Axiomatic studies of multiwinner voting rules go back to Felsenthal and Maoz  and Debord , but a systematic work on the topic has began only recently and on several different fronts. New results appear within social choice theory, computer science, artificial intelligence, and a number of other fields (see the work of Faliszewski et al. 
for more details on the history as well as recent progress). The reason for this explosion of interest from a number of research communities is the wide range of applications of multiwinner voting rules on the one hand, and the corresponding richness and diversity of the spectrum of those rules on the other. Typically, social-choice theorists study normative properties of various multiwinner rules, computer scientists investigate feasibility of computing the election results, and researchers working within artificial intelligence use multiwinner elections as a versatile tool (e.g., useful in genetic algorithms, for ranking search results , or for providing personalized recommendations ). Yet, there is a growing interplay between these areas and an increased need for a new level of comprehension of results obtained in all of them. In this paper we partially address this need by linking syntactic features of certain families of committee scoring rules with their normative properties. The syntactic features of the rules are useful, e.g., for establishing their computational properties [74, 31], or for viewing those rules as achieving certain optimization goals (which allows one to consider these rules as tools for certain tasks from artificial intelligence and operation research). The normative properties, on the other hand, are useful for understanding the ‘behavior’ of these rules and the settings for which they may be appropriate.
The model of multiwinner elections studied in this paper is as follows. We are given a set of candidates, a collection of voters—each with a preference order in which the candidates are ranked from the best to the worst—and an integer . A multiwinner rule maps this input to a subset of candidates (i.e., a committee; we discuss tie-breaking later) that, in some sense, best reflects the voters’ preferences. For example, the Single Non-Transferable Vote rule (the SNTV rule) chooses candidates that are top-ranked most frequently, whereas the Bloc rule selects candidates that are ranked most frequently among top positions (equivalently, under Bloc each voter names members of his or her favorite committee, and those that are mentioned most often are selected). Naturally, there are many other multiwinner rules to choose from, defined in various ways.
In this paper we focus on the class of committee scoring rules, introduced by Elkind et al.  as multiwinner generalizations of classic positional scoring rules. The main idea of committee scoring rules is essentially the same as in the single-winner case: Each voter gives each committee a score based on the positions of members of this committee in the voter’s ranking, scores from individual voters are aggregated into the societal scores of the committees, and the committee(s) with the highest score wins. Committee scoring rules appear to form a remarkably rich class that includes both very simple rules, such as SNTV and Bloc, and rather sophisticated ones, such as the rule of Chamberlin and Courant  or variants of the Proportional Approval Voting rule . As these rules tend to be very different in nature, they are suitable for different purposes, such as selecting a diverse committee, selecting a committee that proportionally represents the electorate, or selecting a committee consisting of individually best candidates. This richness is the main strength of the class of committee scoring rules, but to choose rules for given settings wisely, it is important to understand the internal structure of the class. Understanding this structure is the main goal of the current paper.
So far, researchers have identified the following subclasses of committee scoring rules (we provide their formal definitions in Sections 2 and 3; here we give intuitions only). (Weakly) separable rules, introduced by Elkind et al. , are those rules where we compute a separate score for each candidate (using a single-winner scoring rule) and then pick candidates with the top scores (for example, using Plurality scores leads to SNTV).111If the underlying single-winner scoring rule does not depend on the size of the committee (as in the case of SNTV) then the rule is referred to as separable. If there is such dependence (as in the case of Bloc), then the rule is weakly separable. Representation-focused rules, also introduced by Elkind et al. , are similar in spirit to the Chamberlin–Courant rule, whose aim is to ensure that in the elected committee each voter’s most preferred committee member (his or her representative) is ranked as high as possible. On the other hand, top--counting rules, introduced by Faliszewski et al. , capture rules where each voter evaluates the quality of a committee by the number of members of that committee that he or she ranks among the top ones; Bloc is a prime example of a top--counting rule. Finally, the class of OWA-based rules—introduced by Skowron et al. , also studied in the approval-based election model [2, 5, 49]—contains all the previously mentioned classes. Under these rules a voter calculates the score of a committee as the ordered weighted average (OWA) of the scores of the candidates in that committee.222See the original work of Yager  for a general discussion of OWAs, and, e.g., the works of Kacprzyk et al.  or Goldsmith et al.  for their other applications in voting. In this paper we also introduce the class of decomposable committee scoring rules that strictly contains all the OWA-based ones, has interesting applications, and appears to be easier to work with axiomatically.
