Basic Properties of ABC Rules

Abstract

In this chapter, we consider basic axiomatic properties of ABC rules. These properties describe the behaviour of such rules and offer insights into the nature of specific ABC rules. Important axiomatic properties include Pareto efficiency, committee monotonicity, candidate monotonicity, consistency as well as axioms pertaining to strategic voting.

In the previous chapter we have seen a wide array of ABC rules. Considering how much they differ in their definitions, it can be expected that they differ also in the properties they exhibit. In this chapter we consider basic properties of ABC rules. These properties describe the behaviour of such rules and offer insights into the nature of specific ABC rules. Table 3.1 offers an overview of most properties discussed in this chapter. This table also includes a rough dichotomy of the rules concerning their computational complexity. Rules that are in P can be computed efficiently, whereas rules that are ${{\textrm{NP}}}$-hard are computationally more demanding; we discuss this dichotomy and further complexity results in Sect. 5.1.

Table 3.1 Basic properties of ABC rules

3.1 Anonymity, Neutrality, and Resoluteness

Anonymity and neutrality are two of the most basic properties in the social choice literature [1, 20, 23]. Anonymity states that the identity of voters should not influence the outcome: it should be irrelevant whether voter i approves A(i) and voter j approves A(j) or vice versa. Formally, an ABC rule $\mathcal {R}$ satisfies anonymity if for all election instances (A, k) with voter set N and bijections $\pi :N\rightarrow N$ it holds that $\mathcal {R}(A, k)=\mathcal {R}(A\circ \pi , k)$. All but one rule introduced in Chap. 2 satisfy anonymity; the exception is Greedy Monroe which uses a fixed tiebreaking order on voters.¹ A typical example of a voting rule that fails anonymity is any dictatorial rule (a rule considering only the preferences of a single distinguished voter, e.g., of voter 1).

Neutrality is the counterpart to anonymity but applies to candidates: it states that all candidates should be treated equally. Formally, an ABC rule $\mathcal {R}$ satisfies neutrality if for all election instances (A, k) with candidate set C and bijections $\pi :C\rightarrow C$ it holds that $\mathcal {R}(A, k)=\mathcal {R}(\pi ^*\circ A, k)$, where $\pi ^*$ is the natural extension of $\pi$ to a bijection from $\mathcal {P}(C)$ to $\mathcal {P}(C)$ defined by $\pi ^*(X)=\{\pi (c) : c\in X\}$ for each $X \subseteq C$. The rules that fail neutrality are usually those that require some form of tiebreaking.

The third and equally fundamental property we discuss here is resoluteness. Recall that an ABC rule is resolute if it always returns exactly one winning committee. An ABC rule can either be resolute or neutral, but not both. To see this, consider an approval profile where all voters approve candidates $\{a,b\}$ and $k=1$: either a rule returns two winning committees or decides in favour of one of the two candidates. Clearly, any rule can be made resolute by imposing a tiebreaking between winning committees. Conversely, if a resolute rule is defined by a tiebreaking order over candidates (this includes all rules in Chap. 2 that fail neutrality), it can be made neutral by returning all committees that win according to some tiebreaking order. In this way, one can trade neutrality against resoluteness.

Finally, we mention that an in-depth treatment of the interplay between anonymity, neutrality, and resoluteness—albeit in the setting of single-winner elections—can be found in the work of Ozkes and Sanver [25] and Campbell and Kelly [5].

3.2 Pareto Efficiency and Condorcet Committees

Pareto efficiency² is a very general concept to compare two outcomes given the preferences of individuals: outcome Y dominates outcome X if (1) every individual weakly prefers outcome Y to X (i.e., everyone likes Y at least as much as X), and (2) there is at least one individual that strictly prefers Y to X. Pareto efficiency, broadly speaking, means that dominated outcomes are avoided. This concept can be directly translated to our setting by defining when a voter prefers committee $W_1$ to $W_2$. This requires a so-called set extension, i.e., a way how to extend preferences over individual items to sets of items; we refer the reader to the survey of Barberá et al. [4] for a comprehensive overview. Here, we use the Pareto efficiency definition by Lackner and Skowron [17] and assume that $W_1$ is preferred to $W_2$ if $W_1$ contains more approved candidates.

Definition 3.1

A committee $W_1$ dominates a committee $W_2$ if

• 1. every voter has at least as many approved candidates in $W_1$ as in $W_2$ (for $i\in N$ it holds that $|A(i)\cap W_1|\ge |A(i)\cap W_2|$), and

• 2. there is one voter with strictly more approved candidates (there exists $j\in N$ with $|A(j)\cap W_1|> |A(j)\cap W_2|$).

