Proportionality and Strategyproofness in Multiwinner Elections

Dominik Peters University of Oxford Oxford, UK [email protected]

ABSTRACT for applications in areas outside the political sphere, such as in Multiwinner voting rules can be used to select a fixed-size com- group recommendation systems. mittee from a larger set of candidates. We consider approval-based To formalise the requirement of proportionality in this party-free committee rules, which allow voters to approve or disapprove can- setting, it is convenient to consider the case where input preferences didates. In this setting, several voting rules such as Proportional are given as approval ballots: each voter reports a set of candidates Approval Voting (PAV) and Phragmén’s rules have been shown to that they find acceptable. Even for this simple setting, there isa produce committees that are proportional, in the sense that they rich variety of rules that exhibit different behaviour [31], and this proportionally represent voters’ preferences; all of these rules are setting gives rise to a rich variety of axioms. strategically manipulable by voters. On the other hand, a generalisa- One natural way of selecting a committee of 푘 candidates when tion of Approval Voting gives a non-proportional but strategyproof given approval ballots is to extend Approval Voting (AV): for each voting rule. We show that there is a fundamental tradeoff between of the 푚 candidates, count how many voters approve them (their these two properties: we prove that no multiwinner voting rule can approval score), and then return the committee consisting of the 푘 simultaneously satisfy a weak form of proportionality (a weakening candidates whose approval score is highest. Notably, this rule can of justified representation) and a weak form of strategyproofness. produce committees that fail to represent large groups of voters. Our impossibility is obtained using a formulation of the problem Consider, for example, an instance where 푘 = 3, and where 5 voters in propositional logic and applying SAT solvers; a human-readable approve candidates 푎, 푏 and 푐, while 4 other voters approve only the version of the computer-generated proof is obtained by extracting candidate 푑. Then AV would select the committee {푎,푏, 푐}, leaving a minimal unsatisfiable set (MUS). We also discuss several related almost half of the electorate unrepresented. Intuitively, the latter axiomatic questions in the domain of committee elections. group of 4 voters, consisting of more than a third of the electorate, should be represented by at least 1 of the 3 committee members. KEYWORDS Aziz et al. [3] introduce an axiom called justified representation (JR) which formalises this intuition that a group of 푛/푘 voters should computational social choice; multiwinner elections; proportionality; not be left without any representation; a stronger version of this strategyproofness; computer-aided methods; SAT solving axiom called proportional justified representation (PJR) has also been introduced and studied [39]. While AV fails these axioms, there Note. [September 2020] This version is a slight revision of the are appealing rules which satisfy them. An example is Proportional paper that appeared in the conference proceedings. The conference Approval Voting (PAV), first proposed by Thiele [46]. The intuition version contained a gap in the proof of Lemma 5.5. To fix this gap, it behind this rule is that voters prefer committees which contain was necessary to introduce an additional axiom, weak efficiency, to more of their approved candidates, but that there are decreasing the impossibility theorems; see Section 3.3. I thank Boas Kluiving, marginal returns; specifically, let us presume that voters gain 1 ‘util’ 1 Adriaan de Vries, Pepijn Vrijbergen for pointing out the error. in committees that contain exactly 1 approved candidates, 1+ 2 utils 1 1 with 2 approved candidates, and in general 1 + 2 + · · · + 푟 utils with 1 INTRODUCTION 푟 approved candidates. PAV returns the committee that maximises The theory of multiwinner elections is concerned with designing utilitarian social welfare with this choice of utility function. PAV and analysing procedures that, given preference information from satisfies a strong form of justified representation [3]. arXiv:2104.08594v1 [cs.GT] 17 Apr 2021 a collection of voters, select a fixed-size committee consisting of 푘 When voters are strategic, PAV has the drawback that it can often members, drawn from a larger set of 푚 candidates. Often, we will be manipulated. Indeed, suppose a voter 푖 approves candidates 푎 be interested in picking a representative committee whose members and 푏. If 푎 is also approved by many other voters, PAV is likely to together cover the diverse interests of the voters. We may also aim include 푎 in its selected committee anyway, but it might not include for this representation to be proportional; for example, if a group 푏 because voter 푖 is already happy enough due to the inclusion of 푎. of 20% of the voters have similar interests, then about 20% of the However, if voter 푖 pretends not to approve 푎, then it may be utility- members of the committee should represent those voters’ interests. maximising for PAV to include both 푎 and 푏, so that 푖 successfully Historically, much work in mathematical social science has tried manipulated the election.∗ Besides PAV, there exist several other to formalise the latter type of proportionality requirement, in the proportional rules, such as rules proposed by Phragmén [14, 30], form of finding solutions to the apportionment problem, which arises but all of them can be manipulated using a similar strategy. in settings where voters express preferences over parties which are comprised of many candidates [5]. More recently, theorists have ∗For a specific example, consider 푃 = (푎푏푐,푎푏푐,푎푏푐,푎푏푑,푎푏푑) for which 푎푏푐 is the focussed on cases where there are no parties, and preferences are unique PAV-committee for 푘 = 3. If the last voter instead reports to approve 푑 only, expressed directly over the candidates [23]. The latter setting allows then the unique PAV-committee is 푎푏푑. , Dominik Peters

