Dramatis Personae: ABC Rules

Abstract

In this chapter, we define the basic ingredients of approval-based committee (ABC) voting: candidates, voters, preferences, and committees. Most importantly, we present the main characters of this book: ABC voting rules.

In this chapter, we define the basic ingredients of approval-based committee (ABC) voting: candidates, voters, preferences, and committees. Most importantly, we present the main characters of this book: ABC voting rules. We introduce and define the most important ABC rules and discuss the main classes they belong to. These include Thiele methods and their sequential variants, Monroe’s rule, Phragmén’s rules and its derivatives, as well as non-standard ABC rules.

2.1 The Formal Model

We now define the basic ingredients of approval-based committee (ABC) voting: candidates, voters, preferences, committees, and ABC rules.

2.1.1 Candidates, Voters, and Preferences

Let C be a finite set of available candidates (also called alternatives). We assume that voters’ preferences are available in the form of approval preferences, i.e., voters distinguish between alternatives they approve and those that they disapprove—due to this dichotomy such preferences are also called dichotomous preferences. Hence a voter’s preference over candidates can be represented by a set of approved alternatives. Let $N\subseteq \mathbb {N}$ denote a finite set of voters.

An approval profile is the collection of all voters’ preferences; formally it is a function $A: N \rightarrow \mathcal {P}(C)$. We say that $A(i) \subseteq C$ is voter i’s approval ballot. Throughout the book, we use $n=|N|$ to denote the number of voters and $m=|C|$ to denote the number of alternatives. Further, we write N(c) to denote the subset of voters that approve candidate c, i.e., $N(c)=\{i\in N: c\in A(i)\}$.

Example 2.1

An academic society chooses a steering committee. Such a committee consists of four persons ($k=4$) and there are seven candidates competing for these positions, $C=\{a,b,c,d,e,f,g\}$. All members of the society are eligible to vote and may provide approval ballots to indicate their preference. In total, 12 ballots have been submitted ($N=[12]$):

$$\begin{aligned} \begin{array}{llllllll} &{}A(1) :\{ a,b\} &{} &{}\qquad A(2) :\{ a,b\} &{} &{}\qquad A(3) :\{ a,b\} &{}&{}\qquad A(4) :\{ a,c\}\\ &{} A(5) :\{ a,c\} &{}&{}\qquad A(6) :\{ a,c\} &{}&{} \qquad A(7):\{ a, d\} &{}&{}\qquad A(8):\{ a, d\} \\ &{} A(9):\{ b, c, f\} &{}&{}\qquad A(10):\{ e\} &{} &{}\qquad A(11):\{ f\} &{} &{}\qquad A(12):\{ g\} \text {.} \end{array} \end{aligned}$$

Figure 2.1 shows a graphical representation of this approval profile. In this figure, each column correspond to one voter (one approval set) and each candidate appears in only one row—each candidate is approved by the voters that appear below the boxes that represent the candidate. Colours are used to distinguish different candidates.

Fig. 2.1

Graphical representation of the approval profile from Example 2.1. Each candidate is represented by one or several boxes that appear in a single row in the figure, and that are marked with a candidate-specific colour. A voter approves those candidates whose corresponding boxes appear above the voter. For example, voter 1 approves candidates a and b and voter 4 approves candidates a and c

Sometimes, we are only interested in how often a specific approval set occurs in an approval profile and thus ignore the names (identifiers) of the voters who cast the approval ballots. In such cases, we do not specify the concrete mapping from N to approval sets but use the following notation:

$$\begin{aligned} \begin{array}{llllllll} &{}3 \times \{ a,b\} &{} &{}\qquad 3 \times \{ a,c\} &{}&{}\qquad 2 \times \{ a, d\} &{}&{}\qquad 1 \times \{ b, c, f\}\\ &{} 1 \times \{ e\} &{} &{}\qquad 1 \times \{ f\} &{} &{} \qquad 1 \times \{ g\} \text {.} \end{array} \end{aligned}$$

The reader may ponder which steering committee of size $k=4$ should be selected given this approval profile—there is certainly more than one sensible choice. In the following chapter, we will see how different voting rules decide in this situation.

We do not make assumptions about the size of approval ballots, as we assume that it is the voters’ decision how many candidates she approves. In applications, however, there is sometimes an upper limit on how many candidates can be approved (often the desired committee size). Such a requirement has hardly any effect on the results presented in this book. In a richer model where voters have underlying, non-dichotomous (i.e., non-binary) preferences, such a restriction would become more relevant; this effect has been analysed by Xiao et al. [47] and Godziszewski et al. [20]. The main conclusion is that it is typically better to give the voters freedom in choosing how many candidates they wish to approve.

2.1.2 Committees and ABC Rules

As we have seen in Example 2.1, committees are sets of candidates. Typically, we are interested in committees of a specific size, which we denote by k. The input for choosing such a committee is an election instance $E=(A, k)$ consisting of a preference profile A and a desired committee size k. Note that given A, we can derive N and C from this function: N is the domain of A and—under the mild assumption that all candidates are approved by at least one voter—C is the union of all function values, i.e., $C=\bigcup _{i\in N} A(i)$. Thus we do not mention N and C in this notation.

Let us now define the key concept of this book: approval-based committee voting rules (short: ABC rules). An ABC rule is a voting method for choosing committees, i.e., an ABC rule takes an election instance as input and outputs one or more size-k subsets of candidates. We refer to these size-k subsets as winning committees.

If an ABC rule outputs more than one committee, we say that these committees are tied. An ABC rule is resolute if it always outputs exactly one committee. In practical settings, it is often undesirable to have more than one winning committee. Consequently, in many concrete voting systems a tiebreaking method is included so that a resolute outcome is guaranteed. This tiebreaking method is typically a random process. As we assume that an ABC rule is a deterministic process, we further assume that all randomisation is done before the election (or at least before the ABC rule is applied). Under this assumption, a randomised tiebreaking method corresponds to a fixed (linear) tiebreaking order over committees; if more than one committee is winning, this tie is resolved by picking the winning committee that is maximal in the tiebreaking order. In this sense, our model incorporates voting systems that rely on randomised tiebreaking.¹

Some of the ABC rules defined in the following are resolute, i.e., they always return a single winning committee, and some are irresolute. Most rules can be defined either way; we have chosen the more natural definition for each rule.

For the following definitions, we assume that we are given an election instance $E=(A,k)$ with a voter set N and a candidate set C.