A committee that is not dominated by any other committee (of the same size) is called Pareto optimal.

An ABC rule $\mathcal {R}$ satisfies strong Pareto efficiency if $\mathcal {R}$ never outputs dominated committees. An ABC rule $\mathcal {R}$ satisfies weak Pareto efficiency if for all election instances (A, k) it holds that if $W_2\in \mathcal {R}(A,k)$ and $W_1$ dominates $W_2$, then $W_1\in \mathcal {R}(A,k)$.

Table 3.1 summaries which rules satisfy Pareto efficiency.³ It may be surprising that rather few ABC rules satisfy this kind of Pareto efficiency. Indeed, among the rules introduced in Chap. 2 only Thiele rules, SAV, and MAV satisfy weak Pareto efficiency [17], and among those, e.g., AV, PAV, and SAV satisfy strong Pareto efficiency (but not CC and MAV, for details see Proposition A.1). (Although, we recall that these results rely of course on our chosen set extension.)

To see an example how a rule may fail Pareto efficiency, it is instructive to consider Monroe’s rule:

Example 3.1

([17, Example 3]) Consider the approval profile

$$\begin{aligned}&2 \times \{ a \} \qquad 1 \times \{ a,c\} \qquad 1 \times \{ a,d\} \qquad 10 \times \{ b,c \} \qquad 10 \times \{ b,d \}\text {.} \end{aligned}$$

For $k=2$, Monroe selects $\{c, d\}$ as the (only) winning committee with a Monroe-score of 22. Committee $\{c, d\}$ is however dominated by $\{a,b\}$: every voter approves a candidate in $\{a,b\}$ but only 22 voters approve one in $\{c, d\}$. Thus, every voter is either equally satisfied or better off with committee $\{a, b\}$. This example shows that Pareto efficiency clashes with Monroe’s goal to assign representatives to groups of similar size.

One may wonder whether it is sensible to improve an ABC rule $\mathcal {R}$ that is not Pareto efficient in the following way: given an election instance E, if $W\in \mathcal {R}(E)$ is dominated by another committee, then instead output all Pareto optimal committees that dominate W. There are two main objections against this idea: First, this modification may destroy other axiomatic properties (e.g., Pareto efficiency and perfect representation, which is discussed in Sect. 4.3, are incompatible). Second, finding Pareto improvements is a computationally hard task:

Theorem 3.1

(Aziz and Monnot [3, Theorem 2]) Given an election instance (A, k) and committee W, it is ${{\textrm{coNP}}}$-complete to determine whether W is Pareto optimal.

As a consequence of Theorem 3.1, we cannot expect to obtain polynomial-time computable, Pareto efficient ABC rules by modifying existing rules as described above. Note, however, that polynomial-time computable, Pareto efficient ABC rules exist, e.g., AV and SAV. Thus, finding a Pareto optimal committee is possible in polynomial-time.

A related property to Pareto efficiency has been proposed by Darmann [6]: a committee W is a Condorcet committee if for every other committee $W'$, for a majority of voters $V\subseteq N$ ($|V|>\nicefrac {|N|}{2}$) it holds that $|A(i) \cap W| > |A(i) \cap W'|$ for all $i\in V$. Similarly to Theorem 3.1, deciding whether a given committee W is a Condorcet committee is ${{\textrm{coNP}}}$-complete. However, in contrast to Pareto optimality, it is also ${{\textrm{coNP}}}$-complete to decide whether a Condorcet committee exists [6]. To the best of our knowledge, it has not been analysed which ABC rules output a Condorcet committee if it exists.

3.3 Committee Monotonicity

Committee monotonicity (also referred to as house monotonicity or committee enlargement monotonicity) is a property that is highly desirable in some settings: if the committee size k is increased to $k+1$, then a winning committee of size k should be a subset of a winning committee of size $k+1$. Since this property is particularly useful for resolute rules, we define it exclusively for resolute rules. Appropriate definitions for irresolute rules can be found, e.g., in papers of Elkind et al. [9] and of Kilgour and Marshall [14] (called upward- and downward-accretive in the latter work).

Definition 3.2

A resolute ABC rule $\mathcal {R}$ is committee monotone if for all election instances (A, k) it holds that $W \subseteq W^{\prime }$, where W is the single winning committee in $\mathcal {R}(A,k)$, and $W'$ is the single winning committee in $\mathcal {R}(A,k+1)$.

