The Complexity of Fully Proportional Representation for Single-Crossing Electorates

The Complexity of Fully Proportional Representation for Single-Crossing Electorates

Piotr Skowron
University of Warsaw
Warsaw, Poland
p.skowron@mimuw.edu.pl
Lan Yu
Nanyang Technological University
Singapore
jen.lan.yu@gmail.com
Piotr Faliszewski
AGH University
Krakow, Poland
faliszew@agh.edu.pl
Edith Elkind
Nanyang Technological University
Singapore
eelkind@ntu.edu.sg
Abstract

We study the complexity of winner determination in single-crossing elections under two classic fully proportional representation rules—Chamberlin–Courant’s rule and Monroe’s rule. Winner determination for these rules is known to be NP-hard for unrestricted preferences. We show that for single-crossing preferences this problem admits a polynomial-time algorithm for Chamberlin–Courant’s rule, but remains NP-hard for Monroe’s rule. Our algorithm for Chamberlin–Courant’s rule can be modified to work for elections with bounded single-crossing width. To circumvent the hardness result for Monroe’s rule, we consider single-crossing elections that satisfy an additional constraint, namely, ones where each candidate is ranked first by at least one voter (such elections are called narcissistic). For single-crossing narcissistic elections, we provide an efficient algorithm for the egalitarian version of Monroe’s rule.

1 Introduction

Parliamentary elections, i.e., procedures for selecting a fixed-size set of candidates that, in some sense, best represent the voters, received a lot of attention in the literature. Some well-known approaches include first-past-the-post system (FPTP), where the voters are divided into districts and in each district a plurality election is held to find this district’s representative; party-list systems, where the voters vote for parties and later the parties distribute the seats among their members; SNTV (single nontransferable vote) and Bloc rules, where the voters cast -approval ballots and the rule picks candidates with the highest approval scores (here is the target parliament size, and for SNTV and for Bloc); and a variant of STV (single transferable vote). In this paper, we focus on two voting rules that, for each voter, explicitly define the candidate that will represent her in the parliament (such rules are said to provide fully proportional representation), namely, Chamberlin–Courant’s rule [CC83] and Monroe’s rule [Mon95]. Besides parliamentary elections, the winner determination algorithms for these rules can also be used for other applications, such as resource allocation [Mon95, SFS13b] and recommender systems [LB11].

Let us consider an election where we seek a -member parliament chosen out of candidates by voters. Both Chamberlin–Courant’s and Monroe’s rule work by finding a function that assigns to each voter the candidate that is to represent in the parliament. This function is required to assign at most candidates altogether.111Under Monroe’s rule we are required to pick exactly winners. Some authors also impose this requirement in the case of Chamberlin–Courant’s rule, but allowing for smaller parliaments appears to be more consistent with the spirit of this rule and is standard in its computational analysis (see, e.g., [LB11, BSU13, SFS13b, SFS13a, YCE13]). In any case, this distinction has no bite if there are at least candidates that are ranked first by some voter, which is usually the case in political elections. Further, under Monroe’s rule each candidate is either assigned to about voters or to none. The latter restriction does not apply to Chamberlin–Courant’s rule, where each selected candidate may represent an arbitrary number of voters, and, as a consequence, the parliament elected in this manner may have to use weighted voting in its proceedings. Finally, each voter should be represented by a candidate that this voter ranks as high as possible.

To specify the last requirement formally, we assume that there is a global dissatisfaction function , , such that is a voter’s dissatisfaction from being represented by a candidate that she views as -th best. (A typical example is Borda dissatisfaction function given by .) In the utilitarian variants of Chamberlin–Courant’s and Monroe’s rules we seek assignments that minimize the sum of voters’ dissatisfactions; in the egalitarian variants (introduced recently by Betzler et al. [BSU13]) we seek assignments that minimize the dissatisfaction of the worst-off voter.