2.2 Thiele Methods

In the single-winner setting, i.e., if $k=1$, there are few reasonable voting rules when presented with approval ballots. The arguably most natural rule is Approval Voting. Approval Voting selects those alternatives that are approved by the maximum number of voters, all of which are (co-)winners according to this rule. Most ABC rules introduced in this chapter are equivalent to Approval Voting for the case $k=1$ (we discuss notable exceptions in Sect. 2.7). There is, however, one ABC rule that extends the reasoning of Approval Voting to $k>1$ in the most natural manner; this rule is therefore called Multi-Winner Approval Voting (short: AV).²

Rule 1

(Multi-Winner Approval Voting, AV) This ABC rule selects the k candidates which are approved by most voters. Formally, the AV-score of an alternative $c\in C$ is defined as ${{\textrm{score}_{{{\textrm{AV}}}}}}(A, c) = |N(c)| = |\{i\in N: c\in A(i)\}|$ and AV selects committees W that maximise ${{\textrm{score}_{{{\textrm{AV}}}}}}(A, W) = \sum _{c\in W}{{\textrm{score}_{{{\textrm{AV}}}}}}(A, c)$.

Example 2.2

Let us consider the instance of Example 2.1:

$$\begin{aligned}&3 \times \{ a,b\}{} & {} 3 \times \{ a,c\}{} & {} 2 \times \{ a, d\}{} & {} 1 \times \{ b, c, f\}&1 \times \{ e\}{} & {} 1 \times \{ f\}{} & {} 1 \times \{ g\} \text {.} \end{aligned}$$

To compute winning committees according to AV, we count how often each alternative is approved: a: 8 times, b: 4, c: 4, d: 2, e: 1, f: 2 and g: 1. We want to select the four most-approved alternatives. These are a, b, c, and there is a tie between d and f (both having the fourth highest number of approvals). Hence, AV returns two tied committees: the sets $W_1=\{a,b,c,d\}$ and $W_2=\{a,b,c,f\}$. It is noteworthy that $W_1$ leaves three voters completely unsatisfied with the chosen alternatives, whereas $W_2$ results in only two completely unsatisfied voters.

We continue with an ABC rule that can be seen as the exact opposite of AV. Whereas AV disregards whether some voters completely disagree with a committee, the Approval Chamberlin–Courant rule grants as many voters as possible at least one approved alternative in the committee. This rule was first mentioned by Thiele³ [44], and then independently introduced in a different context by Chamberlin and Courant [12].

Rule 2

(Approval Chamberlin–Courant, CC) The CC rule outputs all committees W that maximise ${{\textrm{score}_{{{{\textrm{CC}}}}}}}(A, W) = |\{i\in N: A(i)\cap W\ne \emptyset \}|$.

Example 2.3

Considering again the instance of Example 2.1, there is exactly one committee that grants each voter (at least) one approved candidate: $W=\{a, e, f, g\}$. This is the winning committee according to Approval Chamberlin–Courant. While this committee indeed provides some satisfaction for every voter, it includes alternatives (e and g) that are approved only by single voters.

The two ABC rules we discussed so far—AV and CC—can be seen as extreme points in the spectrum of ABC rules captured by the class of Thiele methods. This class, introduced by Thiele in the late 19th century [44], encompasses all rules that maximise the sum of the voters’ individual satisfaction, subject to a chosen definition of how satisfaction is measured. The unifying assumption is that a voters’ satisfaction with a committee W is solely determined by the number of approved candidates in this committee, i.e., voter i’s satisfaction is determined by a function $w\left( |W \cap A(i)|\right)$. By choosing different w-functions, a very broad spectrum of ABC rules can be covered.

Rule 3

(Thiele methods, w-Thiele⁴) A Thiele method is parameterized by a non-decreasing function $w:\mathbb N\rightarrow \mathbb R$ with $w(0)=0$. The score of a committee W given a profile A is defined as

$$\begin{aligned} {{\textrm{score}_{w}}}(A, W) = \sum _{i \in N} w\left( |W \cap A(i)|\right) ; \end{aligned}$$

the w-Thiele method returns committees with maximum score.

Indeed, AV is the w-Thiele method with $w(x)=x$, and CC is the w-Thiele method with $w(x)=\min (1, x)$. This is an immediate consequence of the respective definitions.

The following Thiele method is arguably one of the most important: Proportional Approval Voting, in short PAV. Also this rule was defined in Thiele’s original paper [44]. The definition (and properties) of PAV crucially depend on the sequence of harmonic numbers.

Rule 4

(Proportional Approval Voting, PAV) Let ${{\textrm{h}}}(x) = \sum _{j=1}^{x} \nicefrac {1}{j}$ denote the sequence of harmonic numbers. PAV is ${{\textrm{h}}}$-Thiele, i.e., it is the w-Thiele rule with $w(x)={{\textrm{h}}}(x)$. In other words, PAV assigns to each committee W the PAV-score, ${{\textrm{score}_{{{\textrm{PAV}}}}}}(A, W) = \sum _{i \in N} {{\textrm{h}}}\left( |W \cap A(i)|\right)$, and returns all committees with maximum score.

Fig. 2.2

Defining w-functions for three Thiele methods: Multi-Winner Approval Voting (AV), Proportional Approval Voting (PAV), and Approval Chamberlin–Courant (CC)

By using the sequence of harmonic numbers ${{\textrm{h}}}(\cdot )$, we introduce a flattening satisfaction function for voters, akin to the law of diminishing returns. As a consequence, PAV balances the (justified) demands of large groups with the conflicting goal of satisfying small groups. Indeed, as we will see in Chap. 4, Proportional Approval Voting achieves this balance in a proportional fashion. Figure 2.2 shows a visualisation of the defining w-functions of different Thiele methods:

$$\begin{aligned} w_{{{\textrm{AV}}}}(x) = x{} & {} \qquad w_{{{\textrm{PAV}}}}(x) = \sum _{i=1}^{x} \nicefrac {1}{i}{} & {} \qquad w_{{{{\textrm{CC}}}}}(x) = {\left\{ \begin{array}{ll}0 &{} \text {if }x = 0,\\ 1 &{} \text {if }x\ge 1 \text {.}\end{array}\right. } \end{aligned}$$

Note that also visually the function defining PAV is “in between” AV and CC.

Example 2.4

Given the instance of Example 2.1:

$$\begin{aligned}&3 \times \{ a,b\}{} & {} 3 \times \{ a,c\}{} & {} 2 \times \{ a, d\}{} & {} 1 \times \{ b, c, f\}&1 \times \{ e\}{} & {} 1 \times \{ f\}{} & {} 1 \times \{ g\} \text {,} \end{aligned}$$

PAV selects the committee $W=\{a,b,c, f\}$. For one voter (the one that approves $\{b,c,f\}$) this committee contains three approved alternatives, for six voters this committee contains two approved alternatives, for three voters W contains one approved alternative, and two voters are not at all satisfied with W. Thus, we have ${{\textrm{score}_{{{\textrm{PAV}}}}}}(A, W) = (1+\nicefrac 1 2+\nicefrac 1 3) + 6 \cdot (1+\nicefrac 1 2) + 3 \cdot 1 = \nicefrac {83}{6}$ and this value is optimal. Coincidentally, W is one of the two committees produced by AV, namely the one with fewer dissatisfied voters. It appears that PAV strives for a compromise between AV and CC—this is an intuition that we will discuss in more detail later (Sect. 4.5).