To see why committee monotonicity can be an essential requirement in some applications, consider the following situation. A group can jointly acquire k items and uses an ABC rule to fairly select those. Once these k items are purchased, it turns out that one additional item can be afforded. If the used ABC rule is committee monotone, it is clear which item to acquire next. However, if the rule is not committee monotone, then the selection for k + 1 items might contain several items that were not contained in the selection of k items, a useless recommendation.

Another example is a hiring process where it is not determined up-front how many candidates are to be hired. Here it is useful that a committee monotone rule actually produces a ranking of candidates: which one should be hired if only one position is available, which one if a second position is to be filled, etc. This connection between committee monotone ABC rules and rankings has been explored in-depth by Skowron et al. [32].

However, committee monotonicity also reduces the flexibility of voting rules and thus comes at a price. For example, we will see in Chap. 4 that committee monotone rules are typically less proportional (although a formal proof for this statement is missing). Thus, if the setting does not dictate committee monotonicity, it may be advantageous to set this axiom aside. A more elaborate discussion of this topic can be found in the paper of Elkind et al. [9].

Table 3.1 shows which of the considered rules are committee monotone, assuming that these rules are made resolute by fixing a tiebreaking order among candidates. AV, seq-PAV, seq-CC, rev-seq-PAV, seq-Phragmén, and SAV are committee monotone; this follows immediately from their corresponding definitions. Counterexamples for the remaining rules can be found in Appendix A, Proposition A.2.

3.4 Candidate and Support Monotonicity

Candidate monotonicity deals with a seemingly obvious requirement: if the support of a candidate increases (i.e., more voters approve this candidate), then this cannot harm the candidate’s inclusion in a winning committee. However, this property is not satisfied by some ABC rules, in particular, if we demand such a monotonicity to also hold for groups of candidates. In addition, there is a difference depending on whether an existing voter changes her ballot, or if a new voter enters the election.

Candidate monotonicity axioms for ABC rules have been considered in a number of papers [2, 13, 16], but the paper by Sánchez-Fernández and Fisteus [27] should be highlighted for the most in-depth analysis.⁴

Further, we write $A_{+X}$ to denote the profile A with one additional voter approving X, i.e., $A_{+X}=(A(1),\dots ,A(n),X)$, and $A_{i+X}$ to denote the profile A where voter i additionally approves the candidates from X.

Definition 3.3

(Sánchez-Fernández and Fisteus [27]) An ABC rule $\mathcal {R}$ satisfies support monotonicity without additional voters if for every election instance (A, k), $i\in N$, and candidate set $X\subseteq C$ it holds that

• 1. if $X\subseteq W$ for all $W\in \mathcal {R}(A,k)$, then $X\subseteq W'$ for all $W'\in \mathcal {R}(A_{i+X},k)$, and

• 2. if $X\subseteq W$ for some $W\in \mathcal {R}(A,k)$, then $X\subseteq W'$ for some $W'\in \mathcal {R}(A_{i+X},k)$.

An ABC rule $\mathcal {R}$ satisfies support monotonicity with additional voters if for any election instance (A, k) and candidate set $X\subseteq C$ the properties above hold for $A_{+X}$ instead of $A_{i+X}$.

If an ABC rule satisfies these axioms only for singleton sets ($X=\{c\}$), we speak of candidate monotonicity with/without additional voters.⁵

The analysis of ABC rules with respect to these axioms is mostly due to Janson [13], Sánchez-Fernández and Fisteus [27], and Mora and Oliver [22]. We summarise the results in Table 3.1. There, the symbol means that support monotonicity is satisfied, “ ” means that candidate monotonicity is satisfied but not support monotonicity, and means that the rule fails even candidate monotonicity. Detailed counterexamples related to support monotonicity can be found in Proposition A.3 in the appendix.

If one is interested in ABC rules that are—in a sense—fair to candidates, then candidate monotonicity (both with and without additional voters) is generally a desirable property. Hence, the fact that Monroe, Greedy Monroe, and the Method of Equal Shares fail the axiom can be seen as a serious argument against these rules. Monroe and the Method of Equal Shares, however, have other distinguished advantages (discussed in Chap. 4) that may override this downside. In settings where a fair treatment of candidates is not necessary (e.g., because candidates represent inanimate objects to be chosen), candidate monotonicity should not be a concern.

3.5 Consistency

Consistency is an axiom describing whether a rule behaves consistently with respect to disjoint groups: if the outcome of an election is the same for two disjoint groups, then a voting rule should arrive at this outcome also if these two groups are joined into a single electorate. This axiom is a straightforward adaption of consistency as defined for single-winner rules by Smith [34] and Young [36] and was first discussed in the context of ABC rules by Lackner and Skowron [18]. In the following, for two profiles A and $A'$ we write $A+A'$ to denote the joint profile where A and $A'$ are concatenated.

