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 : 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 < 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 : 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 : = 3, and where 5 voters in propositional logic and applying SAT solvers; a human-readable approve candidates 0, 1 and 2, while 4 other voters approve only the version of the computer-generated proof is obtained by extracting candidate 3. Then AV would select the committee f0,1, 2g, 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 =/: 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 ¸ · · · ¸ A utils with 1 INTRODUCTION A 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 : When voters are strategic, PAV has the drawback that it can often members, drawn from a larger set of < candidates. Often, we will be manipulated. Indeed, suppose a voter 8 approves candidates 0 be interested in picking a representative committee whose members and 1. If 0 is also approved by many other voters, PAV is likely to together cover the diverse interests of the voters. We may also aim include 0 in its selected committee anyway, but it might not include for this representation to be proportional; for example, if a group 1 because voter 8 is already happy enough due to the inclusion of 0. of 20% of the voters have similar interests, then about 20% of the However, if voter 8 pretends not to approve 0, then it may be utility- members of the committee should represent those voters’ interests. maximising for PAV to include both 0 and 1, so that 8 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 % = ¹012,012,012,013,013º for which 012 is the focussed on cases where there are no parties, and preferences are unique PAV-committee for : = 3. If the last voter instead reports to approve 3 only, expressed directly over the candidates [23]. The latter setting allows then the unique PAV-committee is 013. , Dominik Peters That voting rules are manipulable is very familiar to voting 퐴8 of 퐶, so that ; < 퐴8 ( 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 f0,1g is written 01. An (approval) profile is a function trivial voting rule is manipulable. However, in the approval-based % : # !B assigning every voter an approval ballot. For brevity, multiwinner election setting, we have the tantalising example of we write a profile % as an =-tuple, so that % = ¹% ¹1º,...,% ¹=ºº. For Approval Voting (AV): this rule is strategyproof in the sense that example, in the profile ¹01, 012,3º, voter 1 approves candidates 0 voters cannot induce AV to return a committee including more and 1, voter 2 approves 0, 1, and 2, and voter 3 approves 3 only. approved candidates by misrepresenting their approval set. This Let : be a fixed integer with 1 6 : 6 <.A committee is a subset raises the natural question of whether there exist committee rules of 퐶 of cardinality :. We write C: for the set of committees, and that combine the benefits of AV and PAV: are there rules that are again for brevity, the committee f0,1g is written as 01. An (approval- # simultaneously proportional and strategyproof? based) committee rule is a function 5 : B !C: , 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 5 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 5 to profiles % with j 8 2# % ¹8ºj > tionality we use is much weaker than JR. It requires that if there is :, so that it is possible to fill the committee with candidates who a group of at least =/: voters who all approve a certain candidate 2, 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 2, then 2 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 : candidates that contains additional approved candidates. In particular, our no- with highest approval score, that is, the : candidates 2 for which tion of strategyproofness only requires that the committee rule be jf8 2 # : 2 2 % ¹8ºgj 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 , ⊆ 퐶 with j, j = : 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 j% ¹8º \ , j The impossibility theorem is obtained using computer-aided 8 2# 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.