Other Thiele methods that have been studied in the literature are the class of p-geometric rules [42], threshold procedures [19, 23], and Sainte-Laguë Approval Voting (SLAV) [25].

Thiele methods pick committees that maximise a certain welfare of the voters and thereby belong to a broader class of welfarist rules.

Definition 2.1

A welfare vector induced by a committee W specifies, for each voter, her satisfaction from W (measured as the number of candidates she approves in W):

$$\begin{aligned} \textrm{welf}(W) = (|A(1) \cap W|, |A(2) \cap W|, \ldots , |A(n) \cap W|) \text {.} \end{aligned}$$

An ABC rule $\mathcal {R}$ is welfarist if there is a function $f:\mathbb N^N \rightarrow \mathbb R$, mapping welfare vectors to scores, such that for each instance (A, k) we have

$$\begin{aligned} \mathcal {R}(A, k) = \mathop {{{\,\mathrm{arg\,max}\,}}}_{W\subseteq C\text { with } |W|=k} f(\textrm{welf}(W)). \end{aligned}$$

In this definition, $f(\textrm{welf}(W))$ can be viewed as the welfare that voters gain from W. For Thiele methods, $f(\textrm{welf}(W)) = {{\textrm{score}_{w}}}(A, W)$, i.e., welfare is the sum of the voters’ w-scores. The class of welfarist rules also allows for an aggregation other than summation. For example, one can define $f(\textrm{welf}(W))$ as the satisfaction of the least-satisfied voter—akin to egalitarian aggregation [29]. Another example of a welfarist rule is a dictatorial rule which compares welfare vectors lexicographically given a fixed order of voters: the first voter in this order is a dictator and only if the dictator is indifferent between two outcomes, the second-in-place may decide, and so on.

These other forms of aggregation have been studied in the context of multi-winner elections with ranking-based preferences (for the egalitarian aggregation see the work of Aziz et al. [4] and Skowron et al. [41]; for OWA-based aggregation see the work of Elkind and Ismaili [13] and Faliszewski et al. [17]). For approval ballots, we are aware of only two works that consider such aggregations. Computational properties of CC and Monroe rules based on the egalitarian aggregation are considered by Betzler et al. [5]. Amanatidis et al. [1] consider OWA-based aggregation but for other types of welfare of individual voters. Specifically, the satisfaction of voters with a committee is measured via the Hamming distance, which is in contrast to the definition of $\textrm{welf}(W)$. The most important rule based on the Hamming distance is Minimax Approval Voting, which we discuss in Sect. 2.7.

2.3 Sequential Variants of Thiele Methods

Thiele methods are defined via optimisation statements: given an objective function, Thiele methods return all committees that maximise this function. Instead of computing the true optimum (which is computationally hard, as we will see in Chap. 5), one can define sequential procedures that construct an approximate solution. We define here two classes of sequential procedures: sequential and reverse sequential Thiele methods. Both classes have been introduced in Thiele’s original paper [44] (see Janson’s survey for further historical remarks [22]). Furthermore, both classes can be seen as greedy approximation algorithms for Thiele methods; we return to this analogy in Sect. 5.2.3.

Let us begin with sequential Thiele methods: starting with an empty committee, they add committee members one by one, in each step the one that increases the objective function the most.

Rule 5

(Sequential w-Thiele, seq-w-Thiele) For each w-Thiele method, we define its sequential variant, seq-w-Thiele, as follows. We start with an empty committee $W_0 = \emptyset$. In each round $r\in \{1,\dots ,k\}$, we compute $W_r=W_{r-1} \cup \{c\}$, where c is a candidate that maximises ${{\textrm{score}_{w}}}(A, W_{r-1} \cup \{c\})$, i.e., the candidate that improves the committee’s score the most. If more than one candidate yields a maximum score, we break ties according to some given tie-breaking order. The seq-w-Thiele rule returns $W_k$.

Two sequential Thiele methods will be of particular interest here: sequential $w_{{\textrm{PAV}}}$-Thiele and sequential $w_{{{\textrm{CC}}}}$-Thiele. We refer to these two rules as seq-PAV and seq-CC. In contrast, the sequential variant of AV (seq-$w_{{\textrm{AV}}}$-Thiele) is not relevant to us as it is equivalent to AV. This is because the AV-score (${{\textrm{score}_{{{\textrm{AV}}}}}}$) of candidates is not influenced by the other candidates in the committee.

Example 2.5

Since the instance of Example 2.1 yields the same result for PAV and seq-PAV (and also for CC and seq-CC), we take a look at a different profile:

$$\begin{aligned}&3 \times \{ a,b \}{} & {} \qquad 6 \times \{ a, d\}{} & {} \qquad 4 \times \{ b \}{} & {} \qquad 5 \times \{c\}{} & {} \qquad 5 \times \{c, d\} \text {.} \end{aligned}$$

For $k=2$, PAV selects the committee $\{a,c\}$ with a PAV-score of 19. (Each voter except those that approve only candidate b has exactly one approved candidate in the committee.) Let us contrast this result with seq-PAV. All sequential Thiele methods with $w(1)>0$, including seq-PAV, select the candidate with the largest number of approvals in the first round—the winner according to (single-winner) Approval Voting. Thus, d is selected in the first round as it gives an AV-score of 11. In the second round, we choose between a (increasing the score by 6) and b (increasing the score by 7) and c (increasing the score by 7.5). Hence, seq-PAV returns the committee $\{c, d\}$ with a PAV-score of 18.5.

Similarly to sequential Thiele methods, reverse sequential Thiele methods build committees sequentially, but here one starts with the set of all candidates and sequentially removes the candidate that contributes the least to the committee’s score.⁵

Rule 6

(Reverse Sequential w-Thiele, rev-seq-w-Thiele) For each w-Thiele method, we define its reverse sequential variant, rev-seq-w-Thiele, as follows. We start with $W_m = C$, the set of all candidates. Each round, the candidate with the least marginal contribution to the score is removed. To be precise, in each round r from $m-1$ down to k, we compute $W_{r}=W_{r+1} \setminus \{c\}$, where c is a candidate that maximises ${{\textrm{score}_{w}}}(A, W_{r+1} \setminus \{c\})$, i.e., the candidate whose removal decreases the committee’s score the least. If more than one candidate does that, we break ties according to some given tie-breaking order. The rev-seq-w-Thiele rule returns $W_k$.