All these classes have been defined purely in terms of the syntactic features of the functions used to calculate the scores of the committees.333A notable exception is the class of top--counting rules which were discovered while characterizing those committee scoring rules that satisfy the fixed-majority property . These syntactic features are important if, for example, one wants to assess some computational properties of the rules (e.g., it is known that weakly separable rules are polynomial-time computable , that representation-focused rules tend to be -hard to compute [65, 51, 74], and that the structure of the functions used within OWA-based rules affects the ability to compute their results approximately ). Such syntactic features are also essential when we view committee scoring rules as specifying optimization goals for particular applications (for example, since under the Chamberlin–Courant rule each voter’s score depends solely on his or her representative in the elected committee, this rule is particularly suitable in the context of deliberative democracy , for targeted advertising [51, 52], or for certain facility location problems ). Nonetheless, these syntactic features do not tell us much about the behavior of the rules.
Our first result reinforces the syntactic hierarchy of committee scoring rules. We show that the class of committee scoring rules strictly contains the class of decomposable rules, which, in turn, strictly contains the class of OWA-based ones, and that the class of OWA-based rules strictly contains the classes of (weakly) separable rules, representation-focused rules, and top--counting rules. For each pair of the latter three classes, we show that their intersection contains exactly one, previously-known, non-trivial voting rule. See Figure 1 for a visualization of the syntactic hierarchy of committee scoring rules.
Our second, and the main, result establishes a link between several levels of the syntactic hierarchy and respective normative properties. In other words, we establish axiomatic characterizations of some of the studied subclasses of committee scoring rules. Until now, the only result of this form, which is due to Faliszewski et al. , was a characterization of fixed-majority consistent committee scoring rules as those top--counting rules whose scoring functions satisfy (a relaxed variant of) the convexity property. Here, our main result is that many of the syntactic properties of our rules nicely correspond to certain types of monotonicity. Specifically, we focus on the committee enlargement monotonicity444This notion is also known as committee monotonicity  and enlargement consistency . We chose a name that is more informative than the former, but which is not tied to the realm of resolute rules, as the latter. In the literature on apportionment rules, a related property is often called house monotonicity [66, 6]. property, which requires that if we increase the size of the committee sought in the election, then the new winning committee should be a superset of the old winning committee, and on variants of the non-crossing monotonicity property, which requires that if we shift forward some members of a winning committee within any vote in a way that does not affect the positions of the remaining members of this committee, then this committee should still win. We show that committee enlargement monotonicity characterizes exactly the class of separable rules among committee scoring rules, and that non-crossing monotonicity characterizes the class of weakly separable ones. Then we introduce top-member monotonicity (a variant of non-crossing monotonicity restricted within each vote to shifting only the highest-ranked member of the winning committee) and show that together with narrow-top consistency (which requires that if there are at most candidates that are ever ranked in the top position within a vote, then these candidates should belong to the winning committee) it characterizes the class of representation-focused rules. Finally, we show that if a committee rule is prefix-monotone (i.e., satisfies a yet another restricted variant of non-crossing monotonicity) then it must be decomposable.
The paper is organized as follows. In Section 2 we describe the model of multiwinner elections, define the class of committee scoring rules, provide their basic properties, and show several examples of committee scoring rules. Section 3 is devoted to structural properties of classes of committee scoring rules. Here we build the hierarchy of the classes and show results regarding containments and intersections among these classes. In Section 4 we switch to axiomatic properties of the rules in the classes of the hierarchy and give several axiomatic characterizations of those classes. Finally, we discuss related work in Section 5 and conclude in Section 6.
2 Multiwinner Elections and Committee Scoring Rules
In this section we set the stage for the discussions provided throughout the rest of the paper by providing preliminary definitions as well as introducing the class of committee scoring rules. For each positive integer , we write to denote . By we mean the set of nonnegative real numbers.
An election is a pair , where is a set of candidates and is a collection of voters. Each voter has a preference order , expressing his or her ranking of the candidates, from the most desirable one to the least desirable one. Given a voter and a candidate , by we mean the position of in ’s preference order (the top-ranked candidate has position , the next one has position , and so on).
A multiwinner voting rule is a function that given an election and a committee size , , returns a family of size- subsets of , i.e., the set of committees that tie as winners of the election (we use the nonunique-winner model or, in other words, we assume that multiwinner rules are irresolute). We provide a few concrete examples of multiwinner rules in Section 2.2.
Most of the multiwinner rules that we study are based on single-winner scoring functions. A single-winner scoring function for candidates is a nonincreasing function that assigns a score value to each position in a preference order. Given a preference order and a candidate , by the -score of (given by voter ) we mean the value . The two most commonly used scoring functions are the Borda scoring function,
and the -approval scoring function,
In particular, is known as the Plurality scoring function.