Chamberlin–Courant’s and Monroe’s rules have a number of attractive properties, which distinguish them from other multiwinner rules. Indeed, they elect parliaments that (at least in some sense) proportionally represent the voters, ensure that candidates who are not individually popular cannot make it to the parliament even if they come from very popular parties, and take minority candidates into account. In contrast, FPTP can provide largely disproportionate results, party-list systems cause members of parliament to feel more responsible to the parties than to the voters, SNTV and Bloc tend to disregard minority candidates, and STV is believed to put too much emphasis on voters’ top preferences.

Unfortunately, Chamberlin–Courant’s and Monroe’s rules do have one flaw that makes them impractical: It is -hard to compute their winners [PRZ08, LB11, BSU13]. Nonetheless, these rules are so attractive that there is a growing body of research on computing their winners exactly (e.g., through integer linear programming formulations [PB98], by means of fixed-parameter tractability analysis [BSU13], by considering restricted preference domains [BSU13, YCE13]) and approximately [LB11, SFS13b, SFS13a]. We continue this line of research by considering the complexity of finding exact Chamberlin–Courant and Monroe winners for the case where voters’ preferences are single-crossing. Our results complement those of Betzler et al. [BSU13] for single-peaked electorates.

Recall that voters are said to have single-crossing preferences if it is possible to order them so that for every pair of candidates the voters who prefer to form a consecutive block on one side of the order and the voters who prefer to form a consecutive block on the other side. For example, it is quite natural to assume that the voters are aligned on the standard political left-right axis. Given two candidates and , where is viewed as more left-wing and is viewed as more right-wing, the left-leaning voters would prefer to and the right-leaning voters would prefer to . While real-life elections are typically too noisy to have this property, it is plausible that they may be close to single-crossing, and it is important to understand the complexity of the idealized model before proceeding to study nearly single-crossing profiles (in the context of single-peaked elections this agenda has been successfully pursued by Faliszewski et al. [FHH11]).

Our main results are as follows: for single-crossing elections winner determination under Chamberlin–Courant’s rule is in (for every dissatisfaction function, and both for the utilitarian and for the egalitarian version of this rule), but under Monroe’s rule it is -hard. Our hardness result for Monroe’s rule applies to the utilitarian setting with Borda dissatisfaction function. Our algorithm for Chamberlin–Courant’s rule extends to elections that have bounded single-crossing width (see [CGS12, CGS13]). Our proof proceeds by showing that for single-crossing elections Chamberlin–Courant’s rule admits an optimal assignment that has the contiguous blocks property: the set of voters assigned to an elected representative forms a contiguous block in the voters’ order witnessing that the election is single-crossing. This property can be interpreted as saying that each selected candidate represents a group of voters who are fairly similar to each other, and we believe it to be desirable in the context of proportional representation.

The -hardness result for Monroe’s rule motivates us to search for further domain restrictions that may make this problem tractable. To this end, we focus on the egalitarian version of Monroe’s rule and, following the example of Cornaz et al. [CGS12], consider elections that, in addition to being single-crossing, are narcissistic, i.e., have the property that every candidate is ranked first by at least one voter. In parliamentary elections, narcissistic profiles are very natural: we expect all candidates to vote for themselves. We provide a polynomial-time algorithm for the egalitarian version of Monroe’s rule for all elections that belong to this class. Our algorithm is based on the observation that for single-crossing narcissistic elections under the egalitarian version of Monroe’s rule there is always an optimal assignment that satisfies the contiguous blocks property. We show, however, that this result does not extend to general single-crossing elections or to the utilitarian version of Monroe’s rule: in both cases, requiring the contiguous blocks property may rule out all optimal assignments.

In a sense, our result for single-crossing narcissistic elections is not new: it can be shown that such elections are single-peaked (this result is implicit in the work of Barberà and Moreno [BM11]), and Betzler et al. [BSU13] provide a polynomial-time algorithm for the egalitarian version of Monroe’s rule for single-peaked electorates. However, our algorithm has two significant advantages over the one of Betzler et al.: First, it has considerably better worst-case running time, and second, it produces assignments that have the contiguous blocks property. In contrast, if we formulate the analogue of the contiguous blocks property for single-peaked elections, by considering the ordering of the voters that is induced by the axis (see Section 5 for details), we can construct an election where no optimal assignment has the contiguous blocks property; this holds both for Monroe’s rule and for Chamberlin–Courant’s rule (and both for the egalitarian version and for the utilitarian version of either rule).