In the remainder of the book, we will only encounter reverse sequential PAV (rev-seq-PAV) from the class of Reverse Sequential w-Thiele methods.

Example 2.5

(continued) For rev-seq-PAV, we start with the full set of candidates $W_4=\{a,b,c,d\}$ and remove the candidate with the least marginal contribution: removing a decreases the score by 4.5, removing b decreases the score by 5.5, c by 7.5, and d by 5.5. Thus, a is removed and $W_3=\{b,c,d\}$. Now, we again compute the marginal contributions: for b it is 7, for c it is 7.5, and for d it is 8.5. We obtain $W_2=\{c,d\}$, which is the winning committee. We see that for this instance seq-PAV and rev-seq-PAV yield the same winning committee. This does not hold in general.

An election instance where PAV, seq-PAV, and rev-seq-PAV all yield different winning committees can be found in Janson’s survey [22, Example 13.3]. The example is due to Thiele [44] and is significantly larger than the examples presented here.

As we have mentioned in Sect. 2.2, most ABC rules coincide with Approval Voting for $k=1$. Reverse Sequential PAV is an exception. This is, however, not a consequence of the underlying assumptions how ballots are interpreted, but a consequence of how the rule is computed (i.e., in a reverse fashion).

Example 2.6

To see that rev-seq-PAV is a non-standard method, consider the profile:

$$\begin{aligned}&1 \times \{ a, b\}&1 \times \{ a, b,c\}&1 \times \{ a, b,d\}&2 \times \{ a, c,d\}&\,\,1 \times \{ b\}&1 \times \{ c\}&1 \times \{ d\} \text {.} \end{aligned}$$

In the first round, the marginal contribution of a is $\nicefrac 1 2 + 4\cdot \nicefrac 1 3$; the marginal contribution from the other candidates is at least 2. Thus, candidate a is removed in the first round, even though it has the highest approval score.

Finally, let us mention a paper by Faliszewski et al. [18] which considers and compares several heuristic algorithms for approximating multi-winner rules (e.g., via simulated annealing). This line of work has not yet been extended specifically to Thiele methods, though the ideas in their work can be applied to the ABC setting.

2.4 Monroe’s Rule

Monroe’s rule [27] is an ABC rule⁶ related to the Chamberlin–Courant rule. It also aims at maximising the number of voters who are represented by at least one candidate in the elected committee. The main difference is that each committee member can represent at most $\nicefrac 1 k$-th of the voters.

Rule 7

(Monroe) Given a committee W, a Monroe assignment for W is a function $\phi :N \rightarrow W$ such that each candidate $c \in W$ is assigned roughly the same number of voters, i.e., for all $c\in W$ it holds that $\lfloor \nicefrac {n}{k} \rfloor \le |\phi ^{-1}(c)| \le \lceil \nicefrac {n}{k} \rceil$. The candidate $\phi (i)$ can be viewed as the representative of voter i. Let $\Phi (W)$ be the set of all possible Monroe assignments for W. The Monroe-score of a committee W is defined as the number of voters that have a representative assigned that they approve (given an optimal Monroe assignment), i.e., ${{\textrm{score}_{{{{\textrm{Monroe}}}}}}}(A, W) = \max _{\phi \in \Phi (W)} |\{i\in N: \phi (i) \in A(i)\}|$. Monroe returns all committees with a maximum Monroe score.

Example 2.7

Consider again the profile of Example 2.1:

$$\begin{aligned}&A(1) :\{ a,b\}{} & {} \,\, A(2) :\{ a,b\}{} & {} \qquad A(3) :\{ a,b\}{} & {} \,\,A(4) :\{ a,c\}\\&A(5) :\{ a,c\}{} & {} \,\, A(6) :\{ a,c\}{} & {} \qquad A(7):\{ a, d\}{} & {} \,\,A(8):\{ a, d\} \\&A(9):\{ b, c, f\}{} & {} A(10):\{ e\}{} & {} \qquad A(11):\{ f\}{} & {} A(12):\{ g\} \text {.} \end{aligned}$$

We first note that the desired committee size $k=4$ divides the number of voters ($n=12$) and hence Monroe assigns exactly 3 voters to each committee member. One optimal Monroe assignment (among many) is shown in Fig. 2.3 and given by $\phi ^{-1}(a)=\{3, 7, 8\}$, $\phi ^{-1}(b)=\{1, 2, 9\}$, $\phi ^{-1}(c)=\{4, 5, 6\}$, $\phi ^{-1}(e)=\{10, 11, 12\}$. The Monroe score of this assignment is ${{\textrm{score}_{{{{\textrm{Monroe}}}}}}}(A, W) = 10$, since only voters 11 and 12 are assigned to a representative (candidate e) that they do not approve. In total there are six winning committees; committee $\{a,b,c,e\}$ is one of them.

Fig. 2.3

An optimal Monroe assignment for Example 2.7: the top row shows the assigned representative for each voter. For example, the assigned representative of voter 1 is b; voter 12 is dissatisfied with her assigned representative e

Monroe’s rule has also a natural sequential version called Greedy Monroe, which was introduced by Skowron et al. [41].⁷ We present Greedy Monroe here in a slightly simpler, more practical fashion, where dissatisfied voters are not assigned to groups.

Rule 8

(Greedy Monroe) This ABC rule proceeds in k rounds: In each round $r\in \{1,\dots , k\}$ Greedy Monroe assigns a candidate to a group of voters $G_r$ of size at most $n_r$ (defined below); this candidate is added to the committee. The maximum size of a group, $n_r$, is defined as follows: for $d\,=\,n\mod \,k$, we set $n_1=\dots =n_d=\lceil \nicefrac {n}{k} \rceil$ and $n_{d+1}=\dots =n_k=\lfloor \nicefrac {n}{k} \rfloor$. In round $r+1$, let $N_{r+1}$ denote the voters that have not yet an assigned committee member, i.e., $N_{r+1}=N\setminus (G_1\cup \dots \cup G_{r})$. Candidate $c_{r+1}$ is chosen as the candidate c that maximises $|\{i\in N_{r+1}: c\in A(i)\}|$ among those not contained in the committee yet (using a tiebreaking order on candidates if necessary). Now, if there are at most $n_{r+1}$ not yet assigned voters that approve $c_{r+1}$, then $G_{r+1}=\{i\in N_{r+1}: c_{r+1}\in A(i)\}$; if there are more than $n_{r+1}$ such voters, a tiebreaking order on voters is used to assign exactly $n_{r+1}$ from these voters to $G_{r+1}$. Greedy Monroe outputs the committee $\{c_1,\dots ,c_k\}$.

