denote the number of votes that party receives, i.e., the number of voters who voted for . The goal is to allocate (parliamentary) seats among the parties. Formally, an instance of the apportionment problem is given by a tuple , where is the vote distribution and is the number of seats to distribute. We use to denote the total number of votes, . Throughout this paper, we assume that for all and . An apportionment method maps every instance to a nonempty set^{1}^{1}1Most apportionment methods allow for ties. In this paper we do not consider any specific tiebreaking rule but rather assume that the apportionment methods might return several tied outcomes. of seat distributions
. A seat distribution is a vector
with . Here, corresponds to the number of seats allocated to party .In our proofs we often argue about seat distributions that result from a given seat distribution by taking away a single seat from a party and giving it to another party. For a seat distribution and two parties and such that , let denote the seat distribution with , , and for all .
2.1.1 Divisor Methods
A rich and very wellstudied class of apportionment methods is defined via divisor sequences.
Definition 1 (Divisor method).
Let be a sequence with for all . The divisor method based on is the apportionment method that maps a given instance to the set of all seat allocations that can result from the following procedure: Start with the empty seat allocation and iteratively assign a seat to a party maximizing where is the number of seats that have already been allocated to party .
Divisor methods are often defined in a procedurally different, but mathematically equivalent way (see Balinski and Young, 1982, Proposition 3.3).^{2}^{2}2Divisor methods with can also be defined. For such methods, which Pukelsheim (2014) calls impervious, the conventions and are used. Examples of impervious divisor methods are the methods due to Huntington and Hill, Adams, and Dean (see Balinski and Young, 1982). Two prominent divisor methods are the D’Hondt method and the SainteLaguë method.
Definition 2 (D’Hondt method).
The D’Hondt method (aka Jefferson method or HagenbachBischoff method) is the divisor method based on . Therefore, in each round, a seat is allocated to a party maximizing , where is the number of seats that have already been allocated to party .
The D’Hondt method was first used in 1791 to apportion seats in the U.S. House of Representatives, and currently it is used as a legislative procedure in over 40 countries.
Definition 3 (SainteLaguë method).
The SainteLaguë method (aka Webster method, Schepers method, or method of major fractions) is the divisor method based on . Therefore, in each round, a seat is allocated to a party maximizing , where is the number of seats that have already been allocated to party .
The SainteLaguë method was first adopted in 1842 for allocating seats in the United States House of Representatives, and currently it is used for parliamentary election in some countries (for instance Latvia, New Zealand, Norway, or Sweden) and for several statelevel legislatures in Germany.
The following example illustrates the two methods defined above.
Example 1.
Consider the instance with four parties and voters such that , and assume that there are seats to be allocated. The outcomes of the D’Hondt method and the SainteLaguë method, which can be computed with the help of Table 1, are and , respectively.
As we can see in the above example, different divisor methods might give different results for some instances of the apportionment problem. In particular, the D’Hondt method slightly favors large parties over small ones in comparison to the SainteLaguë method (Pukelsheim, 2014).
Several other divisor methods such as the HuntingtonHill method (aka the method of equal proportions), the Adams method (aka method of smallest divisors), and the Dean method are studied in the literature on fair representation. We refer the reader to the books of Balinski and Young (1982) and Pukelsheim (2014) for an extensive overview.
2.1.2 The Largest Remainder Method
The largest remainder method is the most wellknown apportionment method that is not a divisor method. Recall that denotes the total number of votes.
Definition 4 (Largest remainder method).
The largest remainder method (aka Hamilton method or HareNiemeyer method) is defined via two steps. In the first step, each party is allocated seats. In the second step, the remaining seats are distributed among the parties so that each party gets at most one of them. To do so, the parties are sorted according to the remainders and the remaining seats are allocated to the parties with the largest remainders.
The largest remainder method was first proposed by Alexander Hamilton in 1792 and it was used as a rule of distributing seats in the U.S. House of Representatives between 1852 and 1900 (e.g., Pukelsheim, 2014). Currently, it is used for parliamentary elections in Russia, Ukraine, Tunisia, Namibia and Hong Kong.
