Proportionality in Committee Selection with Negative Feelings
PProportionality in Committee Selection withNegative Feelings
Nimrod Talmon ∗ and Rutvik Page † Ben-Gurion University, Israel Indian Institute of Information Technology, Nagpur, India
Abstract
We study a class of elections in which the input format is trichoto-mous and allows voters to elicit their negative feelings explicitly. In par-ticular, we study multiwinner elections with a special proclivity to electproportionally representative committees. That is, we design various ax-ioms to deal with negative feelings and suggest some structures to thesepreferences that allow better preference aggregation rules. We proposetwo different classes of axioms designed to aggregate trichotomous pref-erences more efficiently. We propose trichotomous versions of some wellknown multiwinner voting rules and report their satisfiability of our ax-ioms. Hence, with reports of our simulations as evidence, we build uponthe social optimality of our proportionality based axioms to evaluate thequality of voting rules for electing a proportionally representative com-mittee with trichotomous ballots as inputs.
Aggregation of possibly conflicting preferences to take socially optimal decisionsis a central problem in the field of Computational Social Choice and has beenactively pursued by researchers in the AI community [1]. One of the importantscenarios in this field is to elect a committee from a given set of alternatives.In multiwinner elections, the challenge is to select a size- k committee from aset of candidates, given a voter preference profile. Generally, the preferenceelicitation methods available to voters are either ranked ballots or approvalballots. In ranked ballots, the voters put forth complete and strict rankingsover the candidates in a descending order where the most preferred candidateis ranked the highest and the least preferred candidate is ranked the lowest.Another popular way of preference elicitation is approval ballots, in which each ∗ [email protected] † [email protected] a r X i v : . [ c s . G T ] J a n oter only puts forth a subset of candidates that she approves. We study amore general setting of preference elicitation in which each voter divides the setof candidates into three mutually non intersecting subsets. These three subsetscontain the candidates about whom the voters are in approval of, are indifferentabout or are in disapproval of respectively.The design of multiwinner voting rules is a challenging task since its appli-cations range from excellence based rules [2, 3, 4, 5] through selecting a diverseset of candidates [4, 5, 6] to proportional representation [7, 8, 9, 10, 11, 12].Recently, the design of efficient multiwinner voting rules and defining their nat-ural social choice properties [11, 13, 14] has received a considerable attentionfrom the Artificial Intelligence community. As a consequence, a rich variety ofmultiwinner voting rules and social choice axioms are emerging.In various preference aggregation scenarios some voters might wish not onlyto describe their positive feelings towards certain alternatives, but also to de-scribe their negative feelings with the same rigour while preserving the right toremain neutral about some [15, 16]. One prominent example is the veto rulein single winner elections, where each voter specifies one candidate which shedislikes. Another extreme example is approval voting, in which each voter pro-vides an approval set , which is simply a subset of candidates from the availablecandidates; here, it is not clear whether the voter simply does not care aboutthe candidates she did not include in her approval set or perhaps she has, say,strong negative feelings towards them.For instance, inclusion of certain vegetables in grocery shopping list doesnot necessarily mean that those items that are not on the list are despised bythe shopper, neither does it necessarily mean that the shopper is indifferentabout them. Au contraire , while deciding to go to the movies with friends,its not necessary that everyone has a binary preference over every movie beingconsidered, i.e., there might be some friends in the group who are indifferentabout watching some movie they neither like or dislike or there may be otherswho particularly dislike a movie hence emphasizing the fact that there is a cleardistinction between indifference and disliking, which is inadequately capturedby approval voting. Thus, a method of preference elicitation that widens thescope of expressibility of the voter requirements is required in order to capturesocial choice optimally.Political polarization is a common phenomenon in the current era as a di-verse mixture of opinions float via the internet to the voters. Approval ballots(also referred to as dichotomous preferences) provide a way of either approvingor disapproving a particular candidate hence leaving the populace to a morevulnerable position with regards to political polarization. Trichotomous bal-lots provide a way out of this by allowing voters to show their apathy towardssome candidates and hence to some extent reduce the magnitude of politicalpolarization. 2 .1 Our Contributions
In this paper, we initiate a principled study for allowing voters to directly ex-press negative feelings towards certain available options. In the standard modelof social choice there are various input formats, where the most prominent onesare perhaps Approval ballots, Ordinal ballots and Cumulative ballots (Pluralityvoting is very popular, however we view it as a kind of approval voting). Wepropose two broad classes of axioms for evaluating committees selected by elec-tions having input as trichotomous preferences basically differentiated by theliberality of axioms to provide representation to certain voters. Therefore, wepropose new axioms for formalizing the view of proportional representation un-der trichotomous preferences and explore the nature of different voting rules intrichotomous domains. We propose some voting rules tailored for trichotomousinputs and show by simulations the extent to which our rules as well as knownpolynomial time multiwinner voting rules satisfy our axioms and also formalizea preference domain restriction in which some of these axioms seem to be anatural fit (Definition 3.1).