Committee scoring functions generalize single-winner scoring functions to the multiwinner setting in a natural way, by assigning scores to the positions of the whole committees. Formally, given a vote and a committee of size , the committee position of in , denoted , is a sequence that results from sorting the set in the increasing order. We write to denote the set of all such length- increasing sequences of numbers from (in other words, we write to denote the set of all possible committee positions for the case of candidates and committees of size ). Given two committee positions from , and , we say that weakly dominates , , if for each , it holds that (we say that dominates , denoted , if at least one of these inequalities is strict555In previous papers on committee scoring rules, the notions of weak dominance and dominance were conflated. We believe that giving them clear, separate meanings will help in providing more crisp arguments and discussions.). Below we define committee scoring functions formally.
Definition 2.1 (Elkind et al. ).
A committee scoring function for candidates and a committee size is a function such that for each two sequences , if weakly dominates then .
Let be a family of committee scoring functions, where each is a function for candidates and committees of size . Given an election with candidates and a committee of size , we define the -score of to be:
When is clear from the context, we often speak of the score of a committee instead of its -score. Given the above notation, we are ready to define committee scoring rules formally.
Let be a family of committee scoring functions (with one function for each and , ). Committee scoring rule is a multiwinner voting rule that given an election and committee size , outputs all size- committees with the highest -score.
We say that a committee scoring rule is degenerate if there is a number of candidates and a committee size such that is a constant function. As a consequence, a degenerate rule returns all size- committees for every election with candidates. The trivial committee scoring rule is a degenerate rule that returns the set of all size- committees for all elections and all sizes (naturally, it is defined by a family of constant functions).
2.2 Examples of Committee Scoring Rules
Many well-known multiwinner rules are, in fact, committee scoring rules; below we provide several such examples. For each of the rules we provide the family of committee scoring functions used in its definition, discuss these functions intuitively, and mention some applications.
- SNTV, Bloc, and -Borda.
These three rules use the following committee scoring functions:
That is, under the SNTV rule we choose candidates with the highest Plurality scores, under Bloc we choose candidates with the highest -Approval scores, and under -Borda we choose candidates with the highest Borda scores. On the intuitive level, under SNTV each voter names his or her favorite committee member, under Bloc each voter names all the members of his or her favorite committee, and under -Borda each voter ranks all the candidates and assigns them scores in a way which corresponds linearly to their position in the ranking. SNTV and Bloc are sometimes used in political elections (with the former used, e.g., in the parliamentary elections in Puerto Rico, and with the latter often used for various local elections in many countries). -Borda and other rules based on similar scoring schemes are often used to determine finalists of competitions (e.g., the finalists of the Eurovision Song Contest are selected using a system very close to -Borda).
- The Chamberlin–Courant rule.
Under the Chamberlin–Courant rule (the -CC rule), the score that a voter assigns to a committee depends only on how ranks his or her favorite member of (referred to as ’s representative in ). The Chamberlin–Courant rule seeks committees in which each voter ranks his or her representative as high as possible. Formally, the rule uses functions:
This is the variant of the rule originally proposed by Chamberlin and Courant , but, subsequently, other authors (e.g., Procaccia et al. , Betzler at al. , and Faliszewski et al. ) considered other ones, based on other single-winner scoring functions. In particular, we will be interested in the -Approval Chamberlin–Courant rule, (-CC) which is defined through functions:
Intuitively, both variants of the Chamberlin–Courant rule seek committees of diverse candidates that “cover” as broad a spectrum of voters’ views as possible. Lu and Boutilier  considered the rule in the context of recommendation systems.
- The PAV rule.
The Proportional Approval Voting rule (the PAV rule) was originally defined by Thiele  in the approval setting (where instead of ranking the candidates, the voters indicate which ones they accept as committee members; for recent discussions of the rule see the overview of Kilgour  and the works of Aziz et al.  and Lackner and Skowron ). We model it as a committee scoring rule -PAV, where is a parameter, defined using scoring functions of the form:
PAV is particularly well-suited for electing parliaments. Indeed, Brill et al.  have shown that it generalizes the d’Hondt apportionment method, which is used for this purpose in many countries (e.g., in France and Poland). A number of recent works [2, 14, 49, 4] explain why the harmonic sequence used within the PAV scoring function ensures that the elected committee represents the voters proportionally.