That voting rules are manipulable is very familiar to voting 퐴푖 of 퐶, so that ∅ ≠ 퐴푖 ⊊ 퐶; let B denote the set of all ballots. For theorists; indeed the Gibbard–Satterthwaite theorem shows that brevity, when writing ballots, we often omit braces and commas, so for single-winner voting rules and strict preferences, every non- that the ballot {푎,푏} is written 푎푏. An (approval) profile is a function trivial voting rule is manipulable. However, in the approval-based 푃 : 푁 → B assigning every voter an approval ballot. For brevity, multiwinner election setting, we have the tantalising example of we write a profile 푃 as an 푛-tuple, so that 푃 = (푃 (1), . . . , 푃 (푛)). For Approval Voting (AV): this rule is strategyproof in the sense that example, in the profile (푎푏, 푎푏푐,푑), voter 1 approves candidates 푎 voters cannot induce AV to return a committee including more and 푏, voter 2 approves 푎, 푏, and 푐, and voter 3 approves 푑 only. approved candidates by misrepresenting their approval set. This Let 푘 be a fixed integer with 1 ⩽ 푘 ⩽ 푚.A committee is a subset raises the natural question of whether there exist committee rules of 퐶 of cardinality 푘. We write C푘 for the set of committees, and that combine the benefits of AV and PAV: are there rules that are again for brevity, the committee {푎,푏} is written as 푎푏. An (approval- 푁 simultaneously proportional and strategyproof? based) committee rule is a function 푓 : B → C푘 , assigning to The contribution of this paper is to show that these two demands each approval profile a unique winning committee. Note that this are incompatible. No approval-based multiwinner rule satisfies both definition assumes that 푓 is resolute, so that for every possible requirements. This impossibility holds even for very weak versions profile, it returns exactly one committee. In our proofs, wewill Ð of proportionality and of strategyproofness. The version of propor- implicitly restrict the domain of 푓 to profiles 푃 with | 푖 ∈푁 푃 (푖)| ⩾ tionality we use is much weaker than JR. It requires that if there is 푘, so that it is possible to fill the committee with candidates who a group of at least 푛/푘 voters who all approve a certain candidate 푐, are each approved by at least one voter. Since we are aiming for and none of them approve any other candidate, and no other voters a negative result, this domain restriction only makes the result approve 푐, then 푐 should be part of the committee. Strategyproof- stronger. ness requires that a voter cannot manipulate the committee rule Let us define two specific committee rules which will be useful by dropping candidates from their approval ballot; a manipulation examples throughout. would be deemed successful if the voter ends up with a committee Approval Voting (AV) is the rule that selects the 푘 candidates that contains additional approved candidates. In particular, our no- with highest approval score, that is, the 푘 candidates 푐 for which tion of strategyproofness only requires that the committee rule be |{푖 ∈ 푁 : 푐 ∈ 푃 (푖)}| is highest. Ties are broken lexicographically. robust to dropping candidates; we do not require robustness against Proportional Approval Voting (PAV) is the rule that returns the arbitrary manipulations that both add and remove candidates. Ad- set 푊 ⊆ 퐶 with |푊 | = 푘 which maximises ditionally, we impose a mild efficiency axiom requiring that the ∑︁  1 1  rule not elect candidates who are approved by none of the voters. 1 + + · · · + . 2 |푃 (푖) ∩ 푊 | The impossibility theorem is obtained using computer-aided 푖 ∈푁 techniques that have recently found success in many areas of social choice theory [27]. We encode the problem of finding a committee In case of ties, PAV returns the lexicographically first optimum. rule satisfying our axioms into propositional logic, and then use Other important examples that we occasionally mention are a SAT solver to check whether the formula is satisfiable. If the Monroe’s rule, Chamberlin–Courant, Phragmén’s rules, and the formula is unsatisfiable, this implies an impossibility, for afixed sequential version of PAV. For definitions of these rules, we refer to number of voters, a fixed number of candidates, and a fixed 푘. the book chapter by Faliszewski et al. [23]; they are not essential We can then manually prove induction steps showing that the for following our technical results. impossibility continues to hold for larger parameter values. Such techniques were first used by Tang and Lin [44] to give alternative 3 OUR AXIOMS proofs of Arrow’s and other classic impossibilities, and by Geist In this section, we discuss the axioms that will be used in our and Endriss [26] to find impossibilities for set extensions. Brandt impossibility result. These axioms have been chosen to be as weak minimal unsatisfiable and Geist [12] developed a method based on as possible while still yielding an impossibility. This can make them sets human-readable that allows extracting a proof of the base case sound technical and unnatural in isolation. To better motivate them, impossibility. Thus, even though parts of the proofs in this paper we discuss stronger versions that may have more natural appeal. are computer-generated, they are entirely human-checkable. We begin our paper by describing several possible versions of strategyproofness and proportionality axioms. We then explain 3.1 Strategyproofness the computer-aided method for obtaining impossibility results in A voter can manipulate a voting rule if, by submitting a non-truthful more detail, and present the proof of our main theorem. We end by ballot, the voter can ensure that the voting rule returns an outcome discussing some extensions to this result, and contrast our result to that the voter strictly prefers to the outcome at the truthful pro- a related impossibility theorem due to Duddy [17]. file. It is not obvious how to phrase this definition for committee rules, since we do not assume that voters have preferences over 2 PRELIMINARIES committees; we only have approval ballots over candidates. One way to define manipulability in this context isto extend the Let 퐶 be a fixed finite set of 푚 candidates, and let 푁 = {1, . . . , 푛} be preference information we have to preferences over committees. a fixed finite set of 푛 voters. An approval ballot is a proper† subset This is the approach also typically taken when studying set-valued