Definition 3.4

An ABC rule $\mathcal {R}$ satisfies consistency if for every $k\ge 1$ and two profiles $A: N \rightarrow \mathcal {P}(C)$ and $A': N' \rightarrow \mathcal {P}(C)$ with $N \cap N' = \emptyset$, if $\mathcal {R}(A,k) \cap \mathcal {R}(A',k) \ne \emptyset$ then $\mathcal {R}(A + A',k) = \mathcal {R}(A,k) \cap \mathcal {R}(A',k)$.

Monroe’s rule, for example, does not satisfy consistency:

Example 3.2

Let profile A be

$$\begin{aligned}&A(1):\{ a, y\} \qquad A(2):\{ a, y\} \qquad A(3):\{ b, y\} \qquad A(4):\{ b, y\} \end{aligned}$$

and profile $A'$ be

$$\begin{aligned} \begin{array}{llllll} &{}A(5):\{ y\} &{}&{}\qquad A(6):\{ a\} &{}&{}\qquad A(7) = A(8) = A(9) = A(10):\{ a,x\} \\ {} &{} A(11):\{ y\} &{}&{} \qquad A(12):\{ b,y\} &{}&{} \qquad A(13)= A(14) = A(15) = A(16):\{ b, x\}\text {.} \end{array} \end{aligned}$$

For $k=2$, Monroe returns for profile A the winning committees $\{a,b\}$, $\{a,y\}$, and $\{b,y\}$, all of which having a Monroe-score of 4. For profile $A'$, Monroe returns the winning committee $\{a,b\}$, with a Monroe-score of 10; the corresponding Monroe assignment groups voters 5–10 and 11–16. Now, let us consider the profile $A+A'$. Consistency would demand that $\{a,b\}$ is the unique winning committee, as it is the only committee winning in both A and $A'$. Committee $\{a,b\}$ has a Monroe-score of 14 in $A+A'$. This score, however, is not optimal: $\{x,y\}$ has a Monroe-score of 15; the corresponding Monroe assignment groups voters $\{1,\dots ,6,11,12\}$ and $\{7,\dots ,10,13,\dots ,16\}$. Thus, $\{a,b\}$ is not winning and consistency is violated.

Broadly speaking, the only rules satisfying consistency are so-called ABC scoring rules [18]. These are defined similarly to Thiele methods but are more general, as the satisfaction of a voter may depend on the number of candidates approved by this voter:

Definition 3.5

A scoring function is a function $f :\mathbb N\times \mathbb N\rightarrow \mathbb R$ satisfying $f(x,y)\ge f(x',y)$ for $x\ge x'$. Given such a scoring function, we define the score of W in A as

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

The ABC scoring rule defined by a scoring function f returns all committees with maximum score.

By definition, each Thiele method is an ABC scoring rule, whereas SAV is an example of an ABC scoring rule that is not a Thiele method. Further, it follows immediately from the definition of welfarist rules (Definition 2.1) that an ABC scoring rule is welfarist if and only if it is a Thiele method.

Lackner and Skowron [18] axiomatically characterised the class of ABC scoring rules. This characterisation is in a slightly different model than the one we use in this book: the characterisation applies to ABC ranking rules instead of ABC rules (as defined in Sect. 2.1). ABC ranking rules output a weak order over committees (a ranking with ties over committees) instead of just distinguishing between winning and losing committees (as we assume here). However, note that every ABC ranking rule defines an ABC rule (top-ranked committees are winning).

The following characterisation uses two axioms we have not mentioned so far: weak efficiency and continuity. Both are rather weak axioms. Intuitively, weak efficiency requires that approved candidates are preferable to non-approved candidates, and continuity states that a sufficiently large majority can force a committee to win.

Theorem 3.2

(Lackner and Skowron [18]) An ABC ranking rule is an ABC scoring rule if and only if it satisfies anonymity, neutrality, consistency, weak efficiency, and continuity.

As both weak efficiency and continuity are generally satisfied by sensible voting rules, one can conclude that ABC scoring rules essentially capture the class of consistent ABC ranking rules.⁶ In Sect. 4.1, we will discuss how this result can be used to obtain further axiomatic characterisations, e.g., of PAV.

3.6 Strategic Voting

Strategic voting is a phenomenon central to social choice theory. Sometimes, it is preferable for voters to misrepresent their preferences to change the outcome of an election; this is often referred to as “manipulation”. The famous impossibility theorem by Gibbard [12] and Satterthwaite [28], showing that all “reasonable” single-winner voting rules are susceptible to manipulation, is considered one of the main results in the field. The Gibbard–Satterthwaite theorem applies to elections where voters provide linear rankings over alternatives. As our approval-based setting uses a much more restricted form of preferences, strategyproofness is not completely out of the picture.

