1 Introduction
Consider a population with dichotomous (approval) preferences over a set of 300 alternatives. Assume that 50% of the population approves of the first 100 alternatives, 30% of the next 100 alternatives, and 20% of the last 100 ones. Imagine that we want to obtain a ranking of the alternatives that in some sense reflects the preferences of the population. Perhaps the most straightforward approach is to use Approval Voting, i.e., to rank the alternatives from the most frequently approved to the least frequently approved. A striking property of the resulting ranking is that half of the population does not approve any of the first 100 alternatives in the ranking.
In many scenarios, rankings obtained by applying Approval Voting are highly unsatisfactory; rather, it is desirable to interleave alternatives supported by different (sufficiently large) groups. For instance, consider an anonymous user using a search engine to find results for the query “Armstrong”. Even if 50% of the users performing this search would like to see results for Neil Armstrong, 30% for Lance Armstrong, and 20% for Louis Armstrong, it is not desirable to put only results referring to Neil Armstrong in the top part of the ranking shown on the first page; rather, results related to each of the Armstrongs should be displayed in appropriately high positions.
There are numerous applications where such diversity within collective rankings is desirable. For instance, consider recommendation systems which aim at accommodating different types of users (the estimated preferences of these user types should be represented proportionally to their likelihood), a humanresource department providing hiring recommendations when it is unclear how many positions are to be filled, or committee elections with some additional structure (e.g., when we want to elect a committee with a chairman and vicechairman, as well as substitute members). Another example application in which diversity in rankings is relevant is
liquid democracy (Behrens et al., 2014). A defining feature of a liquid democracy is that all participants are allowed—and encouraged—to contribute to the decision making process. In particular, in a context where one (or more) out of several competing alternatives needs to be selected, each participant can propose their own alternative if they are not satisfied with the existing ones. This may lead to situations where a very large number of proposals needs to be considered. Since it cannot be expected that every participant studies all available alternatives before making a decision, the order in which competing alternatives are presented plays a crucial role (Behrens et al., 2014, Chapter 4.10).Indeed, the idea of diversified rankings with respect to the preferences of a population appears in several literatures. For example, in the context of search engines, this aim is often referred to as diversifying search results (Welch, Cho, and Olston, 2011; Santos, MacDonald, and Ounis, 2015; Kingrani, Levene, and Zhang, 2015; Wang, Luo, and Yu, 2016). There are also models which incorporate this idea into online advertising (see the work of Hu et al., 2011, and the references therein). In the context of liquid democracy, Behrens et al. observe that using AV gives rise to what they call the “noisy minorities” problem: relatively small groups of very active participants can “flood” the system with their contributions, creating the impression that their opinion is much more popular than it actually is. This is problematic insofar as other alternatives (that are potentially much more popular) run the risk of being “buried” and not getting sufficient exposure. Behrens et al. suggest that in order to prevent this problem, the ranking mechanism needs to ensure that the order adequately reflects the opinions of the participants.
In this paper, we propose an abstract model applicable to all these applications, and initiate a formal axiomatic study of the problem of finding a proportional collective ranking. In our study, we use tools from the political and social sciences, and in particular we adapt the concept of proportional representation (Balinski and Young, 1982; Monroe, 1995) to the case of rankings. Informally, proportional representation requires that the extent to which a particular preference or opinion is represented in the outcome should be proportional to the frequency with which this preference or opinion occurs within the population. For instance, proportional representation is often a requirement in the context of parliamentary elections (Pukelsheim, 2014), where candidates are grouped into political parties and voters express preferences over parties. If a party receives, say, 20% of the votes, then proportional representation requires that this party should be allocated (roughly) 20% of the parliamentary seats (see Laslier, 2012, for a discussion of arguments for and against proportional representation in a political context).
The concept of proportional representation can be extended to the case of rankings in a very natural way. Intuitively, we say that a ranking is proportional if each prefix of , viewed as a subset of alternatives, satisfies (some form of) proportional representation. We will require that, for each “sufficiently large” group of voters with consistent preferences, a proportional number of alternatives approved by this group is ranked appropriately high in the ranking. The position of such alternatives in the ranking depends on the level of their support, indicated by the size and the cohesiveness of the corresponding group of voters.
Our Contribution.