Naturally, there are many other committee scoring rules, and we will discuss some of them throughout the paper. Nonetheless, the above few suffice to illustrate our main points. There is also a number of other multiwinner rules that are not committee scoring rules, such as STV (see, e.g., the work of Tideman and Richardson ), Monroe , Minimax Approval Voting , or rules which are stable in the sense of Gehrlein . We do not discuss them in this paper, but we provide some literature pointers in Section 5.
2.3 Basic Features of Committee Scoring Rules
The class of committee scoring rules is very rich and there are only a few basic properties shared by all the rules in this class. Below we discuss several such properties that will be useful throughout this paper.
From our point of view, the most important common feature of committee scoring rules is that they are uniquely defined by their scoring functions (up to linear transformations). Formally, we have the following lemma (we provide the proof in the appendix).
Let and be two committee scoring rules defined by committee scoring functions and , respectively. If then for each and , , there are two values, and , such that for each we have that .
Due to Lemma 2.1, to show that two committee scoring rules are distinct it suffices to show that their scoring functions are not linearly related. In particular, this will be very useful when we will be showing that certain rules cannot be represented using scoring functions of a given form.
The second common feature of committee scoring rules is nonimposition, which requires that for every committee there is some election where it wins uniquely. Formally, we have the following definition.
Let be a multiwinner rule. We say that has the nonimposition property if for each candidate set and each subset of , there is an election such that .
Nonimposition is such a basic property that it is hardly surprising that all non-degenerate committee scoring rules satisfy it. We prove the next lemma in the appendix.
Let be a committee scoring rule defined by a family of committee scoring functions . satisfies the nonimposition property if and only if every committee scoring function in is nontrivial.
While at first sight nonimposition and Lemma 2.2 seem hardly exciting, in fact they are sufficient to illustrate intriguing differences between single-winner voting rules and their multiwinner counterparts. For example, one can verify that all nontrivial single-winner scoring rules satisfy the following extended variant of the nonimposition property: For every candidate set and its subset , there is an election where exactly the candidates from tie as winners. Analogous result does not hold for committee scoring rules, even for the case of two committees (in which case it could be dubbed as -nonimposition; the example below is due to Lackner and Skowron ).
Let us fix some committee size and a set containing at least candidates. Consider two disjoint committees and . Let be an arbitrary election where and are tied as winners according to Bloc (such elections exist). We note that each candidate in has exactly the same -Approval score as each candidate in (otherwise at least one of these committees would not be winning). Consequently, every size- committee such that is also winning in , so and are not the two unique winning committees.
The fact that in general -nonimposition does not hold for committee scoring rules is quite disappointing because many results would be far easier to prove if we could assume that it is always possible to construct an election where two arbitrary given committees are the only winning ones. On the other hand, it is possible to construct elections where two size- committees and are the only winning ones, provided that they share candidates (and, indeed, this fact is used in the proof of Lemma 2.1).
There are a few more common properties of committee scoring rules. For example, they all satisfy the candidate monotonicity property which requires that if we shift forward a member of a winning committee then, afterward, this candidate still belongs to some winning committee (but possibly quite a different one; see the work of Bredereck et al. ). Also, all committee scoring rules are consistent in the sense that if two elections and (over the same candidate set) have some common winning committees, then these are exactly the winning committees in an election obtained by merging the voter collections of and . The former property is related to our discussions in Section 4 and the latter one is often useful as a tool when proving various results (and, indeed, it is crucial in characterizing the class of committee scoring rules axiomatically ).
2.4 The T-Shirt Store Example
In Section 2.2 we have provided a number of examples of committee scoring rules and we have discussed some of their applications, focusing mostly on political elections. However, committee scoring rules have far more varied applications (see, e.g., the overview of Faliszewski et al. ), most of which have nothing to do with politics. Below we describe a simplified business-inspired scenario where committee scoring rules may be useful. We use this example to guide our way through the different types of committee scoring rules discussed in this paper.
Consider a T-shirt store that needs to decide which shirts to put on offer. Let be the set of T-shirts that the store can order from its suppliers (). Since the store has limited space, it can only put different T-shirts on display, and it wants to pick them in a way that would maximize its revenue (i.e., the number of T-shirts sold). We assume that every customer knows all the designs (say, from a website) and ranks all T-shirts from the best one to the worst one. Let us say that a customer considers a T-shirt to be “very good” if it is among the top T-shirts (of course, this is an arbitrary choice, made for the sake of simplifying the example).