The paper is organized as follows. In Section 2 we provide the required background, give the definitions of Monroe’s and Chamberlin–Courant’s rules, and define single-crossing and single-peaked elections. Then, in Sections 3 and 4, we discuss the complexity of winner determination under Chamberlin–Courant’s and Monroe’s rules, respectively. We show the limits of the contiguous blocks property approach in Section 5. We conclude the paper in Section 6 by summarizing our results and discussing future research directions.

2 Preliminaries

For every positive integer , we let denote the set . An election is a pair where is a set of candidates and is an ordered list of voters. Each voter has a preference order , i.e., a linear order over that ranks all the candidates from the most desirable one to the least desirable one. For each voter and each candidate , we denote by the position of in ’s preference order (the top candidate has position and the last candidate has position ). We refer to the list as the preference profile.

Given an election and a subset of candidates , we denote by the profile obtained by restricting the preference order of each voter in to . We denote the concatenation of two voter lists and by ; if consists of a single vote we simply write . A list is said to be a sublist of a list (denoted by ) if can be obtained from by deleting voters. An election is said to be a subelection of an election if and for some . Given a subset of candidates , we denote by a fixed ordering of candidates in and by the reverse of this ordering. Given two disjoint sets , we write to denote a vote where all candidates in are ranked above all candidates in .

2.1 Chamberlin–Courant’s and Monroe’s Rules

Both Chamberlin–Courant’s rule and Monroe’s rule rely on the notion of a dissatisfaction function (also known as a misrepresentation function). This function specifies, for each , a voter’s dissatisfaction from being represented by candidate she ranks in position .

Definition 1

For an -candidate election, a dissatisfaction function is a nondecreasing function with .

We will typically be interested in families of dissatisfaction functions, , with one function for each possible number of candidates. In particular, we will be interested in Borda dissatisfaction function . We assume that our dissatisfaction functions are computable in polynomial time with respect to .

Let be a positive integer. A -CC-assignment function for an election is a mapping such that . A -Monroe-assignment function for is a -CC-assignment function that additionally satisfies the following constraints: , and for each either or . That is, both assignment functions select (up to) candidates, and a -Monroe-assignment function additionally ensures that each selected candidate is assigned to roughly the same number of voters. For a given assignment function , we say that voter is represented (in the parliament) by candidate . There are several ways to measure the quality of an assignment function with respect to a dissatisfaction function ; we use the following two:

1. , and

2. .

Intuitively, takes the utilitarian view of measuring the sum of voters’ dissatisfactions, whereas takes the egalitarian view of looking at the worst-off voter only.

We are now ready to define the voting rules that are the subject of this paper.

Definition 2

For every family of dissatisfaction functions , every CC, Monroe, and every , an voting rule is a mapping that takes an election and a positive integer with as its input, and returns a --assignment function for that minimizes (if there are several optimal assignments, the rule is free to return any of them).

Chamberlin and Courant [CC83] and Monroe [Mon95] proposed the utilitarian variants of their rules and focused on Borda dissatisfaction function (though Monroe also considered so-called -approval dissatisfaction functions). Egalitarian variants of both rules have been recently introduced by Betzler et al. [BSU13].

2.2 Single-Crossing and Single-Peaked Profiles

The notion of single-crossing preferences dates back to the work of Mirrlees [Mir71]; we also point the reader to the work of Saporiti and Tohmé [ST06] for some settings where single-crossing preferences are studied. Formally, such elections are defined as follows.

Definition 3

An election , where is a set of candidates and is an ordered list of voters, is single-crossing (with respect to the given order of voters) if for each pair of candidates , such that , there exists a value such that

That is, as we sweep through the list of voters from the first one towards the last one, the relative order of every pair of candidates changes at most once.