Example 2.
Consider again the instance in Example 1. According to the largest remainder method, in the first step, party is allocated seats, gets seats, gets seats, and gets seats. There are three more seats to be distributed among the parties. The three highest fractional parts are for party , for party and for party , thus these three parties are allocated one additional seat each. The resulting seat allocation is .
2.1.3 Properties of Apportionment Methods
The literature on fair representation has identified a number of desirable properties of apportionment methods (Balinski and Young, 1982; Pukelsheim, 2014). In this paper, we focus on properties requiring that the proportion of seats in the resulting apportionment should reflect, as close as possible, the proportion of the votes cast for respective parties.
Definition 5.
An apportionment method respects lower quota if, for every instance , each party gets at least seats. An apportionment method respects quota if each party gets either or seats.
Clearly, any apportionment method respecting quota also respects lower quota. It is well known that the largest remainder method respects quota, but no divisor method does. Moreover, the D’Hondt method is the only divisor method respecting lower quota (Balinski and Young, 1982).
2.2 ApprovalBased Multiwinner Election Rules
We now introduce the setting of approvalbased multiwinner elections. We have a finite set of voters and a finite set of candidates. Each voter expresses their preferences by approving a subset of candidates, and we want to select a committee consisting of exactly candidates. We will refer to element subsets of as size committees, and we let denote the set of candidates approved by voter . Formally, an instance of the approvalbased multiwinner election problem is given by a tuple , where is a preference profile and is the desired committee size. An approvalbased multiwinner election rule (henceforth multiwinner rule) is a function that maps every instance to a nonempty set^{3}^{3}3In order to accommodate tied outcomes, multiple committees might be winning. of size committees. Every element of is referred to as a winning committee.
2.2.1 OWABased Rules
A remarkably general class of multiwinner election rules is defined via ordered weighted averaging (OWA) operators (Thiele, 1895; Yager, 1988; Skowron et al., 2015). A weight sequence is an infinite sequence of real numbers .^{4}^{4}4Originally, OWAbased rules have been defined for a fixed committee size , by a vector of weights rather than by an infinite sequence. Since our focus is on multiwinner election rules that operate on arbitrary committee sizes, we need some way of grouping OWAbased rules for different values of into a single OWAbased rule that works for arbitrary values of the committee size. We assume that the weight vector for the committee size is built by appending a weight to the vector for the committee size . Thus, a single OWAbased rule can be represented by an infinite weight sequence. Another approach would be to represent an OWAbased rule by the collection of vectors of weights (one vector for each possible size of the committee).
Definition 6 (OWAbased rules).
Consider a weight sequence , a committee , and a voter . The satisfaction of from given is defined as . Given an instance of the multiwinner election problem, the based OWA rule selects all size committees that maximize the total satisfaction .
Note that multiplying a weight sequence by a positive constant does not change the way in which a rule operates. Several established multiwinner election rules can be described as OWAbased rules.
Definition 7.
The Chamberlin–Courant rule is the OWAbased rule with weight sequence .
The Chamberlin–Courant rule is usually defined in the context of multiwinner elections where voter preferences are given by ranked ballots, and each voter derives satisfaction only from their most preferred member of the committee (Chamberlin and Courant, 1983). Our definition of the rule is a straightforward adaption to the approval setting: a voter is satisfied with a committee if and only if it contains at least one candidate that the voter approves of.
Definition 8.
Proportional Approval Voting (PAV) is the OWAbased rule with weight sequence .
Though sometimes attributed to Forest Simmons, PAV was already proposed and discussed by the Danish polymath Thorvald N. Thiele in the 19th century (Thiele, 1895).^{5}^{5}5We are grateful to Xavier Mora and Svante Janson for pointing this out to us. According to PAV, each voter cares about the whole committee, but the marginal gain of satisfaction of an already satisfied voter from an additional approved committee member is lower than the gain of a less satisfied voter. The reason for using the particular weight sequence is not obvious. Aziz et al. (2015) and SánchezFernández et al. (2016) provide compelling arguments by showing that is the unique weight sequence such that the based OWA rule satisfies certain axiomatic properties. Theorem 2 in the present paper can be viewed as an additional—though related—argument in favor of the weight sequence .