The contribution of this paper is as follows: (i) We formalize the concept of proportionality of a ranking and introduce a quantitative way measuring it, (ii) we observe that several known multiwinner rules satisfying committee monotonicity can be viewed as rankings rules, (iii) we provide theoretical bounds on the proportional representation of several ranking rules, and (iv) we experimentally evaluate ranking rules with respect to our measures of proportionality.
Related Work
Proportional Representation is traditionally studied in settings where a subset of alternatives (such as a parliament or a committee) needs to be chosen. The setting that is most often studied is that of closed party lists, where alternatives are grouped into pairwise disjoint parties and voters are restricted to select a single party (Gallagher, 1991). In this setting, providing proportional representation reduces to solving an apportionment problem (Balinski and Young, 1982; Pukelsheim, 2014). In an influential paper, Monroe (1995) generalized the concept of proportional representation to settings where voter preferences are given as rankorderings over the set of all alternatives. For approval preferences, concepts capturing proportional representation have recently been introduced by Aziz et al. (2015) and SánchezFernández et al. (2016).
The setting considered in this paper differs from all of the above settings in that we are interested in ranking the alternatives, rather than choosing a subset of them. To the best of our knowledge, proportional rankings based on approval preferences have not been considered in the literature. In the context of linear (i.e., rankorder) preferences, proportional rankings are discussed by Schulze (2011). However, this paper neither proposes a measure of proportionality, nor does it compare the proportionality provided by different rules.
2 Preliminaries
For , we write . For each set , we let and denote the set of all subsets of , and the set of all element subsets of , respectively.
Let be a finite set of voters and a finite set of alternatives. Each voter approves a nonempty subset of alternatives . For each , we write for the set of voters who approve , i.e., . We refer to as the approval score of . A list of approval sets, one for each voter , is called a profile on .
A ranking is a linear order over . For a ranking and for , we denote the th element in by . Thus, can be represented by the list . Given a ranking and , we let denote the subset of consisting of the top elements according to . An (approvalbased) ranking rule maps a profile on to a ranking over .
In what follows, we consider several ranking rules that can be obtained by adapting existing multiwinner rules. An (approvalbased) multiwinner rule takes as input a profile on and an integer and outputs a element subset of , referred to as the winning committee. A generic adaptation of multiwinner rules to ranking rules is possible whenever the respective multiwinner rule has the property that for all (this property is known as committee monotonicity or house monotonicity): whenever this is the case, we can produce a ranking by placing the unique alternative in in position .
We consider the following ranking rules.^{1}^{1}1All rules that we describe may have to break ties at some point in the execution; we adopt an adversarial approach to tiebreaking, i.e., we say that a ranking rule satisfies a property only if it satisfies it for all possible ways of breaking ties. Fix a profile on .
 Approval Voting (AV).

Approval Voting ranks the alternatives in order of their approval score, so that .
 Reweighted Approval Voting (RAV).

This family of rules is based on ideas developed by Danish polymath Thorvald Thiele (Thiele, 1895). It is parameterized by weight vectors, i.e., sequences
of nonnegative reals. For a given weight vector
and a subset of alternatives, define the RAV score of as . The intuition behind this score is that voters prefer sets that contain more of their approved alternatives, but that there are decreasing marginal returns to adding further such alternatives to (if is decreasing). The rule RAV constructs a ranking iteratively, starting with the empty partial ranking . In step , it appends to an alternative with maximum marginal contribution among all yet unranked alternatives. Many interesting rules belong to this family for suitable weight vectors . For example, AV is simply RAV, Sequential Proportional Approval Voting (SeqPAV) is defined as RAV, where , Greedy Chamberlin–Courant is defined as RAV, and for every , the geometric rule is given by RAV.  Reverse SeqPAV.

This rule is a bottomup variant of SeqPAV. It builds a ranking starting with the lowestranked alternative. Initially it sets and . Then, at each step it picks an alternative minimizing , removes it from and prepends it to .
 Phragmén’s Rule.

Phragmén (1895) proposed a committee selection rule based on a load balancing approach: every alternative incurs a load of one unit, and the load of alternative has to be distributed among all voters in . Phragmén’s rule constructs a ranking iteratively, starting with the empty partial ranking . Initially, the load of each voter is . At each step, the rule picks a yet unranked alternative and distributes its associated load of over the voters who approve it; the alternative and the load distribution scheme are chosen so as to minimize the maximum load across all voters. This alternative is then appended to the ranking . (For details, see the work of Janson, 2012, and Mora and Oliver, 2015).