How should the store decide which T-shirts to put on display? This depends on how the customers behave. Consider a customer that ranks the available T-shirts on positions . If this is a very picky customer that only buys a T-shirt if it is the very best among all possible ones (according to his or her opinion) then the number of T-shirts this customer buys is given by . However, if this customer were to buy one copy of each T-shirt he or she considered as “very good,” he or she would buy T-shirts. It is also possible that a customer would buy only one -shirt, provided he or she considered it as “very good.” The number of T-shirts bought by such a customer would be . Depending on which type of customers the store expects to have, it should choose its selection of T-shirts either using SNTV, Bloc, or -Approval Chamberlin–Courant. (Surely, other types of customers are possible as well and we will discuss some of them later. It is also likely that the store would face a mixture of different types of customers, but this is beyond our study.)
3 Hierarchy of Committee Scoring Rules
In this section we describe the classes of committee scoring rules that were studied to date, introduce a new class—the class of decomposable rules—and argue how all these classes relate to each other, forming a hierarchy. In Figure 1 we present the relations between the classes discussed in this section, with examples of notable rules. The classes are defined by setting restrictions on the scoring functions so, in other words, in this section we are interested in the syntactic hierarchy of committee scoring rules. Later, in Section 4, we will consider semantic properties.
3.1 Separable and Weakly Separable Rules
We say that a family of committee scoring functions is weakly separable if there exists a family of (single-winner) scoring functions with such that for every and every committee position we have:
A committee scoring rule is weakly separable if it is defined through a family of weakly separable scoring functions . In other words, if a rule is weakly separable then we can compute the score of each candidate independently, using the single-winner scoring function , and pick the candidates with the highest scores. In consequence, it is possible to compute winning committees for all weakly separable rules in polynomial time, provided that their underlying single-winner scoring functions are polynomial-time computable .666There is a subtlety here as there may be exponentially many winning committees. However, by listing the scores of all the candidates, we provide enough information to, e.g., enumerate all the winning committees in time proportional to the number of these committees, or to perform many other tasks related to winner determination (such as computing the score of a winning committee).
If for all we have , then we say that the family and the corresponding committee scoring rule are separable, without the “weakly” qualification. Thus, separable rules use the same scoring function for each value of the size of a committee to be elected. Interestingly, separable rules have some axiomatic properties that other weakly separable rules lack —we will discuss this further in Section 4.
The notion of (weakly) separable rules was introduced by Elkind et al. ; they pointed out that SNTV and -Borda are separable, whereas Bloc is only weakly separable.
3.2 Representation-Focused Rules
A family of committee scoring functions is representation-focused if there exists a family of (single-winner) scoring functions such that for every and every committee position we have:
This means that the score that a committee receives from a voter depends only on the position of the most preferred member of this committee in the voter’s preference ranking—such a member can be viewed as a representative of the voter in the committee. A committee scoring rule is representation-focused if it is defined through a family of representation-focused scoring functions . The notion of representation-focused rules was introduced by Elkind et al. ; -CC is the archetypal example of a representation-focused committee scoring rule and, in consequence, all the representation-focused rules can be seen as variants of the Chamberlin–Courant rule.
SNTV is both separable and representation-focused, and it is the only non-degenerate committee scoring rule with this property.
SNTV is the only non-degenerate committee scoring rule that is (weakly) separable and representation-focused.
It is easy to verify that SNTV is separable and representation-focused. For the other direction, let be a rule which is separable and representation focused. It follows that for some families of committee scoring functions and , such that and . Every linear transformation of has the same form (i.e., it only depends on ), so by Lemma 2.1 (linearly transforming , if necessary) we can assume that .
Without loss of generality, we can assume that . For each committee positions with , we have that
and, so, we can conclude that . Since is non-degenerate, we have that , and so that . This is sufficient to conclude that is equivalent to SNTV. ∎
Generally, representation-focused rules are -hard to compute (SNTV is one obvious exception). This fact was first shown by Procaccia et al.  in the approval-based setting, and then by Lu and Boutilier  for . Since then, various means of computing the results under the Chamberlin–Courant rule and its variants were studied in quite some detail [9, 20, 75, 78, 59, 34, 48, 28].
3.3 Top--Counting Rules
A committee scoring rule , defined by a family , is top--counting if there exists a sequence of nondecreasing functions , with , such that:
That is, the value depends only on the number of committee members that the given voter ranks among his or her top positions. We refer to the functions as the counting functions. Top--counting rules were introduced by Faliszewski et al. .
It would be quite natural to require that all counting functions for a given committee size were the same, that is, that for each it held that . Following Faliszewski et al. , we formally do not make this requirement, but we expect it to hold for all natural top--counting rules.