Definition 9.
The top rule is the OWAbased rule with weight sequence .
According to the top rule, the winning committee contains the candidates that have been approved by the greatest number of voters.
The following example illustrates the OWAbased rules defined above.
Example 3.
Let and consider the following preference profile with 12 voters:
According to the Chamberlin–Courant rule, the committee is the unique winner in this profile. Indeed, for this committee each voter approves of at least one committee member (voters – approve of , voters and approve of , and voters – approve of ), thus each voter gets satisfaction equal to 1 from this committee. Clearly, this is the highest satisfaction that the voters can get from a committee. The reader might check that is the only committee for which each voter has at least one approved representative.
According to Proportional Approval Voting, is the unique winner. Let us now compute the PAV satisfaction of voters from this committee. Voters 1, 7, 8, 10–12 approve of a single committee member, thus the satisfaction of these voters is equal to 1. Voter 9 does not approve any committee member, thus her satisfaction is equal to 0. The remaining 5 candidates approve of two committee members, so their satisfaction from a committee is equal to . Thus, the total satisfaction of the voters from is equal to and it can be checked that the satisfaction of voters from each other committee is lower.
The top rule, on the other hand, selects committee as the unique winner. Indeed, , and are approved by 7, 6, and 5 voters, respectively, and any other candidate is approved by less than 5 voters.
Other appealing OWAbased rules include the best OWA rule, which is defined by the weight sequence with ones followed by zeros, and the median OWA rule, which is defined by the weight sequence , where the appears at position (Skowron et al., 2015).
2.2.2 Sequential OWABased Rules
Another interesting class of multiwinner election systems consists of sequential (or iterative) variants of OWAbased rules.
Definition 10.
The sequential based OWA rule selects all committees that can result from the following procedure. Starting with the empty committee (), in consecutive steps add to the committee a candidate that maximizes .
Just like OWAbased rules, sequential OWAbased rules have already been considered by Thiele (1895). Sequential Proportional Approval Voting (referred to as “reweighted approval voting” by Aziz et al. (2015)) was used for a short period in Sweden during the early 1900s (Janson, 2016).
Interestingly, a sequential OWAbased rule approximates the original rule whenever the weight sequence is nonincreasing (Skowron et al., 2015). Sequential OWAbased rules are appealing alternatives to their original equivalents for a number of reasons. First, they are computationally tractable, while finding winners for the original OWAbased rules is often hard. Second, the sequential rules often exhibit properties which the original rules do not have; for instance, they (trivially) satisfy committee monotonicity while some of the original versions do not (see, e.g., Elkind et al., 2014)
. Third, sequential methods are easier to describe to nonexperts, as compared to rules formulated as nontrivial combinatorial optimization problems.
Example 4.
Consider the instance from Example 3. Sequential Proportional Approval Voting selects candidate in the first iteration. In order to decide which candidate is selected in the second iteration, one must compare all remaining candidates with respect to their contribution to the total satisfaction of the voters. For instance, by selecting the satisfaction of voters will increase by . One can check that there are two optimal choices in the second step: selecting or selecting . If is selected in the second step, then our procedure will select in the third step (selecting increases the satisfaction of voters by ). If is selected in the second step, then must be chosen in the third step (by choosing we increase the satisfaction of voters by ). Consequently, there are two winning committees according to Sequential Proportional Approval Voting: and .
2.2.3 Monroe’s Rule
The optimization problem underlying the Chamberlin–Courant rule can be thought of in terms of maximizing representation: every voter is assigned to a single candidate in the committee and this “representative” completely determines the satisfaction that the voter derives from the committee. The rule proposed by Monroe (1995) is based on the same idea; However, Monroe requires each candidate to represent the same number of voters. For the sake of simplicity, when considering the Monroe rule we assume that the number of voters is divisible by the size of the committee .
Definition 11.