Definition 3 refers to the ordering of the voters provided by . Alternatively, one could simply require existence of an ordering of the voters that satisfies the single-crossing property. The advantage of our approach is that it simplifies notation, yet does not affect the complexity of the problems that we study: one can compute an order of the voters that makes an election single-crossing (or decide that such an order does not exist) in polynomial time [EFS12, BCW12].

We also consider single-peaked elections [Bla48].

Definition 4

Let be a preference order over candidate set and let be an order over . We say that is single-peaked with respect to if for every triple of candidates it holds that An election is single-peaked with respect to an order over if the preference order of every voter is single-peaked with respect to . An election is single-peaked if there exists an order over with respect to which it is single-peaked.

If an election is single-peaked with respect to some order then we call a societal axis for . There are polynomial-time algorithms that given an election decide if it is single-peaked and if so, compute a societal axis for it [BT86, ELÖ08]. Thus, just as in the case of single-crossing elections, we can freely assume that if an election is single-peaked then we are given a societal axis as well.

3 Chamberlin–Courant’s Rule

We start our discussion by considering the complexity of winner-determination under Chamberlin–Courant’s rule, for the case of single-crossing profiles.

3.1 Single-Crossing Profiles

A key observation in our analysis of Chamberlin–Courant’s rule is that for single-crossing profiles there always exists an optimal -CC-assignment function where the voters matched to a given candidate form contiguous blocks within the voters’ order. In what follows, we will say that assignments of this form have the contiguous blocks property. We believe that this property is desirable from the social choice perspective: it means that voters who are represented by the same candidate are quite similar, which makes it easier for the candidate to act in a way that reflects the preferences of the group he represents. Later, we will see that the contiguous blocks property also has useful algorithmic implications.

Lemma 5

Let be a single-crossing election, where , , and has preference order . Then for every , every dissatisfaction function for candidates, and every , there is an optimal -CC assignment for under --CC such that for each candidate , if then there are two integers, and , , such that . Moreover, for each such that and it holds that .

Proof  Fix a single-crossing election with and , and let be an optimal -CC-assignment function for under --CC. We assume without loss of generality that for each voter in , the candidate is ’s most preferred candidate is . Let be ’s least preferred candidate in . Now consider some voter such that . We have for every voter such that . Indeed, suppose for the sake of contradiction that for . By our choice of we have . On the other hand, we have and , a contradiction with being a single-crossing election. Hence, the voters that are matched to by form a consecutive block at the end of the preference profile.

To see that for each it holds that voters in form a consecutive block, it suffices to delete and the voters that are matched to from the profile, decrease by one, and repeat the same argument.

Lemma 5 suggests a dynamic programming algorithm for Chamberlin–Courant’s rule.

Theorem 6

For every family of polynomial-time computable dissatisfaction functions and for , there is a polynomial-time algorithm that given a single-crossing election and a positive integer finds an optimal -CC assignment for under --CC.

Proof  Let be our input single-crossing election, where , and has preference order , and let be the target parliament size.

For every , , and we define to be the optimal -aggregated dissatisfaction that can be achieved with a -CC-assignment function when considering subelection ), where and (clearly, is single-crossing). It is easy to see that for every , and the following recursive relation holds (in the equation below, we abuse notation and treat as the respective norm on real vectors, i.e., we assume that it maps a list of values to their sum (when ) or their maximum (when )):

 A[i,j,t]=min {A[i,j−1,t],mini∗

The idea of this recursive relation is to guess the first voter to be represented by ; the optimal representation of the preceding voters is found recursively, for assembly size . To take care of the possibility that does not participate in the solution, we also take into account.

The base cases of the above recursion are as follows. For every and , we have . For every and , it holds that

 A[i,j,1]=minj′≤jℓ(α(posv1(cj′)),…,α(posvi(cj′))).

For every , , and , we have (we match each voter to her top candidate). These conditions suffice for our recursion to be well-defined. Using dynamic programming, we can compute in polynomial time (in fact, in time ) the optimal dissatisfaction of the voters and a parliament that achieves it.