†Nothing hinges on the assumption that ballots are proper subsets. Since we are mainly (irresolute) voting rules [25, 45] or probabilistic voting rules [11]. interested in impossibilities, this ‘domain restriction’ slightly strengthens the results. In our setting, there are several ways to extend approval ballots to Proportionality and Strategyproofness in Multiwinner Elections , preferences over committees, and hence several notions of strate- EJR gyproofness. Our impossibility result uses the weakest notion. For the formal definitions, let us introduce the notion of푖-variants. d’Hondt ext. PJR For a voter 푖 ∈ 푁 , we say that a profile 푃 ′ is an 푖-variant of profile 푃 if 푃 and 푃 ′ differ only in the ballot of voter 푖, that is, if 푃 (푗) = 푃 ′(푗) for all 푗 ∈ 푁 \{푖}. Thus, 푃 ′ is obtained after 푖 manipulated in some lower quota ext. JR way, assuming that 푃 was the truthful profile. One obvious way in which one committee can be better than another in a voter’s view is if the former contains a larger number JR on party lists of approved candidates. Suppose at the truthful profile, we elect a committee of size 푘 = 5, of which voter 푖 approves 2 candidates. If 푖 can submit a non-truthful approval ballots which leads to the proportionality election of a committee with 3 candidates who are approved by 푖, then this manipulation would be successful in the cardinality sense. Figure 1: Proportionality axioms and logical implications. Cardinality-Strategyproofness If 푃 ′ is an 푖-variant of 푃, then we do not have |푓 (푃 ′) ∩ 푃 (푖)| > |푓 (푃) ∩ 푃 (푖)|. ballot. Formally, these rules can be manipulated even through re- One can check that AV with lexicographic tie-breaking satisfies porting a proper subset of the truthful ballot. Our final and official cardinality-strategyproofness: it is neither advantageous to increase notion of strategyproofness is a version of subset-strategyproofness the approval score of a non-approved candidate, nor to decrease which only requires the committee rule to resist manipulators who the approval score of an approved candidate. report a subset of the truthful ballot. Alternatively, we can interpret an approval ballot 퐴 ∈ B to Strategyproofness If 푃 ′ is an 푖-variant of 푃 with 푃 ′(푖) ⊂ 푃 (푖), say that the voter likes the candidates in 퐴 (and would like them ′ to in the committee), and that the voter dislikes the candidates then we do not have 푓 (푃 ) ∩ 푃 (푖) ⊋ 푓 (푃) ∩ 푃 (푖). not in 퐴 (and would like them not to be in the committee). The Manipulating by reporting a subset of one’s truthful ballot is voter’s ‘utility’ derived from committee 푊 would be the number sometimes known as Hylland free riding [29, 41]: the manipula- of approved candidates in 푊 plus the number of non-approved tor free-rides on others approving a candidate, and can pretend candidates not in 푊 . Interpreting approval ballots and committees to be worse off than they actually are. This can then induce the as bit strings of length 푚, the voter thus desires the Hamming committee rule to add further candidates from their ballot to the distance between their ballot and the committee to be small. For committee. Aziz et al. [2] study a related notion of ‘excludable two sets 퐴, 퐵, write H(퐴, 퐵) = |퐴 Δ 퐵| = |(퐴 ∪ 퐵)\(퐴 ∩ 퐵)|. strategyproofness’ in the context of probabilistic voting rules. Hamming-Strategyproofness If 푃 ′ is an 푖-variant of 푃, then Interestingly, one can check that PAV cannot be manipulated by we do not have H(푓 (푃 ′), 푃 (푖)) < H(푓 (푃), 푃 (푖)). reporting a superset of one’s ballot; such a manoeuvre never helps. One can check that Hamming-strategyproofness and cardinality- 3.2 Proportionality strategyproofness are equivalent, because for a fixed ballot 푃 (푖), a committee is Hamming-closer to 푃 (푖) than another if and only if We now discuss several axioms formalising the notion that the com- the number of approved candidates is higher in the former. mittee rule 푓 should be proportional, in the sense of proportionally The notions of strategyproofness described so far make sense representing different factions of voters: for example, a ‘cohesive’ if we subscribe to the interpretation of an approval ballot as a di- group of 10% of the voters should be represented by about 10% chotomous preference, with the voter being completely indifferent of the members of the committee. The version of proportionality between all approved candidates (or being unable to distinguish be- used in our impossibility is the last axiom we discuss. All other tween them). In some settings, this is not a reasonable assumption. versions imply the one leading to impossibility; thus, this version For example, suppose 푖 approves {푎,푏, 푐}; still it might be reason- is the weakest notion among the ones discussed here. Figure 1 able for 푖 to prefer a committee containing just 푎 to a committee shows a Hasse diagram of all discussed axioms. Approval Voting containing both 푏 and 푐, maybe because 푖’s underlying preferences (AV) fails all of them, as can be checked for the example profile ( ) are such that 푎 is preferred to 푏 and 푐, even though all three are 푃 = 푎푏푐, 푎푏푐,푑 and 푘 = 3, where AV returns 푎푏푐. approved. However, 푖 should definitely prefer a committee that in- We say that a profile 푃 is a party-list profile if for all voters ∈ ( ) ( ) ( ) ∩ ( ) ∅ cludes a strict superset of approved candidates. For example, a com- 푖, 푗 푁 , either 푃 푖 = 푃 푗 , or 푃 푖 푃 푗 = . For example, ( ) ( ) mittee containing 푎 and푏 should be better than a committee contain- 푎푏,푎푏,푐푑푒,푐푑푒, f is a party-list profile, but 푎푏, 푐, 푐, 푎푏푐 is not. A ing only 푎. This is the intuition behind superset-strategyproofness, party-list profile induces a partition of the set 퐶 of candidates into which is a weaker notion than cardinality-strategyproofness. disjoint parties, so that each voter approves precisely the members ′ of exactly one party. The problem of finding a proportional com- Superset-Strategyproofness If 푃 is an 푖-variant of 푃, then we mittee given a party-list profile has been extensively studied as the do not have 푓 (푃 ′) ∩ 푃 (푖) ⊋ 푓 (푃) ∩ 푃 (푖). problem of apportionment. Functions 푔 : {party-list profiles} → C푘 Interestingly, PAV and other proportional rules are often manip- are known as apportionment methods; thus any committee rule ulable in a particularly simple fashion: a manipulator can obtain a induces an apportionment by restricting its domain to party-list better outcome by dropping popular candidates from their approval profiles15 [ ]. Many proportional apportionment methods have been , Dominik Peters introduced and defended over the last few centuries. Given a com- The proportionality axiom we use in our impossibility combines mittee rule 푓 , one way to formalise the notion that 푓 is proportional features of the JR-style axioms with the apportionment-extension is by requiring that the apportionment method induced by 푓 is pro- axioms. Consider the following axiom. portional. JR on party lists Suppose 푃 is a party-list profile, and some Given a party-list profile 푃, let us write 푛 (퐴) = |{푖 ∈ 푁 : 푃 (푖) = 푃 ballot 퐴 ∈ B appears at least 푛 times in 푃. Then 푓 (푃) ∩퐴 ≠ ∅. 