Assume that divides and consider a size committee . A balanced allocation of the voters to the candidates is a function such that for all . The satisfaction of a voter from is equal to one if approves of , and zero otherwise. The total satisfaction of voters provided by , denoted , is defined as the satisfaction from the best balanced allocation, i.e., . The Monroe rule selects all committees maximizing .
Example 5.
Consider the instance from Example 3 and consider committee , which is a winning committee under the Chamberlin–Courant rule. Even though every voter has an approved committee member (voters – approve of , voters and approve of , and voters – approve of ), this committee has a total satisfaction of 10 only. This is because the function , defined as
does not satisfy the Monroe criterion (candidate represents more voters than candidates and ). Under the Monroe rule, is the unique winning committee. For this committee, the optimal valid assignment function maps voters – to their approved candidate , voters – to their approved candidate and voters – to (only three of those four voters approve of ). Thus, the winning committee has a score of .
2.2.4 Phragmén’s Rules
In the late 19th century, Swedish mathematician Lars Edvard Phragmén proposed several methods for selecting committees based on approval votes (Phragmén, 1894, 1895, 1896). Here, we formulate two particularly interesting variants (see Brill et al., 2017).
The motivation behind Phragmén’s rules is to find a committee whose “support” is distributed as evenly as possible among the electorate. For every candidate in the committee, one unit of “load” needs to be distributed among the voters approving this candidate. The goal is to find a committee of size
for which the maximal load of a voter (or the variance of load distribution) is as small as possible.
Definition 12.
Consider an instance of the multiwinner election problem. A load distribution for is a matrix such that

,

for all , and

if .
Every load distribution corresponds to a size committee .
The multiwinner election rules maxPhragmén and varPhragmén are defined as optimization problems over the set of all load distributions.
Definition 13.
The rule maxPhragmén maps an instance to the set of size committees corresponding to load distributions minimizing . The rule varPhragmén maps an instance to the set of size committees corresponding to load distributions minimizing .
Example 6.
Let and consider the following preference profile with 5 voters:
It can be checked that maxPhragmén selects the committee and that varPhragmén selects the committee . Optimal load distributions corresponding to these committees are illustrated in Figure 1. Load distributions minimizing the maximal voter load (like the one depicted on the left in Figure 1) have a maximal voter load of and a variance of , and the load distribution minimizing the variance of voter loads (depicted on the right in Figure 1) has a maximal voter load of and a variance of .
3 Apportionment Via Multiwinner Election Rules
In this section we demonstrate how approvalbased multiwinner rules can be employed as apportionment methods. For a given instance of the apportionment problem, this procedure involves three steps:

Translate into an instance of the multiwinner problem.

Apply the multiwinner rule to .

Translate committee(s) in into seat distribution(s) for .
We now describe each step in detail.
Step 1.
Given an instance of the apportionment problem, we construct an instance of the multiwinner problem as follows. For each , we introduce a set consisting of candidates, and a set consisting of voters. Each voter in approves of all candidates in (and of no other candidates). Furthermore, we define , , and . Intuitively, is the set of members of party and is the set of voters who voted for party .
Step 2.
We can now apply multiwinner rule to in order to find the set of winning committees.
Step 3.
For every winning committee , we can extract a seat distribution for the instance in the following way: the number of seats of a party is given by the number of candidates from in the committee , i.e., .
The next example illustrates this threestep procedure.
Example 7 (Sequential PAV as an apportionment method).
Consider the instance of the apportionment problem. We construct an instance of the multiwinner election setting with candidates; each party has a set consisting of candidates. Further, there are voters: voters approve the ten candidates in , voters approve the ten candidates in , voters the ten candidates in , and voters the ten candidates in . The Sequential PAV method selects the winning committee by selecting in consecutive steps candidates from , , , , , , , , , . Thus, the corresponding seat allocation is .
For a given multiwinner rule , we let denote the apportionment method defined by steps 1 to 3. We say that a multiwinner election rule satisfies (lower) quota if the corresponding apportionment method satisfies the respective property.