3.2 Extension to Profiles with Bounded Single-Crossing Width

Following the ideas of Cornaz, Galand, and Spanjaard [CGS12, CGS13], we can extend our algorithm for Chamberlin–Courant’s rule to profiles with so-called bounded single-crossing width.

Definition 7

A set , , is a clone set in an election if each voter in ranks the candidates from consecutively (but not necessarily in the same order).

Definition 8

We say that an election has single-crossing width (respectively, single-peaked width) at most if there exists a partition of into sets such that (a) for each the set is a clone set in and , and (b) if we contract each in each vote to a single candidate , then the resulting preference profile is single-crossing (respectively, single-peaked).

Profiles with small single-crossing width may arise, e.g., in parliamentary elections where the candidates are divided into (small) parties and the voters have single-crossing preferences over the parties, but not necessarily over the candidates. Using the same techniques as Cornaz et al., we obtain the following result.

Proposition 9

For every family of polynomial-time computable dissatisfaction functions and for every , there is an algorithm that given an election with , whose single-crossing width is bounded by , a partition of into clone sets that witnesses this width bound, and a positive integer , finds an optimal -CC assignment for under --CC, and runs in time .

Proof sketch  Let be our input election, and let be a partition of witnessing that the single-crossing width of is at most ; assume that the order of the sets is such that the preference order of the first voter in is of the form . We first observe that Lemma 5 generalizes easily to elections with a given partition into clone sets. Specifically, there exists an optimal -CC assignment for under --CC, where for each clone set , if (that is, if at least one candidate from is assigned to some voter) then: (a) there are two integers, and , , such that , and (b) for each such that and it holds that . That is, the voters matched to the candidates from a given clone set form a consecutive block within the voter order.

Now it is easy to modify our dynamic programming algorithm for profiles with bounded single-crossing width. We guess an integer , a subset of (the candidates from to join the assembly), and a voter such that voters are represented by the candidates from . Note that assigning the candidates from to these voters optimally is easy under Chamberlin–Courant’s rule: each voter gets her most preferred candidate from . An optimal representation for is found recursively (for an appropriately smaller assembly). To implement guessing, we try all possible choices of , all possible subsets of , and all possible choices of , and we use dynamic programming to implement the recursive calls efficiently, just as in the perfectly single-crossing case. Since there are only possibilities to consider at each guessing step and , we obtain the desired bound on the running time.

Naturally, for this result to be useful, we need an efficient algorithm that computes single-crossing width of a profile and an appropriate division into clone sets. Fortunately, such an algorithm is provided by Cornaz et al. [CGS13]. (Interestingly, a very similar problem of finding a division into clones that results in a single-crossing election with as many candidates as possible is -hard [EFS12]). As a consequence, we have the following corollary (see the books [Nie06, DF99] for an introduction to fixed-parameter complexity theory).

Corollary 10

For every family of polynomial-time computable dissatisfaction functions and for every , the problem of winner determination for --CC is fixed-parameter tractable with respect to the single-crossing width of the input profile.

4 Monroe’s Rule

The results of Betzler et al. [BSU13] suggest that winner determination under Monroe’s rule tends to be harder than winner determination under Chamberlin–Courant’s rule. In this section, we show that this is also the case for single-crossing profiles: we prove that for the utilitarian variant of Monroe’s rule with Borda dissatisfaction function (perhaps the most natural variant of Monroe’s rule) computing winners is -hard, even for single-crossing elections. We then complement this hardness result by showing that for the egalitarian version of Monroe’s rule winner determination is easy if we additionally assume that the preferences are narcissistic.

4.1 Hardness for General Single-Crossing Profiles

This section is devoted to proving that winner determination under Monroe’s rule is NP-hard. The main idea of the proof is to reduce the problem of winner determination for unrestricted profiles to the case of single-crossing profiles.

Theorem 11

Finding a set of winners under --Monroe voting rule is -hard, even for single-crossing elections.