The following example illustrates the ranking rules defined above.
Example 1.
Consider the following profile with 7 voters and 6 alternatives:
Assume lexicographic tiebreaking. Since the approval scores of the alternatives are equal to 6, 5, 5, 5, 3, and 2, respectively, Approval Voting returns the ranking .
SeqPAV in the first iteration selects an alternative with the highest approval score, i.e., . In the second iteration, the marginal contribution of alternatives to the PAVscore are equal to , , , , and , respectively. Thus, SeqPAV selects . In the next iterations, the rule appends , , , and finally to the ranking.
Let us now consider Reverse SeqPAV. Removing alternative from decreases the PAVscore of the voters by . Removing alternatives from decreases the PAVscore of the voters by , , , and , repectively. Thus, alternative is put in the last position of the ranking produced by Reverse SeqPAV. The whole ranking returned by this rule is .
The reader can easily verify that the geometric rule and the Greedy Chamberlin–Courant rule return the rankings and , respectively.
Finally, Phragmén’s rule returns the ranking . The corresponding load distribution is illustrated in Figure 1.
3 Measures of Proportionality
In this section, we define a measure of proportionality for rankings and then extend it to ranking rules. In what follows, let be a profile on with .
Given a group of voters and a set of alternatives , a natural measure of the group’s “satisfaction” provided by is the average number of alternatives in that are approved by a voter in . Thus, we define
We refer to as the average representation of with respect to . To extend this idea to rankings, we consider the case where the subset is an initial segment of a given ranking , i.e., for some . Intuitively, every group wants to have an average representation that is as large as possible, for all . Now, whether a group of voters deserves to be represented in the top positions of a ranking depends on two parameters: its relative size and its cohesiveness, i.e., the number of alternatives that are unanimously approved by the group members. This motivates the following definition.
Definition 1.
(Significant groups) Consider a profile on . For a group , its proportion is given by , and its cohesiveness is given by . Given and , we say that is significant in if and .
The following definition captures the intuitively compelling idea that a group can demand to be proportionally represented in the top positions of the ranking, as long as the number of demanded alternatives does not exceed the cohesiveness of the group.
Definition 2.
(Justifiable demand) The justifiable demand of a group with respect to the top positions of a ranking is defined as
For example, if a group contains 25% of the voters and has a cohesiveness of 3, it has a justifiable demand of 1 with respect to the top four positions, a justifiable demand of 2 with respect to the top eight positions, and a justifiable demand of 3 with respect to the top twelve positions, which is also its maximum justifiable demand.
It would be desirable to find a rule that provides every group with an average representation that meets the group’s justifiable demand. However, the following example shows that this is not always possible.
Example 2.
Let and and consider the profile given by , , , , , and . Consider the ranking . The group has , . Therefore, its justifiable demand with respect to the top two positions is . However, its average representation with respect to the top two positions of is only . Since is completely symmetric with respect to the alternatives, we can find such a group for every other ranking as well.
This example shows that it may not be feasible to provide each group with the level of representation that meets its justifiable demand, but it might be possible to guarantee a large fraction of it. For instance, in the previous example it is possible to ensure that for all groups and for all . This observation leads to the following definition.
Definition 3.
(Optimal ranking) We define the quality of a ranking for a profile as
An optimal ranking for is a ranking in .
We observe that is welldefined, i.e., the set of pairs such that is always nonempty. Indeed, consider an alternative with the highest approval score: as for all , by the pigeonhole principle we have . Moreover, , so . Example 2 illustrates that the quality of an optimal ranking may be less than .
Unfortunately, good rankings are hard to compute.
Theorem 1.
Given a profile , it is hard to decide whether there exists a ranking with .
Proof.
We give a reduction from Vertex Cover. An instance of this problem is given by a graph and an integer . It is a ‘yes’instance if there is a subset of vertices such that each edge in contains some vertex in . We can assume that is regular, as Vertex Cover remains hard in this case (Garey and Johnson, 1979).