퐴}| for the number of voters approving party 퐴. An apportionment 푘 method 푔 satisfies lower quota if for every party-list profile 푃, each This axiom only requires JR to hold for party-list profiles; thus, it 푘 only requires that we represent large-enough groups of voters who party 퐴 in 푃 gets at least ⌊푛푃 (퐴)· 푛 ⌋ seats, that is, |푔(푃) ∩ 퐴| ⩾ 푘 all report the exact same approval ballot [see also 7]. As an example, ⌊푛푃 (퐴)· ⌋. This notion gives us our first proportionality axiom. 푛 this axiom requires that 푓 (푎푏,푎푏,푐푑,푐푑) ∈ {푎푐,푎푑,푏푐,푏푑}, because Lower quota extension The apportionment method induced 푛 = 4 = the ballots 푎푏 and 푐푑 both appear at least 푘 2 2 times. by 푓 satisfies lower quota. Our official proportionality axiom is still weaker, and onlyre- This axiom is satisfied by PAV, the sequential version of PAV, by quires us to represent singleton parties with large-enough support. Monroe’s rule if 푘 divides 푛, and Phragmén’s rule [15]. Proportionality Suppose 푃 is a party-list profile, and some We can strengthen this axiom by imposing stronger conditions singleton ballot {푐} ∈ B appears at least 푛 times in 푃. Then on the induced apportionment method. For example, the apportion- 푘 푐 ∈ 푓 (푃). ment method induced by PAV and by Phragmén’s rule coincides with the d’Hondt method (aka Jefferson method, see [15] for a defi- This axiom should be almost uncontroversial if we desire our com- nition), so we could use the following axiom. mittee rule to be proportional in any sense. A group of voters who d’Hondt extension The apportionment method induced by 푓 all approve just a single candidate is certainly cohesive (there are is the d’Hondt method. no internal disagreements), it is clear what it means to represent this group (add their approved candidate to the committee), and Aziz et al. [3] introduce a different approach of defining a propor- the group is uniquely identified (because no outside voters approve tionality axiom. Instead of considering only the case of party-list sets that intersect with the group’s approval ballot). all profiles, they impose conditions on profiles. The intuition behind Since our proportionality axiom only refers to the apportionment their axioms is that sufficiently large groups of voters that have method induced by 푓 , our impossibility states that no reasonable similar preferences ‘deserve’ at least a certain number of represen- apportionment method admits an extension to the ‘open list’ setting tatives in the committee. They introduce the following axiom: (where voters are not bound to a party) which is strategyproof. ′ Justified Representation (JR) If 푃 is a profile, and 푁 ⊆ 푁 A type of axiom related to proportionality are diversity require- ′ 푛 Ñ is a group with |푁 | ⩾ and 푖 ∈푁 ′ 푃 (푖) ≠ ∅, then 푓 (푃) ∩ ments. These typically require that as many voters as possible Ð 푘 푖 ∈푁 ′ 푃 (푖) ≠ ∅. should have a representative in the committee, but they do not 푛 insist that groups of voters be proportionally represented [18, 23]. Thus, JR requires that no group of at least 푘 voters for which there is at least one candidate 푐 ∈ 퐶 that they all approve can remain The Chamberlin–Courant rule [16] is an example of a rule select- unrepresented: at least one of the voters in the group must approve ing diverse committees. Lackner and Skowron [33] propose the at least one of the committee members. This axiom is satisfied, for following formulation of this requirement for the approval setting: example, by PAV, Phragmén’s rule, and Chamberlin-Courant [3], Disjoint Diversity Suppose 푃 is a party-list profile with at most but not by the sequential version of PAV unless 푘 ⩽ 5 [3, 39]. 푘 different parties. Then 푓 (푃) contains at least one member One may think that JR is too weak: even if there is a large majority from each party. of voters who all report the same approval set, JR only requires that one of their candidates be a member of the committee. But this group Our main result (Theorem 5.1) also holds when replacing propor- may deserve several representatives. The following strengthened tionality by disjoint diversity, since all profiles in its proof where version of JR is due to [39]. It requires that a large group of voters proportionality is invoked feature at most 푘 different parties. for which there are several candidates that they all approve should be represented by several committee members. 3.3 Efficiency Proportional Justified Representation (PJR) For any profile We will additionally impose a mild technical condition, which can 푃 and each ℓ = 1, . . . , 푘, if 푁 ′ ⊆ 푁 is a group with |푁 ′| ⩾ ℓ · 푛 be seen as an efficiency axiom.The axiom will only be used inone Ñ Ð 푘 and | 푖 ∈푁 ′ 푃 (푖)| ⩾ ℓ, then |푓 (푃) ∩ 푖 ∈푁 ′ 푃 (푖)| ⩾ ℓ. of the induction steps (Lemma 5.5). Ð This axiom is also satisfied by PAV and Phragmén’s rule [14, 39]. Weak Efficiency If 푃 is a profile with | 푖 ∈푁 푃 (푖)| ⩾ 푘, and 푐 Brill et al. [15] show that if a rule satisfies PJR, then it is also a lower is a candidate who is approved by no voters, then 푐 ∉ 푓 (푃). quota extension. A yet stronger version of JR is EJR, introduced by Thus, a rule satisfying weak efficiency should fill the committee Aziz et al. [3]; EJR requires that there is at least one group member with candidates who are approved by some voters, rather than who has at least ℓ approved committee members. electing candidates approved by no one. A similar axiom of the Extended Justified Representation (EJR) For any profile 푃 same name is used by Lackner and Skowron [33]. As we declared in = ′ ⊆ | ′| · 푛 and each ℓ 1, . . . , 푘, if 푁 푁 is a group with 푁 ⩾ ℓ 푘 Section 2, in our proofs we will always restrict attention to profiles 푃 ′ | Ñ ′ 푃 (푖)| ℓ |푓 (푃) ∩ 푃 (푖)| ℓ 푖 ∈ 푁 Ð and 푖 ∈푁 ⩾ , then ⩾ for some . with | 푖 ∈푁 푃 (푖)| ⩾ 푘, so that weak efficiency applies to all relevant This axiom is satisfied by PAV [3], but not by Phragmén’s rule [14]. profiles. Proportionality and Strategyproofness in Multiwinner Elections ,