The proof of this theorem is somewhat involved. We first need the following two lemmas.

Lemma 12

Consider an election with , . Let and be two disjoint sets of candidates such that . For each , there is a single-crossing election with candidate set and voter list such that for each , and the profile , where has preference order and has preference order , is also single-crossing.

Proof  Set and . Fix a candidate . We build the election as follows. We set ’s preference order to be

 a1≻a2≻… ≻amn≻b1≻b2≻… ≻bposv1(i)−1≻ci≻bposv1(i)≻⋯≻bmn.

For , we build the preference order of voter based on the preference order already constructed for . Given that the preference order of is of the form

 a1≻a2≻… ≻ax≻b1≻b2≻⋯≻by≻ci ≻by+1≻⋯≻bmn≻ax+1≻⋯≻amn,

we construct the preference order of either by moving some of the candidates from to precede or by moving some of the candidate from to follow . Specifically, we do the following. First, we compute ; note that . If then we set ’s preference order to be

 a1≻a2≻… ≻ax≻b1≻b2≻⋯≻by+d≻ci ≻by+d+1≻⋯≻bmn≻ax+1≻⋯≻amn.

If then we set ’s preference order to be

 a1≻a2≻… ≻ax−d≻b1≻b2≻⋯≻by≻ci ≻by+1≻⋯≻bmn≻ax−d≻⋯≻amn.

If then we set ’s preference order to be the same as ’s. Clearly, to construct each vote it suffices to shift forward or backward a block of at most candidates and since both and contain candidates, doing so is always possible. Finally, it is clear that we never change the relative order of the candidates within and within , and that the resulting profile is single-crossing, even if we prepend and append to it.

Lemma 13

For every pair of positive integers such that divides , and every set of candidates, there is a single-crossing profile with voters such that each candidate is ranked first by exactly voters.

Proof  We build a list of voters, where each , , contains voters. The preference order of each voter in is It is clear that a thus-constructed profile is single-crossing. Note that the preference order of the last voter in this profile is the reverse of the preference order of the first voter.

We extend the notation introduced in Lemma 13 to apply to orders of candidates. That is, if is an order of candidates in , then by we denote the election that we would construct in Lemma 13 if the first voter’s preference order was (i.e., if we took to be the top candidate according to , to be the second one, and so on). By we denote an order of the candidates in such that produces an election where the last voter has preference order . With these lemmas and notation available, we are ready to give our proof of Theorem 11.

Proof of Theorem 11  Let be an instance of the problem of finding winners under --Monroe rule, and let be the election considered in . Set and . We assume that is divisible by and that (computing --Monroe winners is still -hard under these assumptions [BSU13, SFS13b]). We will show how to construct in polynomial time an instance of the problem of finding winners under --Monroe where the election is single-crossing so that it is easy to extract the set of winners for from the set of winners for .

We construct in the following way. First, we define the candidate set to be the union of the following disjoint sets (we provide names of the candidates only where relevant and abbreviate to ):

1. , where ;

2. , where for each ;

3. , where for each ;

4. , where ;

5. , where for each ;

6. , where for each ;

7. , where ;

8. .

The ordered list of voters consists of the following five sublists (we only give names to those voters to whom we will refer directly later; whenever sufficient, we only give the number of voters in a given list):

1. , ;

2. , ;

3. , ;

4. , ;

5. , .

We give the preferences of the voters in Table 1. In the thus-defined profile our goal is to find a parliament of size . Consequently, each selected candidate should be assigned to voters.

We claim that optimal solutions for satisfy the following conditions:

1. Each candidate is a winner and is assigned to those voters from that rank in position (note that only one of these candidates can be assigned to (some of the) voters in ).

2. Each candidate is a winner and is assigned to those voters from that rank in position .

3. Each candidate is a winner and is assigned to voters from (exactly voters from have some candidate from assigned to them); each such voter ranks in position .

4. Exactly candidates from are winners. Each of them is assigned to voters in and to one voter in that ranks him highest.