Example 2.8

In our running example (Example 2.1) given by

$$\begin{aligned} \begin{array}{llllllll} &{} A(1) :\{ a,b\} &{} &{}\qquad A(2) :\{ a,b\} &{}&{}\qquad A(3) :\{ a,b\} &{}&{}\qquad A(4) :\{ a,c\}\\ &{} A(5) :\{ a,c\} &{}&{}\qquad A(6) :\{ a,c\} &{}&{}\qquad A(7):\{ a, d\} &{}&{}\qquad A(8):\{ a, d\} \\ &{} A(9):\{ b, c, f\} &{}&{}\qquad A(10):\{ e\} &{}&{}\qquad A(11):\{ f\} &{} &{}\qquad A(12):\{ g\} \text {,} \end{array} \end{aligned}$$

Greedy Monroe first picks candidate a as it is approved by most voters. We assume that ties among voters are broken in increasing order, so $G_1=\{1, 2, 3\}$. Now c is chosen since it is the only candidate with four supporters among the remaining voters ($N_2=\{4,\dots ,12\}$). The corresponding group of voters is $G_2=\{4,5,6\}$ (again choosing voters with smaller indices first). Now there are two candidates left that are approved by two voters in the remaining set ($N_3=\{7,\dots ,12\}$): candidates d and f. We choose d by alphabetic tiebreaking and so we set $G_3=\{7,8\}$. Finally, there is one candidate that has two supporting voters in $N_4=\{9,\dots ,12\}$: f is approved by voters 9 and 11; thus $G_4=\{9, 11\}$. A Monroe assignment corresponding to this committee $\{a,c,d,f\}$ is, e.g., given by $\phi ^{-1}(a)=\{1,2,3\}$, $\phi ^{-1}(c)=\{4,5,6\}$, $\phi ^{-1}(d)=\{7,8,10\}$, and $\phi ^{-1}(f)=\{9, 11, 12\}$. In this instance, Greedy Monroe was able to find a committee with an optimal Monroe score, but this does not hold in general.

2.5 Phragmén’s Rules

Phragmén⁸ introduced a number of voting rules, most of which are based on a form of cost-sharing (or load balancing). The core idea is that placing a candidate in the winning committee incurs a cost, or load, that has to be shouldered by the voters who approve this candidate. The goal is to choose a committee that allows for as equal as possible a distribution of its cost. In this way, the preferences of as many voters as possible are taken into account.

Phragmén’s main proposal is called Phragmén’s Sequential Rule (seq-Phragmén). Even though Phragmén’s Sequential Rule can be considered one of the most appealing ABC rules, it remained unknown to many social choice researchers until recently. Few publications before 2017 mention Phragmén’s methods; notable exceptions are a survey by Janson [21] (in Swedish) and a paper by Mora and Oliver [28] (in Catalan). Since 2017 several papers have proven Phragmén’s method to be a particularly strong ABC rule, in particular being a proportional ABC rule that is both polynomial-time computable and committee monotone.

We present two (equivalent) formulations of seq-Phragmén. The first is conceptually simpler, while the second gives a clearer picture how the rule is computed in practice.

Rule 9

(Phragmén’s Sequential Rule, seq-Phragmén) This ABC rule is based on the assumption that placing a candidate in the winning committee incurs a cost (or a load) of 1, which is distributed among the set of voters that approve this candidate.

Continuous formulation: We assume that each voter has a budget which constitutes his or her voting power. Voters start with a budget of 0 and this budget continuously increases as time advances. At time t, the budget of each voter is t. As soon as a group of voters that jointly approve a candidate has a total budget of 1, the joint candidate is added to the winning committee. Then the budget of all involved voters is reset to 0; only voters that do not approve the selected candidate keep their current budget. This process continues until the committee is filled. If at some point two candidates could be added to the committee at the same time, a tie-breaking order is used to decide which candidate is selected.

Discrete formulation: seq-Phragmén works in rounds; each round one candidate is added to the committee. Let $y_r(v)$ denote the load assigned to (or cost contributed by) voter v after round $r\le k$. We naturally start with $y_0(v)=0$ for all $v\in N$. Let $\{c_1,\dots ,c_{r-1}\}$ be the candidates added to the committee in rounds 1 to $r-1$. To determine the next candidate $c_{r}$ to add, we compute for each candidate $c\in C\setminus \{c_1,\dots ,c_{r-1}\}$ the maximum load that would arise from adding $c_{r}$:

$$\begin{aligned} \ell _{r}(c)=\frac{1 + \sum _{i\in N(c)} y_{r-1}(i)}{|N(c)|}; \end{aligned}$$

the load of voters in N(c) would increase to this amount if c were added to the committee. Note that the load is distributed so that all voters approving c end up with the same total load; this is so to minimise the maximum load. Now, to keep the maximum load as small as possible, seq-Phragmén chooses the candidate c with a minimum $\ell _{r}(c)$, i.e.,

$$\begin{aligned} c_{r} = \mathop {{{\,\mathrm{arg\,min}\,}}}_{c\in C\setminus \{c_1,\dots ,c_{r-1}\}} \ell _{r}(c). \end{aligned}$$

If two or more candidates yield the same maximum load, a tie-breaking method is required (typically some fixed order on C). After choosing $c_{r}$, the voter loads are adapted accordingly:

$$y_{r}(i) = {\left\{ \begin{array}{ll} \ell _{r}(c_{r}) &{} \text {if }i\in N(c_{r}),\\ y_{r-1}(i) &{} \text {if }i\notin N(c_{r}).\end{array}\right. }$$

The rule returns the winning committee $\{c_1,\dots ,c_k\}$.

To see that these two formulations are equivalent, note that for a winning committee $W=\{c_1,\dots ,c_k\}$ (selected in this order) the maximum loads in each round $\ell _r(c_r)$ directly corresponds to the time points at which sufficient budget was available to pay for $c_r$. From this point of view, the discrete formulation is only the explicit calculation of time points at which sufficient budget is available to place a new candidate in the committee.

Example 2.9

Let us again consider our running example (Example 2.1):

$$\begin{aligned}&3 \times \{ a,b\}&3 \times \{ a,c\}&2 \times \{ a, d\}\\ {}&1 \times \{ b, c, f\}&1 \times \{ e\}&1 \times \{ f\}&1 \times \{ g\} \text {.} \end{aligned}$$

We use the continuous formulation to describe the method, but it is easy to repeat the calculations using the discrete formulation. Figure 2.4 shows a visualisation of the procedure, which we will now explain step by step. The first time sufficient budget is available to add a candidate to the committee is at time $t_1=\nicefrac 1 8$. At this point, voters $\{1,\dots , 8\}$ can jointly pay for candidate a. Now the budgets of voters 1 to 8 are reset to 0; the remaining voters have a budget of $\nicefrac 1 8$ each.