3.1 OWABased Apportionment Methods
In this section, we consider apportionment methods induced by OWAbased rules. Fix a weight sequence and let denote the based OWA rule. For every instance of the apportionment setting, contains all seat distributions maximizing the total voter satisfaction , which is given by
(1) 
Here, denotes the satisfaction of a voter from party .
We let denote the marginal increase in when assigning the th seat to party . Due to (1), we have . Taking away a seat from party and giving it to party results in a change of total voter satisfaction of
Optimality of implies that for all with .
Whenever the weight vector is nonincreasing, the based OWA rule induces the same apportionment method as its sequential variant.^{6}^{6}6OWAbased rules with nonincreasing weight sequences are very natural, especially when viewed in the context of apportionment. OWAbased rules with increasing sequences induce apportionment methods that allocate all seats to the single party that receives the most votes. The same seat distribution can be also obtained by using the OWAbased rule with a constant weight vector (see Proposition 5).
Proposition 1.
Let be a weight sequence with for all . The apportionment method induced by the based OWA rule coincides with the apportionment method induced by the sequential based OWA rule.
Proof.
Fix a weight sequence satisfying for all . Let be the based OWA rule and let be the sequential based OWA rule. We show that the apportionment methods induced by and coincide, i.e., .
Consider an instance of the apportionment setting. Then contains all seat distributions maximizing the total voter satisfaction , which is given by (1).
Now consider the apportionment method based on the sequential based OWA rule. This method starts with the empty seat allocation and iteratively assigns a seat to a party such that the marginal increase in total voter satisfaction is maximized. Recall that denotes the marginal increase in when assigning the th seat to party . Due to (1), this quantity is independent of for and equals . For , method iteratively chooses a party with and sets .
For every seat allocation , the total voter satisfaction can be written as . In particular, is independent of the order in which seats are allocated to parties. Furthermore, our assumption implies that the marginal satisfaction is monotonically decreasing in for every party . Therefore, any seat distribution maximizing (i.e., any ) can be iteratively constructed by applying method . ∎
An immediate corollary is that OWAbased rules with a monotonically decreasing weight sequence can be computed efficiently in the apportionment setting.
Further, we can use Proposition 1 to show that every OWAbased rule with a nonincreasing weight sequence induces a divisor method. For a weight sequence , let be the sequence defined by for all . For example, yields .
Theorem 1.
Let be a weight sequence with for all . The apportionment method induced by the based OWA rule coincides with the divisor method based on .
Proof.
Fix a weight sequence with for all . Let denote the based OWA rule, and let denote the sequential based OWA rule. It follows from Proposition 1 that coincides with . Let furthermore denote the divisor method based on . We are going to show that coincides with .
Both and work iteratively. At each iteration, assigns a seat to a party maximizing , where is the number of votes for party and is the number of seats allocated to party in previous iterations. Method , on the other hand, assigns a seat to a party maximizing the marginal increase in total voter satisfaction. The marginal increase in total voter satisfaction when giving an additional seat to party equals .
Using , we can observe that the quantities used in divisor method exactly coincide with the quantities used in the method . Thus, both methods assign seats in exactly the same way. ∎
3.2 Proportional Approval Voting as an Extension of the D’Hondt Method
A particularly interesting consequence of Theorem 1 is that Proportional Approval Voting, the OWA rule with , induces the D’Hondt method.
Corollary 2.
The apportionment method induced by PAV coincides with the D’Hondt method.
The observation that PAV reduces to the D’Hondt method in the partylist setting occasionally occurs (without proof) in the literature (e.g., see Pereira, 2016).^{7}^{7}7In fact, already Thiele (1895) stated this equivalence, but only in the special case when perfect proportionality is possible; see the survey by Janson (2016). Theorem 1 shows that this is just one special case of the general relationship between OWAbased multiwinner rules and divisor methods.
Since the D’Hondt method satisfies lower quota, the same holds for PAV (and Sequential PAV) in the apportionment setting.
Corollary 3.
PAV and Sequential PAV satisfy lower quota.
The fact that PAV satisfies lower quota can also be established by observing that every multiwinner rule satisfying extended justified representation (Aziz et al., 2015) or proportional justified representation (SánchezFernández et al., 2016) also satisfies lower quota. See Appendix A for details.