Top--counting rules include, for example, the Bloc rule, -PAV, and -CC, where Bloc uses the linear counting functions , -PAV uses counting functions , and -CC uses counting functions:
As an extreme example of a top--counting rule, Faliszewski et al.  introduced the Perfectionist rule, which uses counting functions:
Perfectionist is extreme in the sense that a voter assigns a point to a committee exactly if he or she ranks all the members of this committee as best ones.
Let us recall our T-shirt store example (Example 2.2). Consider a particularly snobbish customer, who is willing to buy a shirt from a store only if he or she views all the available shirts as very good (recall that we defined “very good” to mean being ranked among top positions). Then if are the positions of the available shirts in the customer’s ranking, the number of shirts that the store should expect to sell to such a customer is:
Thus if the store expects such customers, then it should use the Perfectionist rule to choose its merchandise (and, possibly, should also increase its prices!).
Bloc is the only nontrivial rule that is both weakly separable and top--counting, and -CC is the only nontrivial rule that is both representation-focused and top--counting.
Bloc is the only nontrivial rule that is weakly separable and top--counting.
-CC is the only nontrivial rule that is representation-focused and top--counting.
It is easy to verify that -CC is top--counting and representation-focused. For the other direction, let be a rule which is both top--counting and representation focused. It follows that for two functions, and , such that, and , where . Since any linear transformation of has the same form, by the uniqueness we can assume that .
For each let denote the sequence . For each we have that:
By the same reasoning, we can prove that for each we have . Since the rule is nontrivial, we know that for some it holds that . This is sufficient to claim that is equivalent to -CC. ∎
Faliszewski et al.  show that top--counting rules tend to be -hard to compute, but point out several polynomial-time computable exceptions, including Bloc and Perfectionist. They also observe that for rules with concave counting functions there are polynomial-time constant-factor approximation algorithms, whereas for rules with convex counting functions such algorithms may be missing (under standard complexity-theoretic assumptions).
3.4 OWA-Based Rules
Skowron et al.  introduced a class of multiwinner rules based on ordered weighted average (OWA) operators. Similar rules for approval-based ballots were first considered in the 19th century by Thiele  and more recently were studied by Aziz et al. [2, 5] and Lackner and Skowron  (see also the discussion by Kilgour ). Elkind and Ismaili  use OWA operators to define a different class of multiwinner rules, which we do not consider in this paper.
We provide intuition for the OWA-based rules by using our T-shirts store example.
Let us say that a customer views a T-shirt as “good enough” if it
is among the top of the shirts available on the market. Suppose
that a customer identifies the best T-shirt available in the store
and buys it with probability 1, provided it is “good enough”. Then he
or she also finds the second best T-shirt and buys it with
of the shirts available on the market. Suppose that a customer identifies the best T-shirt available in the store and buys it with probability 1, provided it is “good enough”. Then he or she also finds the second best T-shirt and buys it with probability(again, provided that it is “good enough”), the third best shirt with probability , and so on, all the way to the ’th best T-shirt, which he or she buys with probability (if it is “good enough”). If are the positions (in the customer’s preference order) of the T-shirts that the store puts on display, then the expected number of T-shirts he or she buys is given by the function:
Thus, to maximize its revenue, the store should find a winning committee for the election where the T-shirts are the candidates, the voters are the customers, and where we use committee scoring rule based on . This multiwinner voting rule is -PAV, a variant of the Proportional Approval Voting rule.
Now let us define OWA-based rules formally. An OWA operator of dimension is a sequence of nonnegative real numbers.
Let be a sequence of OWA operators such that has dimension . Let be a family of single-winner scoring functions. Then, and define a family of committee scoring functions such that for each we have:
We refer to committee scoring rules defined through in this way as OWA-based.
It is known that weakly separable, representation-focused, and top--counting rules are OWA-based. The first class is defined using OWA operators , the second one uses OWA operators , and the last one contains rules that use -Approval single-winner scoring functions and any OWA operator (the argument that shows this is due to Faliszewski et al. [31, Proposition 3] and requires a bit more effort than for the previous two classes). As a corollary to the preceding propositions, we get the following.
Each of the classes of separable, top--counting, and representation-focused rules is strictly contained in the class of OWA-based rules.