A second candidate can be added to the committee at time $t_2=\nicefrac {11}{32}$. Voters 1, 2, 3, 9 approve candidate b; their respective budgets are $(\nicefrac {7}{32},\nicefrac {7}{32},\nicefrac {7}{32},\nicefrac {11}{32})$ (note that voters 1, 2, and 3 have budgets that are by $\nicefrac 1 8$ lower than that of voter 9). At this time, also voters 4, 5, 6, 9 (who all approve candidate c) have a joint budget of 1. We use alphabetic tiebreaking and select b.

Candidate c is then added as a third candidate at time $t_3=\nicefrac {55}{128}$. At this point, voters 4, 5, and 6 have budgets of $\nicefrac {39}{128}$, and voter 9 has a budget of $\nicefrac {11}{128}$; that is in total 1. Note that these numbers follow from the fact that voters 4–6 already paid $\nicefrac 1 8$ each for selecting candidate a and voter 9 paid $\nicefrac {11}{32}$ for selecting candidate b.

Finally, at time $t_4=\nicefrac 5 8$ the last candidate, d, is added to the committee. At this point, the two voters approving d (voters 7 and 8) have budgets of $\nicefrac 5 8 - \nicefrac 1 8 = \nicefrac 1 2$, in total 1. Thus, seq-Phragmén returns the committee $\{a,b,c,d\}$. When repeating this calculation using the discrete formulation, one obtains the final loads $y_4 = (t_2, t_2, t_2, t_3, t_3, t_3, t_4, t_4, t_3, 0,0,0)$.

Fig. 2.4

A visualisation of seq-Phragmén (upper part) applied to the election instance of Example 2.1 (lower part). In the upper part all regions of the same colour (corresponding to the same candidate) have an area of 1, which is the budget spent on this candidate

Phragmén also discussed optimisation-based analogues of seq-Phragmén. These rules are based on choosing a committee that optimises an objective function (in a similar way as Thiele methods optimise an objective function). We will discuss the most notable optimisation-based method: leximax-Phragmén⁹ [8, 22, 37].

Rule 10

(Phragmén’s Leximax Rule, leximax-Phragmén) Each candidate in the committee incurs a load (or cost) of 1 which has to be distributed among voters approving this candidate. Given a committee $W=\{c_1,\dots ,c_k\}$, a valid load distribution for W is a function $\ell _W:W\times N\rightarrow [0,1]$ which satisfies (1) if $\ell _W(c, i) > 0$ then voter i approves c, and (2) $\sum _{i \in N} \ell _W(c, i) = 1$ for all $c\in W$. Let $\bar{\ell }_W=\left( \sum _{c\in W} \ell _W(c,i) \right) _{i\in N}$ denote the vector of total loads assigned to the voters.

To compare two (valid) load distributions, we use a lexicographic order. Given a valid load distribution $\ell _W$ for W, let $sort (\bar{\ell }_W)$ denote the tuple $\bar{\ell }_W$ sorted from largest to smallest. Let $\ell _W$ and $\ell _{W'}$ denote two valid load distributions for committees W and $W'$, respectively. We say that $\ell _W$ is lexicographically smaller than $\ell _{W'}$ if there exists an index $j\le |N|$ such that the first j entries of $sort (\bar{\ell }_W)$ and $sort (\bar{\ell }_{W'})$ are equal and the $(j+1)$-st entry of $sort (\bar{\ell }_W)$ is strictly smaller than the $(j+1)$-st entry of $sort (\bar{\ell }_{W'})$.

Let $\ell _W^{\min }$ denote a lexicographically smallest valid load distribution for committee W. Then, leximax-Phragmén returns all committees W for which $\ell _W^{\min }$ is lexicographically minimal in the set $\{\ell _{W'}^{\min } : W'\subseteq C\text { and }|W'|=k \}$. Note that if leximax-Phragmén returns two committees $W_1$ and $W_2$, then $sort (\bar{\ell }_{W_1}^{\min } )= sort (\bar{\ell }_{W_2}^{\min })$.

Example 2.10

In our running example, leximax-Phragmén behaves differently than seq-Phragmén. When looking for a committee that has the lexicographically smallest load distribution, we find committee $W=\{a,b,c,f\}$ with $\bar{\ell }_W^{\min }=(\nicefrac 3 8, \nicefrac 3 8, \nicefrac 3 8, \nicefrac 3 8, \nicefrac 3 8, \nicefrac 3 8, \nicefrac 3 8, \nicefrac 3 8, \nicefrac 1 2, 0, \nicefrac 1 2, 0)$. This load distribution is depicted in Figure 2.5. Committee W is the only winning committee; for example, committee $W'=\{a,b,c,d\}$ (the winning committee of seq-Phragmén) has $\bar{\ell }_{W'}^{\min }=(\nicefrac 3 7, \nicefrac 3 7, \nicefrac 3 7, \nicefrac 3 7, \nicefrac 3 7, \nicefrac 3 7, \nicefrac 1 2, \nicefrac 1 2, \nicefrac 3 7, 0, 0, 0)$, which is lexicographically larger.

Fig. 2.5

A visualisation of leximax-Phragmén (upper part) applied to the election instance of Example 2.1 (lower part). In the upper part all regions of the same colour (corresponding to the same candidate) have an area of 1, which is the budget spent on this candidate

2.6 Phragmén-Like Rules

We now discuss a very recent addition to the zoo of ABC rules: the Method of Equal Shares [32, 33] (this method had been originally named Rule X). This rule can be viewed as a variant of seq-Phragmén, where the voters are given some budget upfront, rather than receiving it continuously. This rule is polynomial-time computable and even surpasses the proportionality guarantees of seq-Phragmén.

Rule 11

(Method of Equal Shares) The rule proceeds in two phases. The first phase consists of at most k rounds; in each round one candidate is added to the committee. In the second phase the committee is completed in one of several possible ways.