4 THE COMPUTER-AIDED APPROACH ALGORITHM 1: Encode Problem for SAT Solving To obtain our impossibility result, we have used the computer-aided Input: Set 퐶 of candidates, set 푁 of voters, committee size 푘. technique developed by Tang and Lin [44] and Geist and Endriss Question: Does a proportional and strategyproof committee rule [26]. This approach is based on using a computer search (usually exist? ∈ B푁 in form of a SAT solver) to establish the base case of an impossi- for each profile 푃 do if 푃 is a party-list profile then bility theorem, and then using (manually proved) induction steps allowed[푃 ] ← {퐶 ∈ C : 퐶 provides JR to singleton parties} to extend the theorem to bigger values of 푛 and 푚. Tang and Lin 푘 else [44] used this technique to give proofs of Arrow’s and other classic allowed[푃 ] ← C푘 impossibility theorems in social choice, and Geist and Endriss [26] for each committee 퐶 ∈ allowed[푃 ] do used it to find new impossibilities in the area of set extensions. introduce propositional variable 푥푃,퐶 A paper by Brandt and Geist [12] used this approach to prove an for each profile 푃 ∈ B푁 do Ô impossibility about strategyproof tournament solutions; an impor- add clause 퐶∈allowed[푃 ] 푥푃,퐶 minimal Ó tant technical contribution of their paper was the use of add clauses 퐶≠퐶′∈allowed[푃 ] (¬푥푃,퐶 ∨ ¬푥푃,퐶′ ) unsatisfiable sets to produce human-readable proofs of the base for each voter 푖 ∈ 푁 do case. This technology was also used to prove new impossibilities for each 푖-variant 푃′ of 푃 with 푃′ (푖) ⊆ 푃 (푖) do about the no-show paradox [13], about half-way monotonicity [38], for each 퐶 ∈ allowed[푃 ] and 퐶′ ∈ allowed[푃′] do and about probabilistic voting rules [10]. A recent book chapter by if 퐶′ ∩ 푃 (푖) ⊋ 퐶 ∩ 푃 (푖) then Geist and Peters [27] provides a survey of these results. add clause (¬푥푃,퐶 ∨ ¬푥푃′,퐶′ ) The “base case” of an impossibility theorem proves that no voting pass formula to SAT solver rule exists satisfying a certain collection of axioms for a fixed num- return whether formula is satisfiable ber of voters and alternatives, and (in our case) a fixed committee size 푘. Fixing these numbers, there are only finitely many possible 푚
2푚푛 steps; however, this is not always straightforward to do, and in some rules, and we can in principle iterate through all 푘 possibilities and check whether any satisfies our axioms. However, this search cases, impossibilities do not hold for all larger parameter values space quickly grows out of reach of a naïve search. [e.g., 38]. In many cases, the induction step on 푛 is most-difficult In many cases, we can specify our axiomatic requirements in to establish. We also run into trouble proving this step, and our propositional logic, and use a SAT solver to check for the existence impossibility is only proved for the case where 푛 is a multiple of 푘. of a suitable voting rule. Due to recent dramatic improvements in Another challenge is to find a proof of the obtained impossi- solving times of SAT solvers, this approach often makes this search bility, and preferably one that can easily be checked by a human. feasible, even for moderately large values of 푛 and 푚 [13]. Many SAT solvers can be configured to output a proof trace which How can we encode our problem of finding a proportional and contains all steps used to deduce that the formula is unsatisfiable; strategyproof committee rule into propositional logic? This turns but these proofs can become very large. Recent examples are SAT- out to be straightforward. Our formula will be specified so that generated proofs of a special case of the Erdős Discrepancy Con- every satisfying assignment explicitly encodes a committee rule sat- jecture [32] which takes 13GB, and of a solution to the Boolean isfying the axioms. We generate a list of all (2푚)푛 possible approval Pythagorean Triples Problem [28] which takes 200TB. Clearly, hu- profiles, and for each profile 푃 and each committee푊 , we introduce mans cannot check the correctness of these proofs. We use the method introduced by Brandt and Geist [12] via min- a propositional variable 푥푃,푊 with the intended interpretation that imal unsatisfiable sets (MUS). An MUS of an unsatisfiable proposi- ⇐⇒ ( ) 푥푃,푊 is true 푓 푃 = 푊 . tional formula in conjunctive normal form is a subset of its clauses We then add clauses that ensure that any satisfying assignment which is already unsatisfiable, but minimally so: removing any fur- encodes a function (so that 푓 (푃) takes exactly one value), we add ther clause leaves a satisfiable formula. Thus, every clause inan clauses that ensure that only proportional committees may be re- MUS corresponds to a ‘proof ingredient’ which cannot be skipped. turned, and we iterate through all profiles 푃 and all 푖-variants of MUSes of formulas derived from voting problems like ours are of- it, adding clauses to ensure that no successful manipulations are ten very small, only referring to a few dozen profiles. This can be possible. The details are shown in Algorithm 1; the formulation explained through the ‘local’ nature of the axioms used: propor- we use is slightly more efficient by never introducing the variable tionality constrains the behaviour of the committee rule at a single 푥푃,푊 in case that the committee 푊 is not proportional in profile 푃. profile, and strategyproofness links the behaviour at two profiles. Now, given numbers 푛, 푚, and 푘, Algorithm 1 encodes our prob- MUSes can be found using MUS extractors, which have become lem, passes the resulting propositional formula to a SAT solver [1, 9] reasonably efficient. We used MUSer2 [8] and MARCO [34]. Once and reports whether the formula was satisfiable. If it is satisfiable, one finds a small MUS, it can then be manually inspected tounder- then we know that there exists a propotional and strategyproof stand how the clauses in the MUS fit together. More details of this committee rule for these parameter values, and the SAT solver will process are described in the book chapter by Geist and Peters [27]. return an explicit example of such a rule in form of a look-up table. If the formula is unsatisfiable (like in our case), then we have an 5 THE IMPOSSIBILITY THEOREM impossibility for these parameter values. We are now in a position to state our main result, that there are no A remaining challenge is to extend this impossibility result to proportional and strategyproof committee rules. other parameter values, which is usually done by proving induction , Dominik Peters