Further, we show that PAV is the only OWAbased rule satisfying lower quota.^{8}^{8}8This result also follows from Theorem 1 together with the characterization of the D’Hondt method as the only divisor method satisfying lower quota (Balinski and Young, 1982, Proposition 6.4). We give a direct proof for completeness.
Theorem 2.
PAV is the only OWAbased rule satisfying lower quota.
Proof.
Let be an OWAbased rule such that satisfies lower quota. We will show that is based on a weight sequence with for all , from which we will infer that is equivalent to PAV.
Fix . We will show in two steps.
Step 1.
Given a natural number , we define an instance of the apportionment problem as follows. There are parties and the vote distribution is given by . That is, party gets votes and all other parties get votes. Thus, the total number of votes is . Furthermore, we set .
Consider a seat allocation . Since satisfies lower quota, we have and for all . Thus, is allocated at least one seat, and each of the other parties is allocated at least seats. From the pigeonhole principle we infer that at least one of the parties gets exactly seats. Let be such a party and consider the seat allocation . Since maximizes , we have
It follows that . Since this inequality has to hold for all natural numbers , we infer that .
Step 2.
Next, we construct another instance of the apportionment problem, again parameterized by a natural number . There are parties and the vote distribution is is given by . Thus, . The number of seats is given by .
Consider a seat allocation . Since satisfies lower quota, we infer that . The pigeonhole principle implies that at least one of parties gets no seat. Let be such a party and consider . We have
and thus . Again, this inequality holds for all , and thus .
We have therefore shown that for all . It follows that for some constant . As a consequence, is equivalent to PAV. ∎
Theorem 2 characterizes PAV as the only OWAbased rule respecting lower quota. Since PAV does not respect (exact) quota, it follows that no OWAbased rule respects quota. Due to Proposition 1, the same is true for sequential OWAbased rules.
The loadbalancing rule maxPhragmén also induces the D’Hondt method. Indeed, Phragmén formulated his rule as a generalization of the D’Hondt method to the general (“open list”) case (see Janson, 2016).
Theorem 3.
The apportionment method induced by maxPhragmén coincides with the D’Hondt method.
Proof.
In the apportionment setting, optimal load distributions have a very simple structure. Given a seat distribution , it is clearly optimal to distribute the load of ( for each seat that is allocated to party ) uniformly among the voters of the party. Therefore, the maximal voter load for is given by . Balinski and Young have shown that the D’Hondt method selects seat distributions minimizing this quantity (Balinski and Young, 1982, Proposition 3.10). ∎
We note that Phragmén also proposed a sequential rule based on load distributions. It is straightforward to verify that, in the apportionment setting, this variant coincides with maxPhragmén and thus also induces the D’Hondt method.
3.3 Monroe’s Rule as an Extension of the Largest Remainder Method
We now turn to Monroe’s multiwinner rule. It turns out that it induces the largest remainder method.
Theorem 4.
Assume that the number of seats divides the total number of voters. Then, Monroe’s rule induces the largest remainder method.
Proof.
Let denote Monroe’s rule. Recall that Monroe’s rule assigns voters to candidates in a balanced way, so as to maximize the total voter satisfaction. In the apportionment setting, a voter in is satisfied if and only if she is assigned to a candidate in . The apportionment method selects seat distributions maximizing the total voter satisfaction.
For a seat allocation , let denote the total voter satisfaction provided by . We can write , where is the total satisfaction that voters in derive from . For a given instance of the apportionment problem, can be expressed as
Since the case occurs if and only if , we have and
(2) 
We show that coincides with the largest remainder method for all instances such that divides . Fix such an instance and let . The proof consists of three steps.
Step 1.
We first show that for all parties . Assume for contradiction that this is not the case and let be a party with . We infer that and, by the pigeonhole principle, that there exists a party such that . Thus, and . Therefore,
contradicting our assumption that .
Step 2.
We next show that for all parties . Assume for contradiction that this is not the case and let be a party with . Similarly as before, we infer that and that there exists a party with . Thus,