For the first phase, we assume each voter is initially given a budget of $\nicefrac {k}{n}$. Let $x_r(i)$ denote the budget of voter i after round r; thus $x_0(i) = \nicefrac {k}{n}$. As with seq-Phragmén, putting a candidate in the committee incurs a cost of 1. In round $r+1$, we consider the set of candidates that have not yet been placed in the committee and whose supporters can afford to pay for them, i.e., all candidates c for which $\sum _{i \in N(c)}x_r(i) \ge 1$. Let this set be $C_r\subseteq C$. If $C_r$ is empty, then we conclude the first phase and move to phase two. Otherwise, for each candidate $c\in C_r$ we ask what is the minimal budget $\rho (c)$ such that each voter approving c pays at most $\rho (c)$ and all voters who approve c pay 1 in total, i.e., what is the minimal value $\rho (c)$ that satisfies:

$$\begin{aligned} \sum _{i \in N(c)} \min (\rho (c), x_{r}(i)) = 1 \text {.} \end{aligned}$$

(Such a $\rho (c)$ always exists, since otherwise c would not be contained in $C_r$.) We select the candidate c that minimises $\rho (c)$ (using some fixed tiebreaking if necessary), and reduce the budget of voters who approve c accordingly—for each $i \in N$ we set

$$\begin{aligned} x_{r+1}(i) = {\left\{ \begin{array}{ll} x_i(r) - \rho (c) &{} \text {if } c\in A(i) \text { and } x_i(r) \ge \rho (c),\\ 0 &{} \text {if } c\in A(i) \text { and } x_i(r) < \rho (c),\\ x_i(r) &{} \text {if }c\notin A(i) \text {,} \end{array}\right. } \end{aligned}$$

i.e., voters who approve c either pay $\rho (c)$ or their remaining budget.

The second phase is only relevant if fewer than k candidates have been put in the committee W so far. If $|W| < k$, we have to add $k - |W|$ additional candidates to W. Many properties of the Method of Equal Shares do not depend on the specific way in which these $k - |W|$ candidates are selected.¹⁰ A concrete and recommendable way to fill the committee is to use seq-Phragmén but with initial budgets defined in the following fashion: When using the continuous formulation, we set the starting budget of each voter to their budget after the first phase of the Method of Equal Shares; this starting budget increases as usual as time advances. Alternatively, we can use the discrete formulation of seq-Phragmén: if the first phase ends with round $r'$, the starting loads are $y_0(i)= - x_{r'}(i)$. Then seq-Phragmén proceeds as usual until the desired committee size is reached.

The name of the rule corresponds to the two elements of its definition. First, each voter is initially given an equal share of the budget that she can spend for “buying” candidates. when a candidate is selected, its cost is split as equally as possible among the voters who approve the candidate (each voter covers an equal share of the cost of the candidate).

Example 2.11

Consider once again our running example. Each voter is initially given a budget of $\nicefrac {1}{3}$. In the first round candidate a is selected and each of the first 8 voters pays $\nicefrac {1}{8}$ for this. In the second round, $C_2=\emptyset$ since no candidate has sufficiently endowed supporters. For example, the budget of voters who approve b is in total

$$\begin{aligned} 3\cdot (\nicefrac {1}{3} - \nicefrac {1}{8}) + \nicefrac {1}{3} < 1 \end{aligned}$$

and thus insufficient to pay for b. This ends the first phase of the rule.

In the second phase, the voters start receiving additional budget. Voters 1 to 8 start with a budget of $\nicefrac 1 3 - \nicefrac 1 8$; voters 9 to 12 start with a budget of $\nicefrac 1 3$. At time $t_2 = \nicefrac {1}{96}$, voters 1 to 8 have a budget of $\nicefrac {1}{3} - \nicefrac {1}{8} + t_2$ each and voters 9 to 12 have a budget of $\nicefrac {1}{8} + t_2$ each. Hence the voters who approve b (1, 2, 3, 9) have enough money to pay for b:

$$\begin{aligned} 3\cdot (\nicefrac {1}{3} - \nicefrac {1}{8} + t_2) + (\nicefrac {1}{3} + t_2) = 1 \text {.} \end{aligned}$$

The same is true for the voters who approve c. Let us assume that we resolve the tie in favour of b: b is selected and the voters 1, 2, 3 and 9 are left without budget. Next, at time $t_3 = \nicefrac {37}{384}$ candidate c is selected (voters 4–6 contribute $\nicefrac {1}{3}-\nicefrac {1}{8} + t_3$ and voter 9 contributes $t_3-t_2$, with the required total of 1). Finally, at time $t_4 = \nicefrac {7}{24}$ we select d ($2\cdot (\nicefrac {1}{3}-\nicefrac {1}{8}+t_4)=1$). Committee $W=\{a,b,c,d\}$ is the only winning committee. In this example, the Method of Equal Shares returns the same committee as seq-Phragmén.

Since in Example 2.11 only one candidate is selected in the first phase of the Method of Equal Shares, we provide one additional example which better illustrates the first phase of this rule and also shows that seq-Phragmén and the Method of Equal Shares may produce different committees.

Example 2.12

Consider the following approval profile given by

$$\begin{aligned} \begin{array}{lllllll} &{}A(1) = A(2) = A(3) = \{ c,d \} &{} &{} A(4) = A(5) = \{ a,b\} \\ &{} A(6)= A(7) = \{ a,c \} &{} &{} A(8) = \{b,d\}\text {.} \end{array} \end{aligned}$$

The goal is to select a committee of size $k = 3$. Thus, voters start with a budget of $\nicefrac {3}{8}$.

In this example, candidate c is selected in the first round with each approving voter (1, 2, 3, 6, 7) paying $\nicefrac 1 5$. Next, candidate a is selected. Voters 4 and 5 contribute $\nicefrac {13}{40}$, voters 6 and 7 contribute their remaining budget ($\nicefrac {7}{40}$). None of the remaining candidates achieves a total budget of 1 and thus the second phase starts. The starting budgets for seq-Phragmén are $(\nicefrac {7}{40}, \nicefrac {7}{40}, \nicefrac {7}{40}, \nicefrac {1}{20}, \nicefrac {1}{20}, 0, 0, \nicefrac {3}{8})$. At time $t=\nicefrac {1}{40}$ candidate d is selected: voters 1 to 3 can contribute $\nicefrac {7}{40}+t = \nicefrac 1 5$ each and voter 8 can contribute the remaining $\nicefrac {3}{8} + t = \nicefrac 2 5$. Hence, the Method of Equal Shares selects the committee $\{a,c,d\}$. The voters’ payments in the two phases are illustrated in Fig. 2.6.

In contrast, seq-Phragmén picks $\{b, c, d\}$. These candidates are selected in order c, b, d at time $t_1=\nicefrac {1}{5}$, $t_2=\nicefrac {1}{3}$, and $t_3=\nicefrac {29}{60}$, respectively.

Fig. 2.6

A visualisation of the Method of Equal Shares applied to the election instance of Example 2.12 (lower part). In the two upper figures, all regions of the same colour (corresponding to the same candidate) have an area of 1, which is the budget spent on this candidate

Let us discuss three further rules that are related to Phragmén’s rules. The first is the Expanding Approvals Rule [2]. This rule is defined for weak-order preferences and has favourable axiomatic properties in this setting. It is less convincing for approval preferences¹¹ and thus we do not consider it further. The second rule is the maximin support method [40], which is similar to seq-Phragmén. It is an iterative rule based on a form of load balancing, but in contrast to seq-Phragmén all loads can be redistributed each round. A first analysis showed that the maximin support method and seq-Phragmén share many axiomatic properties [40], and a recent manuscript by Cevallos and Stewart [11] shows that the maximin support method provides a constant factor approximation of leximax-Phragmén—in contrast to seq-Phragmén. In the light of the latter paper, one may view the maximin support method as a polynomial-time approximation of leximax-Phragmén (in the same sense as seq-PAV approximates PAV), whereas seq-Phragmén can rather be viewed as a largely independent rule. We focus in this book on seq-Phragmén as it is better studied and conceptually simpler. Still, the maximin support method is an interesting ABC rule that should be analysed in more depth.