Given a vertex cover instance with , we construct an instance of the problem of finding an optimal ranking in the following way. Since the degree of each vertex is exactly 3, we can assume that : instances with are trivially ‘no’instances. We set , where is a dummy alternative. For every edge we create a voter that approves . We also add dummy voters who approve only. Consequently, . Since the graph is regular, we have . This defines the profile . We will show that has a vertex cover of size if and only if there exists a ranking with , i.e., for all and .
: Let be a vertex cover of size . Consider a ranking where is ranked first, the next positions are occupied by elements of (in arbitrary order), followed by the remaining alternatives (also in arbitrary order). Consider a group of voters and an integer ; if then trivially , so assume that . We consider the following cases:

. Then , so for each we have .

, . Since the intersection of two or more edges contains at most one vertex and the degree of each vertex is , we have and . For , we obtain ; for , we have . Since and is a vertex cover for , it holds that .

, , i.e., contains a single edge voter. Then for all (recall that ), so for all .
Hence for all , .
: Consider a ranking with for all and . We claim that and is a vertex cover. Note that we have , , so . As voters in only approve , alternative has to appear in the first positions of the ranking. Similarly, for every alternative we have , so and . For , the justifiable demand of is . Now suppose that there exists an edge with . Let ; we have , , so , a contradiction. We conclude that is a vertex cover. ∎
Further, a fairly straightforward reduction from the Maximum Subset Intersection problem (Xavier, 2012) shows that even the problem of deciding whether there exists an significant group of voters is hard.
Proposition 1.
The problem of deciding whether there exists an significant group of voters is complete.
Proof.
It is clear that the problem belongs to . Below we show that it is hard.
Let be an instance of the Maximum kSubset Intersection problem. In we are given a collection of subsets over a set of elements , and two positive integers, and . We ask whether there exists a subcollection of sets from , , such that .
From we can construct an instance of the problem of deciding whether there exists an significant group of voters, in the following way. We associate sets from with alternatives, and the elements from with voters (a voter approves of an alternative if ). We set . It is easy to see that for each alternatives , is the size of the largest group of voters who all approve the alternatives, and that there exists an significant group of voters if and only if there exist alternatives that are all approved by some voters. ∎
The measure can be lifted from individual rankings to ranking rules: we can measure the quality of a ranking rule as the minimum value of , over all possible profiles . However, in our theoretical analysis of ranking rules we take the following approach, which assumes more flexibility on the part of groups of voters that seek to be represented: given a group of proportion and with cohesiveness at least , we ask at which point in the ranking the average satisfaction of this group reaches . This guarantee is given by the function and formally defined as follows.
Definition 4.
(group representation) Let be a function from to . A ranking provides group representation (GR) for profile if for all rational , all , and all voter groups that are significant in it holds that
A ranking rule satisfies group representation (GR) if provides group representation for every profile .
Let us now explore the differences between the two measures, the worstcase quality of the ranking , and group representation. While is just a single number, group representation carries more information. In particular, it makes it possible to express the fact that some groups are better represented in the top parts of the resulting ranking than further below (as would be suggested by being a convex function over ). As a more concrete example, assume that and consider a group that is significant. Since the justified demand of this group is upper bounded by , does not say anything about how far we need to go down the ranking to obtain an average representation greater than (the justifiable demand of is at most and guarantees only half of it), while group representation provides such information for each .
In the next section, we investigate guarantees in terms of group representation provided by the ranking rules that we introduced above.
4 Theoretical Guarantees of Ranking Rules
We start our analysis with the simplest of our rules, namely, Approval Voting. Interestingly, Approval Voting provides very good guarantees to majorities, that is, groups with . However, for minorities it provides no guarantees at all.
Theorem 2.
For rational , Approval Voting satisfies group representation for , but fails it for . For , Approval Voting does not satisfy group representation for any function .
Proof.
We first prove that for Approval Voting satisfies group representation for . Consider a group of voters with , . Let , and let be the ranking returned by Approval Voting. Assume for the sake of contradiction that . Then there exists some alternative that is approved by all voters in , but does not appear in the top positions of . Thus, each alternative in is approved by at least voters and hence by at least voters in . Hence,
This contradiction proves our first claim.