Containment follows from the paragraph above. Strictness follows as we have Bloc as the unique rule in the intersection of top--counting and weakly separable; SNTV as the unique rule in the intersection of weakly separable and representation-focused; and -CC as the unique rule in the intersection of top--counting and representation-focused: it follows that Bloc is not representation-focused; SNTV is not top--counting; and -CC is not weakly separable. We get the claim by noticing that Bloc, SNTV, and -CC, are all OWA-based. ∎
Naturally, there are also OWA-based rules that do not belong to any of the above-mentioned classes. For example, this is the case for -PAV rules (provided that the parameter is not equal to the committee size , e.g., if it is fixed as a constant) or for the related -HarmonicBorda rules (the -HB rules), defined by the following scoring functions ( is a parameter):
The -HarmonicBorda rules were introduced by Faliszewski et al. , who were looking for various means of achieving a compromise between the -Borda rule and the Chamberlin–Courant rule (-HB is -Borda, and as becomes larger and larger, -HB becomes more and more similar to -CC).
Neither -PAV nor -HB is weakly separable, nor representation-focused, nor top--counting, for any choice of constants and .
To prove Proposition 3.5 it suffices to show that the committee scoring functions of these rules cannot be expressed as linear transformations of weakly separable, representation-focused, and top--counting scoring functions, and invoke Lemma 2.1. We omit the details of this simple but somewhat tedious task.
Skowron et al.  have shown that OWA-based rules are typically -hard to compute (with the clear exception of, e.g., weakly separable rules and the Perfectionist rule). They have also linked the properties of the OWA operators with the ability to approximate the rules (generally speaking, if the OWA operators for a given rule are non-increasing then there are polynomial-time constant-factor approximation algorithms for this rule, and otherwise they are typically missing777However, there are exceptions. For example, viewed as an OWA-based rule, Perfectionist uses OWA operators but still is polynomial-time computable. This is because, as a top--counting rule, Perfectionist uses a very restrictive single-winner scoring function, and is not captured by the results of Skowron et al. .).
3.5 Decomposable Rules
We introduce the following class that naturally generalizes the class of OWA-based rules and resort to our T-shirt store example to help the reader rationalize it.
Let be a family of single-winner scoring functions. These functions define a family of committee scoring functions such that for each committee position we have:
We refer to committee scoring rules defined through in this way as decomposable.
At first glance, decomposable rules seem very similar to the weakly separable ones. The difference is that for fixed and and two different values and , for decomposable rules the functions and can be completely different. It is apparent that OWA-based rules are decomposable. We will see that this containment is strict.
Let us recall from Example 3.2 that a customer considers a T-shirt to be “good enough” if it is among the best of all shirts and let us say that a shirt is “great” if it is among the top of all shirts. A customer buys two “great” T-shirts, or one “at least good enough” T-shirt (if there are no two “great” T-shirts on display). Naturally, the customer picks the best T-shirt(s) he can find (respecting the above constraints). If are the positions (in the customer’s preference order) of the T-shirts that the store puts on display, then the number of T-shirts he or she buys is given by function:
Thus, to maximize its revenue, the store should find a winning committee for the election where the T-shirts are the candidates, the voters are the customers, and where we use decomposable committee scoring rule based on .
We refer to decomposable rules defined through committee scoring functions of the form
where are OWA operators and are sequences of integers from , as multithreshold rules (we put no constraints on ; both increasing and decreasing sequences are natural).
The committee scoring rule defined through the multithreshold functions , for , , is not OWA-based.
Let us fix , , , and that satisfy the requirements from the statement of the theorem. For the sake of contradiction, assume that our multithreshold function is OWA-based. By Lemma 2.1 we infer that there exist a committee scoring function of the form:
where are two numbers and is a single-winner scoring function, such that for each committee position it holds that ; this follows because, by Lemma 2.1, the OWA-based committee scoring functions for our rule have to depend on and only, and by applying appropriate linear transformations, we can assume that these functions equal .
Let us now consider two committee positions and . We see that:
and, thus, it must also be the case that:
On the other hand, for committee positions and we have:
Since we have both and , we conclude that . However, for committee positions and we have:
which is a contradiction and completes the proof. ∎
We generally expect decomposable rules to be -hard, but even among these rules there are polynomial-time computable rules (that are not OWA-based). For example, in their discussion of top--counting rules, Faliszewski et al.  mention a multithreshold rule that uses scoring functions that mix SNTV and Perfectionist:
Briefly put, each winning committee under this rule is either an SNTV winning committee or is ranked on top positions by some voter, and it suffices to check all such possibilities (thus, e.g., it is possible to compute some winning committee in polynomial time). One can show that this rule is not OWA-based using the same approach as in Proposition 3.6.
3.6 Beyond Decomposable Rules
Naturally, there are also committee scoring rules that go beyond the class of decomposable rules. Below we provide two examples, starting with one inspired by our T-shirt store.