Finally, Phragmén also introduced a method now referred to as either Phragmén’s first method, Eneström’s method, or method of Eneström–Phragmén¹² [9, 16, 22]. This rule can be viewed as an analogue of Single Transferable Vote (STV) with approval ballots.

Rule 12

(Eneström–Phragmén) This method is based on a quota q, which is typically chosen to be either the Hare quota $q=\frac{n}{k}$ or the Droop quota $q = \frac{n}{k+1}$. Candidates are selected in a sequential fashion. All voters start with a weight of 1. In each round, we compute for each unselected candidate the total weight of approving voters, i.e., the score of an unselected candidate c is the sum of weights of all voters approving c. The candidate with the maximum score is added to the committee (using a tie-breaking if necessary); let this candidate be $c'$ and its score s. Now, the weights are adapted: If $s>q$, then the weights of all voters in $N(c')$ are multiplied by $\frac{s-q}{s}$. Thus, the total weight of voters in $N(c')$ is reduced by q. If $s\le q$, the weights of voters in $N(c')$ are set to 0. This step is repeated until k candidates are selected.

As Eneström–Phragmén is not as well studied as Phragmén’s rules, we do not discuss it further, but we note that further analysis could prove this rule to be of independent interest.¹³

2.7 Non-Standard ABC Rules

As mentioned at the beginning of this chapter, most ABC rules coincide with (single-winner) Approval Voting for $k=1$. If we understand an approval ballot as indicating those alternatives that a voter likes, then for $k=1$ it is indeed very natural to select the most-approved alternative. Thus, we refer to rules that differ from Approval Voting for $k=1$ as non-standard ABC rules. In addition to rev-seq-PAV, which we already showed to be non-standard, we present two further non-standard rules. The first one, Minimax Approval Voting (MAV) introduced by Brams et al. [7], interprets approval ballots as the voter’s exact description of the desired outcome. If a voter approves a set X, then she indicates that all these alternatives should be chosen; any sub- or superset is less desirable. In addition, MAV is an egalitarian rule in the sense that it only pays attention to the least-satisfied voter.

To measure the distance between an approval set and a committee, we rely on the Hamming distance:

Definition 2.2

Given two sets X, Y, we define the Hamming distance between X and Y as the size of their symmetric difference: $d_{\textrm{ham}}(X, Y) = |X \setminus Y| + |Y \setminus X|$.

Rule 13

(Minimax Approval Voting, MAV) MAV selects committees W that minimise the largest Hamming distance among all voters, i.e., MAV minimises $\max _{i \in N}d_{\textrm{ham}}(A(i), W)$.

Example 2.13

To see that MAV does not correspond to Approval Voting for $k=1$, consider the following approval profile:

$$\begin{aligned}&99 \times \{ a\}{} & {} \qquad \qquad 1 \times \{ b,c\}\text {.} \end{aligned}$$

The Hamming distance $d_{\textrm{ham}}$ between the committee $W_1=\{a\}$ and the approval set $\{b,c\}$ is 3. In contrast, for the committee $W_2=\{b\}$ (or $\{c\}$) we have $d_{\textrm{ham}}(\{b,c\}, W_2)=1$ and $d_{\textrm{ham}}(\{a\}, W_2)=2$. Thus, MAV selects either b or c, even though these alternatives are approved by only a single voter.

Remark 1

It is interesting to note that if we replace the $\max$ operator in the definition of MAV by a sum, we obtain the Multi-Winner Approval Voting rule (Rule 1).

Remark 2

MAV, as defined, has a major shortcoming. Consider the following slight modification of Example 2.13:

$$\begin{aligned}&99 \times \{ a\}{} & {} \qquad \qquad 1 \times \{a, b,c\}\text {.} \end{aligned}$$

For all size-1 committees, the Hamming distance to $\{a, b,c\}$ is 2. Hence, all three committees are equally preferable according to MAV—even though candidate a is approved by every voter (and b and c by only one voter). We see that MAV might disregard a unanimous choice. This problem can be remedied by also considering the second-least satisfied voter in case of ties, and the third-least in case there is still a tie, and so on until a difference between the committees is found. More formally, for each committee W, we compute $d_{\textrm{ham}}(A(1), W), d_{\textrm{ham}}(A(2), W), \dots$ and sort this tuple of length |N| in decreasing order; we denote this tuple of distances $D_W$. Instead of considering only the first entry in these tuples, we could lexicographically sort them. That is, a committee $W_1$ is preferred to a $W_2$ if there exists an index $i\le n$ such that $D_{W_1}(i)<D_{W_2}(i)$ and $D_{W_1}(j)=D_{W_2}(j)$ for all $1\le j< i$. In our example, we have $D_{\{a\}}=(2, 0, 0, \dots )$ and $D_{\{b\}}=D_{\{c\}}=(2, 2, 2, \dots )$; with this modification $\{a\}$ is the only winning committee. To the best of our knowledge this modification of MAV has not been studied in the context of voting. However, it is equivalent to the $\textsc {Gmax}$ belief merging operator for the Hamming distance [24].

The second non-standard rule is Satisfaction Approval Voting¹⁴ (SAV). SAV is a variation of AV where each voter has one point and distributes it evenly among all approved candidates. As a consequence, voters who approve more candidates contribute a lesser score to the individual approved candidates.

Rule 14

(Satisfaction Approval Voting, SAV) The SAV-score of a committee W is defined as

$$\begin{aligned} {{\textrm{score}_{{{\textrm{SAV}}}}}}(A,W)=\sum _{i \in N}\frac{|W \cap A(i)|}{|A(i)|}. \end{aligned}$$

SAV returns all committees with a maximum SAV-score.

Note that SAV is not a Thiele method since the total number of candidates that a voter approves influences the SAV-score.

Example 2.14

To see that SAV does not correspond to Approval Voting for $k=1$, consider

$$\begin{aligned}&1 \times \{ a\}{} & {} \qquad \qquad 3 \times \{ b,c,d,e\}\text {.} \end{aligned}$$

The SAV-score of a is 1 and for b, c, d, and e it is $\nicefrac 3 4$. Thus, SAV selects $\{a\}$ even though it is approved by only one voter.