5. The winners from (let us call them ) are also --Monroe winners in and each of them is assigned in to the voters corresponding to those from the -solution.

Let us now show that indeed the optimal solution is of this form. First, we make the following observations:

1. By a simple counting argument, at least of the candidates from must be included in the optimal solution.

2. For each candidate in , if is part of the optimal solution then is ranked in the -th position in the preference order of the voters to which is assigned (candidates from are always ranked first, in the order ).

3. For each candidate , if is included in the optimal solution then each voter to which is assigned ranks in position or worse (this is because every voter’s top positions are taken by the candidates from ).

4. Each voter in ranks each candidate in position worse than

 p1=|H|+∑i|Ei|+∑i|Fi|+|E|>|H|+∑i|Ei|+2m2n+m.
5. Each voter in ranks each candidate in in position better than

 p2=|H|+∑i|Di|+∑i|Fi|+∑i|Gi|+m=|H|+∑i|Ei|+2m2n+m,

but worse than .

6. .

7. For each candidate , there is exactly one voter in that ranks in a position no worse than

 p4 =|H|+∑i|Fi|+∑i

all other voters in rank in a position worse than

 p5=|H|+∑i|Ei|+∑i|Fi|+|E|=|H|+∑i|Ei|+2m2n+m=p2.

Let be an optimal assignment function among those that use exactly candidates from . We claim that satisfies conditions (i)–(iv). This is so, because assigning voters from to candidates other than those in will result in a strictly worse assignment (the assignment would get worse for the candidates in because of points (d), (e), (f) and (g), and it would not improve for the other candidates because of points (b) and (c)). Similarly, each of the selected candidates from should be assigned to exactly one voter from —the one that ranks this candidate highest. Once we assign the winners from to the voters in and to voters in , the optimal way to complete the assignment is to do so as described in conditions (i)–(iv).

Let be an optimal assignment function for that uses exactly candidates from and that satisfies conditions (i)–(iv). We now prove that it also satisfies condition (v). Consider a candidate that is included in the set of winners under . Let be the subcollection of the voters from that are assigned to under (naturally, ). Let be the subcollection of containing the voters corresponding to those in (again, ). Let be the dissatisfaction of the voters in under and let denote the dissatisfaction the voters in would have if they were assigned to (in ). The total dissatisfaction of the voters assigned to under is:

 (|H|+∑j|Ej|+∑j|Fj|+|E|−|Ei|)+s(Vci4)= (|H|+∑j|Ej|+∑j|Fj|+|E|−2m2n−m−(m−i)(2mn+1)nk) +nk(|H|+∑j|Dj|+(m−i)(2mn+1)+mn)+s(Vci))= (nk+1)(|H|+∑j|Ej|+mn)+s(Vci),

which shows that the dissatisfaction of the voters in that are assigned to under differs from the dissatisfaction of the respective voters in , had they been assigned to , only by a value that depends on , , and (but not on ). Thus condition (v) holds.

It remains to show that an optimal assignment function for uses exactly candidates from . Let be an optimal assignment function for that satisfies conditions (i)–(v) (and thus uses exactly candidates from ). Let be an assignment function for that uses more thank candidates from . We will show that the total dissatisfaction under is higher than under . It is easy to see that the average dissatisfaction of the voters in under is lower or equal than that of the voters in under (this follows by contrasting properties (i)–(iii) and observations (b) and (c)).

By the same reasoning as in the proof of property (v), we note that for each candidate , the dissatisfaction of the voters that assigns to can be lower bounded by the dissatisfaction for the case where is assigned to the voter from that ranks highest and to voters from that rank highest. The voter from ranks in a position no better than and each of the voters ranks in position no better than . Thus, under , the dissatisfaction of the voters to whom is assigned can be lower bounded by:

 |H|+∑j|Ej|+∑j|Fj|+|E|−|Ei| +nk(|H|+∑j|Dj|+(m−i)(2mn+1)+mn) =|H|+∑j|Ej|+∑j|Fj|+|E| +nk(|H|+∑j|Dj|