Now, we will show that Approval Voting does not satisfy group representation for . Fix and and let . As , there exist some integers and such that . Let , where and . Consider a profile on that contains two groups of voters and with sizes , such that and have the smallest possible intersection. That is, for the sets and are disjoint, and for we have . Suppose that each voter in approves all alternatives in and each voter in approves all alternatives in . Approval Voting may rank in the first positions. Thus,
Finally, let us prove that for Approval Voting does not satisfy group representation for any function . For the sake of contradiction let us assume that this is not the case. Let us fix , , and let . Using the same idea as in the previous paragraph, we obtain an instance where there is a set of voters with , such that no voter in approves any of the alternatives appearing in top positions of the ranking returned by Approval Voting. This gives a contradiction and proves our claim. ∎
In contrast, Phragmén’s rule, SeqPAV and geometric rules, provide reasonable guarantees for all values of and .
Theorem 3.
Phragmén’s rule satisfies group representation for .
Proof.
Fix , , a profile , and a group of voters such that and . Set . Let denote the load of voter after the th step of Phragmén’s rule. Let , i.e., is the maximum load across all voters from after the th step. Further, define the excess of voter at step as , set , and let
The above definitions are illustrated in Figure 2.
Let and let be ranking returned by Phragmén’s rule. For the sake of contradiction, let us assume that . Then during the first steps of the rule there exists an alternative that is approved by all members of , but has not yet been selected by the rule; denote this alternative by .
First, we prove that for all it holds that . In fact, we will prove a stronger claim:
(1) 
(The claim is stronger because we take the maximum over all voters rather than over the voters from .) Indeed, suppose for the sake of contradiction that this is not the case; let be the first step of the algorithm where Inequality (1) is violated. Since the loads of all voters monotonically increase, we have , i.e., the maximum load strictly increases. However, since , the algorithm could have selected and distributed the associated load in such a way that the maximum load is lower than , a contradiction. Thus, for each .
Second, we prove that
(2) 
Again, for the sake of contradiction assume that . Note that the total load assigned to the voters after steps of the algorithm is equal to . Thus, by the pigeonhole principle, after the th round the load of some voter is at least . Let be the first round where the load of some voter is at least . We have
Thus, the rule could have chosen and distributed the associated load in such a way that the load of each voter is less than , a contradiction. This proves Inequality (2). Since , we infer that
(3) 
Further, let us investigate the relationship between the average representation of voters from and the value . First, observe that : indeed, if that was not the case, the algorithm could have chosen at step and distributed the associated load uniformly among the voters in to achieve a lower maximum load.
Consider a voter . Since , it follows that . Multiplying by , we obtain
(4) 
Also, for all we have
(5) 
We are ready to assess the value :
(6)  
(7)  
(8)  
(9)  
(10)  
(11) 
In the above sequence of inequalities, (7) follows from (5), (8) and (10) follow by simple algebraic operations, (9) is a consequence of (4), and (11) follows from the fact that and . As a result, we get
and thus .
To summarize, we obtained an upper bound of on the sum of squares of the variables from and a lower bound of on their sum. Now, we move to the main step of our proof: we will use these two bounds to assess the total number of representatives of the voters from in top positions.
Let if is positive and otherwise. Note that if , this means that voter gets one more representative at step . It follows that .
Recall that the Cauchy–Schwarz inequality states that for every pair of sequences of real values, and it holds that
(12) 
Applying this inequality, we get
We infer that . Now, recall that , and hence . We have
Here the inequality follows from the fact that , . This gives a contradiction and completes the proof.
Now, let us provide some intuition behind the mathematical formulas and explain why it was useful to consider the sum of squares, . Phragmén’s rule aims at distributing the load among the voters as equally as possible. Intuitively, to show that on average the voters have a significant number of representatives, one needs to show that, on average, when a voter gets an additional representative, she is assigned a relatively small amount of load. In some sense can be viewed as a potential function. In each step, this ‘potential function’ increases by a bounded value. Yet, since the quadratic function is convex, when a voter gets an additional representative, the potential function drops superlinearly with respect to the load . This allows to infer that the increase of the load of a voter getting an additional representative can be bounded; formally, this is accomplished by using the Cauchy–Schwarz inequality. Considering the sum of squares of the values allows us to invoke this inequality, yet we believe that similar bounds can be obtained by considering other types of convex transformations of the values . ∎
Let us recall the relation between arithmetic, geometric, and harmonic means. For each sequence of positive values,
, it holds that:
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