We are going to consider two forms of strategyproofness here: Cardinality-strategyproofness and inclusion-strategyproofness (taken from Peters [26], see the work of Gärdenfors [11] and Taylor [35] for more general discussions of strategyproofness in social choice). Cardinality-strategyproofness assumes that voters are concerned only about the number of approved candidates in the committee (and do not distinguish them), whereas inclusion-strategyproofness assumes that voters may have more complex preferences, so a successful manipulation must produce a committee including all approved candidates that were already included in the original committee.

To simplify the discussion, we assume resoluteness, i.e., we assume a (deterministic) tiebreaking order to resolve ties between committees. To clarify what it means that a voter misrepresents their true preferences, we use the concept of i-variants: Given profiles A and $A'$, both with the same set of voters N, we say that $A'$ is an i-variant of A if $A(j)=A'(j)$ for all $j\in N \setminus \{i\}$ with $j\ne i$. Let us first define both notions for resolute ABC rules.

Definition 3.6

A resolute ABC rule $\mathcal {R}$ satisfies cardinality-strategyproofness if for all profiles A and $A'$ where $A'$ is an i-variant of A and for all $k\ge 1$ it holds that $|\mathcal {R}(A,k)\cap A(i)|\ge |\mathcal {R}(A',k)\cap A(i)|$.

Definition 3.7

A resolute ABC rule $\mathcal {R}$ satisfies inclusion-strategyproofness if for all profiles A and $A'$ where $A'$ is an i-variant of A and for all $k\ge 1$ it holds that $\mathcal {R}(A,k)\cap A(i)$ is not a strict subset of $\mathcal {R}(A',k)\cap A(i)$.

Cardinality-strategyproofness is a stronger notion than inclusion-strategyproofness in the sense that all cardinality-strategyproof ABC rules are also inclusion-strategyproof. This follows from the fact that $|\mathcal {R}(A,k)\cap A(i)|\ge |\mathcal {R}(A',k)\cap A(i)|$ (as required in Definition 3.6) implies that $\mathcal {R}(A,k)\cap A(i)$ cannot be a strict subset of $\mathcal {R}(A',k)\cap A(i)$ (as required in Definition 3.7).

Among the rules considered in this book, only AV satisfies any of the mentioned strategyproofness axioms. Specifically, AV satisfies both inclusion-strategyproofness and cardinality-strategyproofness if AV is made resolute by any tiebreaking order on candidates (for details see Proposition A.4). None of the other ABC rules considered in this paper satisfy these axioms, see Table 3.1 for an overview and Proposition A.4 for details. However, even AV is not strategyproof in a stronger sense when voters have underlying, non-dichotomous preferences (as discussed, e.g., by Niemi [24]).

Both cardinality- and inclusion-strategyproofness can be generalised to irresolute ABC rules via set extensions, i.e., by defining how voters compare sets of committees. For example, Lackner and Skowron [16] propose a rather strong extension based on stochastic dominance. The resulting axiom, called SD-strategyproofness, implies cardinality-strategyproofness. AV satisfies SD-strategyproofness and can even be characterised in the class of ABC scoring rules (Definition 3.5) as the only rule satisfying SD-strategyproofness [16]. We note, however, that under more holistic models, e.g., models where voters have underlying non-dichotomous (non-binary) preferences, even AV is no longer strategyproof (see, e.g., [7, 19, 21, 31]). Another natural extension is the Kelly (or cautious) extension: a voter prefers $\mathcal {R}(A',k)$ to $\mathcal {R}(A,k)$ if every committee in $\mathcal {R}(A',k)$ is preferable to every committee in $\mathcal {R}(A,k)$. A more substantial discussion of strategyproofness of irresolute ABC rules can be found in the paper of Kluiving et al. [15].

We further discuss strategyproofness in Sect. 4.6 in the context of proportionality. We will see that even weak forms of proportionality are incompatible with strategyproofness.

Finally, we note that Scheuerman et al. [29, 30] have conducted a behavioural experiment in which they analysed how the voters vote under non-dichotomous preferences, when they are uncertain about other voters’ preferences, and when AV is used to select the winning candidates. These results suggest that the voters may use different (sometimes suboptimal) heuristics when making decisions which candidates they should approve. This shows that strategic voting in a practical setting can differ substantially from the axiomatic analysis we have presented here.