Theorem 5.1. Suppose 푘 ⩾ 3, the number 푛 of voters is divisible by Consider 푃3 = (푏, 푎, 푐푑). By proportionality, 푓 (푃3) ∈ {푎푏푐, 푎푏푑}. 푘, and 푚 ⩾ 푘 + 1. Then there exists no approval-based committee rule If we had 푓 (푃3) = 푎푏푐, then voter 2 in 푃2.5 could manipulate towards which satisfies weak efficiency, proportionality and strategyproofness. 푃3. Hence 푓 (푃3) = 푎푏푑. Consider 푃 = (푏, 푎푑, 푐푑). By Lemma 5.2, 푏 ∈ 푓 (푃 ). Thus, The assumption that 푘 ⩾ 3 is critical; we discuss the cases 푘 = 1 3.5 3.5 푓 (푃 ) = 푎푏푑, or else voter 2 can manipulate towards 푃 . and 푘 = 2 separately in Section 5.4. The assumption that 푛 be divisi- 3.5 3 Consider 푃 = (푏, 푎푑, 푐). By proportionality, 푓 (푃 ) ∈ {푎푏푐,푏푐푑}. ble by 푘 also appears to be critical; the SAT solver indicates positive 4 4 If we had 푓 (푃 ) = 푏푐푑, then voter 3 in 푃 could manipulate towards results when 푛 is not a multiple of 푘. However, we do not know 4 3.5 푃 . Hence 푓 (푃 ) = 푎푏푐. short descriptions of these rules, and it is possible (likely?) that im- 4 4 Consider 푃 = (푏, 푎푑, 푎푐). By Lemma 5.2, 푏 ∈ 푓 (푃 ). Thus, possibility holds for large 푛 and 푚. Using stronger proportionality 4.5 4.5 푓 (푃 ) = 푎푏푐, or else voter 3 can manipulate towards 푃 . axioms, the result holds for all sufficiently large 푛; see Section 5.3. 4.5 4 Consider 푃 = (푏,푑, 푎푐). By proportionality, 푓 (푃 ) ∈ {푎푏푑,푏푐푑}. The proof of this impossibility was found with the help of com- 5 5 If we had 푓 (푃 ) = 푎푏푑, then voter 2 in 푃 could manipulate to- puters, but it was significantly simplified manually. One convenient 5 4.5 wards 푃 . Hence 푓 (푃 ) = 푏푐푑. first step is to establish the following simple lemma. It uses strate- 5 5 Consider 푃 = (푏, 푐푑, 푎푐). By Lemma 5.2, 푏 ∈ 푓 (푃 ). Thus, gyproofness to extend the applicability of proportionality to certain 5.5 5.5 푓 (푃 ) = 푏푐푑, or else voter 2 can manipulate towards 푃 . profiles that are not party-list profiles. 5.5 5 Consider 푃6 = (푏, 푐푑, 푎). By proportionality, 푓 (푃6) ∈ {푎푏푐, 푎푏푑}. Lemma 5.2. Let 푚 = 푘 + 1. Let 푓 be strategyproof and proportional. If we had 푓 (푃6) = 푎푏푐, then voter 3 in 푃5.5 could manipulate towards Suppose that 푃 is a profile in which some singleton ballot {푐} appears 푃6. Hence 푓 (푃6) = 푎푏푑. 푛 ∈ ( ) = ( ) ∈ ( ) at least 푘 times, but in which no other voter approves 푐. Then 푐 푓 푃 . Consider 푃6.5 푏, 푐푑, 푎푑 . By Lemma 5.2, 푏 푓 푃6.5 . Thus, 푓 (푃6.5) = 푎푏푑, or else voter 3 can manipulate towards 푃6. Proof. Let 푃 ′ be the profile defined by Consider 푃7 = (푏, 푐, 푎푑). By proportionality, 푓 (푃7) ∈ {푎푏푐,푏푐푑}. ( {푐} if 푃 (푖) = {푐}, If we had 푓 (푃7) = 푏푐푑, then voter 2 in 푃6.5 could manipulate towards 푃 ′(푖) = 퐶 \{푐} otherwise. 푃7. Hence 푓 (푃7) = 푎푏푐. Finally, consider 푃7.5 = (푎푏, 푐, 푎푑). By Lemma 5.2, 푐 ∈ 푓 (푃7.5). Then 푃 ′ is a party-list profile, and by proportionality, 푐 ∈ 푓 (푃 ′). Thus, 푓 (푃7.5) = 푎푏푐, or else voter 1 can manipulate towards 푃7. But Thus, 푓 (푃 ′) ≠ 퐶 \{푐}. Now, step by step, we let each non-{푐} voter then voter 3 can manipulate towards 푃1 = (푎푏, 푐,푑), because by our 푗 in 푃 ′ change back their vote to 푃 (푗). By strategyproofness, at each initial assumption, we have 푓 (푃1) = 푎푐푑. Contradiction. □ step the output committee cannot be 퐶 \{푐}. In particular, at the last step, we have 푓 (푃) ≠ 퐶 \{푐}. Thus, 푐 ∈ 푓 (푃), as required. □ 5.2 Induction steps 5.1 Base case We now extend the base case to larger parameter values, by proving induction steps. The proofs all take the same form: Assuming the The first major step in the proof is to establish the impossibility in existence of a committee rule satisfying the axioms for large param- the case that 푘 = 3, 푛 = 3, and 푚 = 4. The proof of this base case is eter values, we construct a rule for smaller values, and show that the by contradiction, assuming there exists some 푓 satisfying the ax- smaller rule inherits the axiomatic properties of the larger rule. This ioms. We start by considering the profile 푃1 = (푎푏, 푐,푑), and break is done, variously, by introducing dummy voters, by introducing some symmetries. (This is a useful strategy to obtain smaller and dummy alternatives, and by copying voters. better-behaved MUSes.) Using proportionality, symmetry-breaking Our first induction step reduces the number of voters. Theun- allows us to assume that 푓 (푃1) = 푎푐푑. The proof then goes through derlying construction works by copying voters, and using the ‘ho- seven steps, applying the same reasoning each time. In each step, mogeneity’ of the axioms of proportionality and strategyproofness. we use strategyproofness to infer the values of 푓 at certain pro- For the latter axiom, we use the fact that in the case 푚 = 푘 + 1, the files 푃2, . . . , 푃7. Finally, we find that strategyproofness implies that preference extension of approval ballots to committees is complete, 푓 (푃1) ≠ 푎푐푑, which contradicts our initial assumption about 푓 (푃1). in that any two committees are comparable. There is no committee rule that satisfies proportional- Lemma 5.3. Lemma 5.4. Suppose 푘 ⩾ 2 and 푚 = 푘 + 1, and let 푞 ⩾ 1 be an = = = ity and strategyproofness for 푘 3, 푛 3, and 푚 4. integer. If there exists a proportional and strategyproof committee rule Proof. Suppose for a contradiction that such a committee rule 푓 for 푞 · 푘 voters, then there also exists such a rule for 푘 voters. existed. Consider the profile 푃1 = (푎푏, 푐,푑). By proportionality, we Proof. For convenience, we write profiles as lists. Given a profile have 푐 ∈ 푓 (푃1) and 푑 ∈ 푓 (푃1). Thus, we have 푓 (푃1) ∈ {푎푐푑,푏푐푑}. 푃, we write 푞푃 for the profile obtained by concatenating 푞 copies By relabelling the alternatives, we may assume without loss of of 푃. Let 푓푞푘 be the rule for 푞 · 푘 voters. We define the rule 푓푘 for 푘 generality that 푓 (푃1) = 푎푐푑. voters as follows: Consider 푃1.5 = (푎푏, 푎푐,푑). By Lemma 5.2, 푑 ∈ 푓 (푃1.5). Thus, 푘 푓 (푃1.5) = 푎푐푑, or else voter 2 can manipulate towards 푃1. 푓푘 (푃) = 푓푞푘 (푞푃) for all profiles 푃 ∈ B . Consider 푃2 = (푏, 푎푐,푑). By proportionality, 푓 (푃2) ∈ {푎푏푑,푏푐푑}. Proportionality. Suppose 푃 ∈ B푘 is a party-list profile in which at If we had 푓 (푃2) = 푎푏푑, then voter 1 in 푃1.5 could manipulate to- 푛 = 푘 = { } wards 푃2. Hence 푓 (푃2) = 푏푐푑. least 푘 푘 1 voters approve 푐 . Then 푞푃 is a party-list profile · 푛 = 푞푛 = { } Consider 푃2.5 = (푏, 푎푐, 푐푑). By Lemma 5.2, 푏 ∈ 푓 (푃2.5). Thus, in which at least 푞 푘 푘 푞 voters approve 푐 . Since 푓푞푘 is 푓 (푃2.5) = 푏푐푑, or else voter 3 can manipulate towards 푃2. proportional, 푐 ∈ 푓푞푘 (푞푃) = 푓푘 (푃). Proportionality and Strategyproofness in Multiwinner Elections ,