In this example, the store does not want to maximize its direct revenue (i.e., the number of T-shirts sold), but the number of happy customers (in hope of increased future revenue). Let us say that a customer is happy if he or she finds at least two “good enough” T-shirts or at least one “great” T-shirt (recall that “at least good enough” shirts are among top of all available ones, and “great” shirts are among the top ). Then the store should use the committee scoring function
We refer to multithreshold rules with summation replaced by the operator as max-threshold rules. Using an approach similar to that from Proposition 3.6, one can show that there are max-threshold rules that are not decomposable (we omit details).
In their search for rules between -Borda and , Faliszewski et al.  introduced the class of -Borda rules, based on the following scoring functions ( is a parameter):
While the motivation for these rules is the same as for the -HarmonicBorda rules, they behave quite differently (see the work of Faliszewski et al.  for a detailed discussion).
There are committee scoring rules that are not decomposable.
Throughout the rest of the paper, we will not venture outside the class of decomposable rules. However, the above two examples show that there are interesting rules there that also deserve to be studied carefully.
4 Axiomatic Properties of Committee Scoring Rules
After exploring the universe of committee scoring rules from a syntactic (structural) perspective, we now consider axiomatic properties of the observed classes. Specifically, we will use two types of monotonicity notions—non-crossing monotonicity (together with its relaxations) and committee enlargement monotonicity—to characterize several of the classes and to gain insights regarding some others. Indeed, various monotonicity concepts have long been used in social choice (with Maskin monotonicity  being perhaps the most important example) and we follow this tradition.
4.1 Non-crossing Monotonicity and Its Relaxations
Elkind et al.  introduced two monotonicity notions for multiwinner rules, namely candidate monotonicity (recall Section 2.3) and non-crossing monotonicity. In the former, we require that if we shift forward a candidate from a winning committee in some vote, then this candidate still belongs to some winning committee after the shift, but possibly to a different one. In the latter monotonicity notion, we require that the whole committee remains winning, but we forbid shifts were members of the winning committee pass each other (i.e., after a shift none of the committee members gets worse and some get better). More formally, we have the following definition.
Definition 4.1 (Elkind et al. ).
A multiwinner rule is non-crossing monotone if for each election and each the following holds: if for some , then for each obtained from by shifting forward by one position in some vote without passing another member of , we still have .
Elkind et al.  have shown that weakly separable rules are non-crossing monotone, and we will now show that the converse is also true. However, before we proceed to the proof, we introduce the following notation (that will also be useful in further analysis):
Consider an arbitrary number of candidates and a size of committee . For each and , let be the set of committee positions from that have their -th element equal to and such that they do not include position . We set .
For example, if and , then , , , and .
Intuitively, is a collection of committee positions in which the -th committee member stands on position and where shifting him or her without passing another committee member is possible. Similarly, is a collection of committee positions in which there is some committee member on position and it is possible to shift him to position without passing another committee member.
Let be a committee scoring rule. is non-crossing monotone if and only if it is weakly separable.
Let be a committee scoring rule defined through a family of scoring functions . Due to the results of Elkind et al. , it suffices to show that if is non-crossing monotone then it is weakly separable. So let us assume that is non-crossing monotone.
Let us fix the number of candidates and the committee size . Let be an election with candidate set and collection of voters , with one voter for each possible preference order. By symmetry, every size- subset of is a winning committee under .
Consider an arbitrary integer , two arbitrary (but distinct) committee positions and from , and an arbitrary vote from the election. Let be the set of candidates that ranks at positions , and let be defined analogously for the case of . Let be the election obtained by shifting in the candidate currently in position one position up. Finally, let and be committee positions obtained from and by replacing the number with (it is possible to do so as and are both from ).
Since, by assumption, is non-crossing monotone, it must be the case that and are winning committees under also in election . The difference of the scores of committee in elections and is , and the difference of the scores of committee in and is . It must be the case that:
However, since the choice of and the choices of and within were completely arbitrary, it must be the case that there is a function such that for each , each sequence , and each committee position obtained from by replacing position with , we have:
and the values of are non-negative.
Our goal now is to construct a single-winner scoring function such that for each committee position it holds that:
We define by requiring that (a) for each , we have (so is a non-increasing function), and (b) is such that (so that indeed correctly describes the -score of the committee ranked at the bottom positions as a sum of the scores of the candidates).
We fix some committee position from . We know that, due to the choice of , for it does hold that . Now we can see that this property also holds for . The reason is that
Thus, for , we have