Strategyproofness. Suppose for a contradiction that 푓푘 is not strat- Finally, we can combine all three induction steps, applying them ′ ′ egyproof, so that there is 푃 and an 푖-variant 푃 with 푓푘 (푃 ) ∩푃 (푖) ⊋ in order, and the base case, to get our main result. 푓 (푃) ∩ 푃 (푖). Because 푚 = 푘 + 1, the committees 푓 (푃 ′) and 푓 (푃) 푘 푘 푘 Proof of the Main Theorem. Let 푘 ⩾ 3, let 푛 be divisible by 푘, must differ in exactly 1 candidate. Since the manipulation wassuc- and let 푚 ⩾ 푘 + 1. Suppose for a contradiction that there does exist cessful, 푓 (푃 ′) must be obtained by replacing a non-approved can- 푘 an approval-based committee rule 푓 which satisfies weak efficiency, didate in 푓 (푃) by an approved one, say 푓 (푃 ′) = 푓 (푃) ∪ {푐}\{푑} 푘 푘 푘 proportionality, and strategyproofness for these parameters. with 푐 ∈ 푃 (푖) ∌ 푑. Now consider 푓 (푞푃), and step-by-step let each 푞푘 By Lemma 5.5 applied repeatedly to 푓 , there also exists such a of the 푞 copies of 푃 (푖) in 푞푃 manipulate from 푃 (푖) to 푃 ′(푖) obtaining rule 푓 ′ for 푘+1 alternatives. By Lemma 5.4 applied to 푓 ′, there exists 푞푃 ′ in the last step. Because 푓 is strategyproof, at each step of 푞푘 a proportional and strategyproof rule 푓 ′′ for 푘 voters. By Lemma 5.6 this process 푓 cannot have exchanged a non-approved candidate 푞푘 applied to 푓 ′′, there must exist a proportional and strategyproof by an approved candidate according to 푃 (푖). This contradicts that rule for committee size 3, for 3 voters, and for 4 alternatives. But 푓 (푃 ′) = 푓 (푃) ∪ {푐}\{푑}. □ 푘 푘 this contradicts Proposition 5.3. □ Our second induction step is the simplest: We reduce the number of alternatives using dummy candidates that no voter ever approves. 5.3 Extension to other electorate sizes This is the only place in the proof where we require the weak One drawback of Theorem 5.1 is the condition on the number of efficiency axiom. voters 푛. For larger values of 푘, practical elections are unlikely to have a number of voters which is exactly a multiple of 푘. The Lemma 5.5. Fix 푛 and 푘, and let 푚 ⩾ 푘. If there exists a weakly impossibility as we have proved it does not rule out that for other efficient, proportional, and strategyproof committee rule for 푚 + 1 values of 푛, there does exist a proportional and strategyproof rule. alternatives, then there also exists such a rule for 푚 alternatives. Indeed, at least for small parameter values, the SAT solver confirms Proof. Let 푓푚+1 be the committee rule defined on the candidate that this is the case. An important open question is whether, for set 퐶푚+1 = {푐1, . . . , 푐푚, 푐푚+1}. Note that every profile 푃 over candi- fixed 푘 ⩾ 3, the impossibility holds for all sufficiently large 푛. date set 퐶푚 = {푐1, . . . , 푐푚 } is also a profile over candidate set 퐶푚+1. In this section, we give one result to this effect, obtained by We then just define the committee rule 푓푚 for the candidate set 퐶푚 strengthening the proportionality axiom. Note that all the axioms by 푓푚 (푃) := 푓푚+1 (푃) for all profiles 푃 over candidate set 퐶푚, where we discussed in Section 3.2 are based on the intuition that a group Ð 푛 we assume that | ∈ 푃 (푖)| ⩾ 푘. By weak efficiency, 푓푚 (푃) ⊆ 퐶푚, of 푘 voters should be represented by one committee member. The 푖 푁 푛 so that 푓푚 is a well-defined rule. It is easy to check that 푓푚 is weakly value “ 푘 ” is known as the Hare quota. An alternative proposal is the efficient, proportional, and strategyproof. It is easy to check that Droop quota, according to which every group consisting of strictly 푛 푓푚 is proportional and strategyproof. □ more than 푘+1 voters should be represented by one committee member. Thus, with Droop quotas, slightly smaller groups already Our last induction step reduces the committee size from 푘 + 1 need to be represented. The strengthened axiom is as follows. to 푘. The construction introduces an additional candidate and an Droop Proportionality Suppose 푃 is any profile, and some additional voter, and appeals to Lemma 5.2 to show that the new { } ∈ B 푛 singleton ballot 푐 appears strictly more than 푘+1 times candidate is always part of the winning committee. Thus, the larger in 푃. Then 푐 ∈ 푓 (푃). rule implicitly contains a committee rule for size-푘 committees. Note that Droop proportionality applies to all profiles and not only Lemma 5.6. Let 푘 ⩾ 2. If there exists a proportional and strate- party-list profiles. With this stronger proportionality axiom, we gyproof committee rule for committee size 푘 + 1, for 푘 + 1 voters, and can show that for fixed 푘 and all sufficiently large 푛, we have an for 푘 + 2 alternatives, then there also exists such a rule for committee incompatibility with strategyproofness. size 푘, for 푘 voters, and for 푘 + 1 alternatives. Proposition 5.7. Let 푘 ⩾ 3, let 푚 ⩾ 푘 + 1, and let 푛 ⩾ 푘2. Then there is no approval-based committee rule satisfying weak efficiency, Proof. Let 푓푘+1 be the committee rule assumed to exist, defined strategyproofness, and Droop proportionality. on the candidate set 퐶푘+2 = {푐1, . . . , 푐푘+2}. We define the rule 푓푘 for committee size 푘 on candidate set 퐶 + = {푐1, . . . , 푐 + } as follows: 푘 1 푘 1 Proof. Suppose such a rule 푓푛 exists. By Lemma 5.5 (suitably 푓푘 (퐴1, . . . , 퐴푘 ) = 푓푘+1 (퐴1, . . . , 퐴푘, {푐푘+2})\{푐푘+2}, reproved to apply to the Droop quota), there also is such a rule for 푚 = 푘 + 1 alternatives, so we may assume that 푚 = 푘 + 1. for every profile 푃 = (퐴 , . . . , 퐴 ) over 퐶 . Notice that this is 1 푘 푘+1 Write 푛 = 푞 · 푘 + 푟 for some 0 ⩽ 푟 < 푘 and some 푞 ⩾ 푘. well-defined and returns a committee of size 푘, since by Lemma 5.2 We will show that there exists a committee rule for 푞 · 푘 voters applied to 푓 , we always have 푐 ∈ 푓 (퐴 , . . . , 퐴 , {푐 }). 푘+1 푘+2 푘+1 1 푘 푘+2 which satisfies proportionality (with respect to the Hare quota) and Proportionality. Let 푃 = (퐴1, . . . , 퐴푘 ) be a party-list profile over 푛 푘 strategyproofness, which contradicts Theorem 5.1. 퐶 + , in which the ballot {푐} occurs at least = = 1 time. Then 푘 1 푘 푘 Fix 푟 arbitrary ballots 퐵 , . . . , 퐵푟 . We define a committee rule 푓 ′ = ( { }) { } 1 푞푘 푃 퐴1, . . . , 퐴푘, 푐푘+2 is a party-list profile, in which 푐 occurs on 푞 · 푘 voters, 푚 alternatives, and for committee size 푘, as follows: 푛+1 = 푘+1 = at least 푘+1 푘+1 1 time; thus, by proportionality of 푓푘+1, we ′ 푓푞푘 (퐴1, . . . , 퐴푞푘 ) = 푓푛 (퐴1, . . . , 퐴푞푘, 퐵1, . . . , 퐵푟 ), have 푐 ∈ 푓푘+1 (푃 ) = 푓푘 (푃). 푞푘 Strategyproofness. If there is a successful manipulation from 푃 to for all profiles 푃 = (퐴1, . . . , 퐴푞푘 ) ∈ B . ′ 푃 for 푓푘 , then there is a successful manipulation from (푃, {푐푘+2}) It is clear that 푓푞푘 inherits strategyproofness from 푓푛: Any suc- ′ to (푃 , {푐푘+2}) for 푓푘+1, contradiction. □ cessful manipulation of 푓푞푘 is also successful for 푓푛. , Dominik Peters

We are left to show that 푓푞푘 satisfies (Hare) proportionality. So 6 RELATED WORK 푞푘 suppose that 푃 = (퐴1, . . . , 퐴푞푘 ) ∈ B is a party-list profile in The closest work to ours is a short article by Duddy [17], who { } 푞푘 = which singleton party 푐 is approved by at least 푘 푞 voters. also proves an impossibility about approval-based committee rules Note that, because 푟 < 푘 ⩽ 푞, involving a proportionality axiom. Duddy’s result is about proba- 푛 푞푘 + 푟 푞푘 + 푞 푞(푘 + 1) bilistic committee rules, which return probability distributions over = < = = 푞, 푘 + 1 푘 + 1 푘 + 1 푘 + 1 the set of committees. Because any deterministic committee rule induces a probabilistic one (which puts probability 1 on the deter- Thus, in the profile 푃 ′ = (퐴 , . . . , 퐴 , 퐵 , . . . , 퐵 ), there are strictly 1 푞푘 1 푟 ministic output), Duddy’s probabilistic result also has implications 푛 { } more than 푘+ voters who approve 푐 . Thus, by Droop propor- 1 ′ for deterministic rules, which we can state as follows. tionality, 푐 ∈ 푓푛 (푃 ) = 푓푞푘 (푃). Thus, 푓푞푘 is (Hare) proportional. □ Theorem 6.1 (Duddy [17]). For 푚 = 3 and 푘 = 2, no approval- Remark. If we want to restrict the Droop proportionality axiom based committee rule 푓 satisfies the following three axioms. to only apply to party-list profiles, we can instead assume in Propo- (1) (Representative.) There exists a profile 푃 in which 푛 voters sition 5.7 that 푚 ⩾ 푘 + 2, and then let 퐵 = ··· = 퐵 = {푐 }, 1 푟 푘+2 approve {푥} and 푛 + 1 voters approve {푦, 푧}, but 푓 (푃) ≠ {푦, 푧}, defining the rule 푓 only over the first 푘 + 1 alternatives. Then the 푞푘 for some 푛 ∈ N and all distinct 푥,푦, 푧 ∈ 퐶. final profile 푃 ′ is a party-list profile. (2) (Pareto-consistent.) If in profile 푃, the set of voters who approve 5.4 Small committees of 푥 is a strict subset of the set of voters who approve of 푦, then 푓 (푃) ≠ {푥, 푧}, for all distinct 푥,푦, 푧 ∈ 퐶. Theorem 5.1 only applies to the case where 푘 ⩾ 3. For the case (3) (Strategyproof.) Suppose profiles 푃 and 푃 ′ are identical, except 푘 = 1, where we elect just a single winner, Approval Voting with that voter 푖 approves {푥,푦} in 푃 but {푥} in 푃 ′. If 푓 (푃) ≠ {푥,푦}, ‡ lexicographic tie-breaking is both proportional and strategyproof. then also 푓 (푃 ′) ≠ {푥,푦}. This leaves open the case of 푘 = 2. The SAT solver indicates that the statement of Theorem 5.1 How does Duddy’s theorem relate to ours? Duddy’s strategyproof- does not hold for 푘 = 2, and that there exists a proportional and ness is weaker than but very similar to our strategyproofness. Our strategyproof rule, at least for small parameter values. However, we result does not require an efficiency axiom. Duddy’s representative can recover an impossibility by strengthening strategyproofness to axiom is noticeably different from the proportionality axioms that superset-strategyproofness, i.e., by allowing manipulators to report we have discussed. Logically it is incomparable to our proportion- arbitrary ballots (rather than only subsets of the truthful ballot). ality axiom; in spirit it may be slightly stronger. Note that not even the strongest of the proportionality axioms that we have discussed Theorem 5.8. Let 푘 = 2, 푚 ⩾ 4, and let 푛 be even. Then there is (i.e., EJR) imply Duddy’s representativeness. It is also worth noting no approval-based committee rule that satisfies weak efficiency, JR that Duddy’s result works for smaller values of 푚 and 푘 than our on party lists and superset-strategyproofness. result, suggesting that Duddy’s axioms are stronger overall. The proof of this result was also obtained via the computer- In computational social choice, there has been much recent in- aided method. However, this proof is long and involves many case terest in axiomatic questions in committee rules. Working in the distinctions, so we omit the details. The proof begins with the context of strict orders, Elkind et al. [18] introduced several axioms starting profile 푃 = (푎푏,푎푏,푐푑,푐푑). By JR on party lists, we have and studied which committee rules satisfy them. Skowron et al. 푓 (푃) ∈ {푎푐,푎푑,푏푐,푏푑}. By relabeling alternatives, we may assume [43] axiomatically characterise the class of committee scoring rules, that 푓 (푃) = 푎푐. The proof then applies strategyproofness to deduce and Faliszewski et al. [22] study the finer structure of this class. For the values of 푓 at other profiles, and arrives at a contradiction. the approval-based setting, Lackner and Skowron [33] characterise Theorem 5.8 requires an even number of voters. This is necessary, committee counting rules, and give characterisations of PAV and of since for 푘 = 2 and odd numbers of voters, AV satisfies both axioms. Chamberlin–Courant. They also have a result suggesting that AV is the only consistent committee rule which is strategyproof. Proposition 5.9. For 푘 = 2, any 푚 ⩾ 3, and 푛 odd, AV satisfies From a computational complexity perspective, there have been proportionality (it even satisfies JR) and is cardinality-strategyproof. several papers studying the complexity of manipulative attacks on multiwinner elections [4, 6, 24, 35, 37]. Other work has studied the Proof. AV is cardinality-strategyproof. Aziz et al. [3, Thm. 3] complexity of evaluating various committee rules. Notably, it is showed that for 푘 = 2 and odd 푛, AV satisfies JR. For completeness, NP-complete to find a winning committee for PAV [4, 42]. we repeat their argument here. Let 푃 be a profile. Suppose there ′ ⊆ | ′| 푛 ∈ ( ) ∈ ′ is some group 푁 푁 with 푁 ⩾ 푘 with 푐 푃 푖 for all 푖 푁 . 7 CONCLUSIONS AND FUTURE WORK Note that |푁 ′| ⩾ 푛 implies |푁 ′| > 푛 , so that 푐 has approval score 푘 2 We have proved an impossibility about approval-based committee > 푛 . Then the highest approval score is also > 푛 , and so there is 2 2 rules. The versions of the proportionality and strategyproofness some 푑 ∈ AV(푃) with approval score > 푛 . Thus, a strict majority 2 axioms we used are very weak. It seems unlikely that, by weakening of voters approve 푑. Since strict majorities intersect, there must ′ the axioms used, one can find a committee rule that exhibits satis- be a voter 푖 ∈ 푁 who approves 푑. Thus 푑 ∈ AV(푃) ∩ Ð ′ 푃 (푖), 푖 ∈푁 fying versions of these requirements. A technical question which whence the latter set is non-empty, and JR is satisfied. □ remains open is whether our impossibility holds for all numbers 푛 ‡It is well-known that AV is strategyproof. Proportionality for 푘 = 1 is equivalent to a of voters, no matter whether it is a multiple of 푘 (see Section 5.3). It 푛 unanimity condition, since 푘 = 푛, and AV satisfies unanimity. would also be interesting to study irresolute or probabilistic rules. Proportionality and Strategyproofness in Multiwinner Elections ,

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