The works of Condorcet (1793) were the first ones to propose a voting rule inwhich the voters were required to partition the candidate set into three groupsaccording to their preferences. Brams [17] proved that elections allowing onlyapproval votes are equivalent to those allowing only disapproval votes and thatdisapproval votes are redundant when there is an absence of lower bound onthe number of approvals a candidate should muster to be declared the winner.Yilmaz [18] presents a normative study of trichotomous voting as a superior al-ternative to approval voting in a way that trichotomous ballots represent voterfeelings more accurately. Falsenthal et al. [19] initiate the study of single winnerelection under trichotomous preferences and present a contrast between approvalvoting and voting with trichotomous ballots and conclude that the latter leadsto voters being more decisive. Hillinger et al.[16], Alcantud et al [15]. andSmaoui et al [20] and Lapresta et al. [21] propose and study utilitarian scoringrules and their axiomatic properties which in a way allow voters to attributeranks to candidates over a range (voters assign scores to candidates from a givenrange) typically selecting that candidate as the winner which has the highestpositive difference between the number of voters approving and disapprovingthe candidate. Baumeister et al. [22] and Zhou et al. [23] study the utilitarianand egalitarian variants of voting rules for committee elections with voter dissat-isfaction with trichotomous preferences. While the former study concentrateson using a distance based approach like Kemeny Distance, the latter general-izes popular voting rules like Chamberlin-Courant Rule, Proportional ApprovalVoting and Satisfaction Approval Voting to trichotomous domains and find theparameterized complexity of winner determination under these rules. Ouafdi etal. [24] draw out a comparison between evaluative voting (trichotomous pref-erences with scores being assigned from the set {
2, 1, 0 } to each candidate3n contrast to the earlier stated {
1, 0, -1 } ) and popular voting methods likeBorda Rule, Plurality and the Approval Rule and investigate the proclivity ofevaluative voting to elect Condorcet committees. Aziz and Lee [25] generalizeProportional Solid Coalitions defined for strict preferences to weak preferencesand show that Proportional Justified Representation [11] is also a specializationof the same. We first provide preliminaries regarding approval ballots and our modeling ofnegative feelings via a generalization to trichotomous preference and then pro-vide preliminaries regarding proportionality axioms for approval ballots that,later, we adapt to our setting.
Given a set of alternatives of size m , A = { a , . . . , a m } , in approval voting eachvoter i in voter set V of size n specifies an approval set A i ⊆ A and the goalis to select a committee W of size exactly k . Usually, the alternatives in A i are understood as the alternatives “approved” by i . Define a dichotomous voterprofile as a vector A Dic = ( A , . . . A n ) such that A i ⊆ A ∀ i ∈ V . Informally, thealternatives in A \ A v can be understood either as (1) alternatives for which v does not have any feelings about, or as (2) alternatives for which v has negativefeelings about (contrasted with the alternative in A v , for which it can be under-stood that v has positive feelings about). In a way, this ambiguity is preciselythe problem, as there is no way for the aggregation mechanism to figure outwhich of the two cases it is for each voter.A natural remedy might be to let each voter specify not only an approvalset, but also a disapproval set; that is, let each voter i specify A + i as well as A − i ,such that A + i , A − i ⊆ A and A + i ∩ A − i = ∅ . Then, for each alternative, a voter i would place the alternatives for which she has positive feelings about in A + i and place the alternatives for which she has negative feelings about in A − i ; theremaining alternatives,i.e., those in A i := A \ ( A + i ∪ A − i ) are those for which i has not feelings at all. We therefore define a trichotomous preference profile asa vector A tri = (( A +1 , A − ) , . . . ( A + n , A − n )). We recall two known axioms of proportional representation in approval votingfor selecting a committee of size k . Definition 2.1.
JR [10] A committee W is said to satisfy Justified Represen-tation if there does not exist group of voters V (cid:48) ⊆ V such that | V (cid:48) | ≥ nk and( ∪ i ∈ V (cid:48) A i ) ∩ W = ∅ efinition 2.2. PJR [11] A committee W is said to satisfy Proportional Jus-tified Representation if there does not exist a group of voter V (cid:48) ⊆ V with size | V (cid:48) | ≥ l nk for l ∈ { , , . . . k } such that | ∩ i ∈ V (cid:48) A i | ≥ l but | ( ∪ i ∈ V (cid:48) A i ) ∩ W | < l .The idea behind JR and PJR is that if voters in large enough groups areinclined to have similar choices, then at least some voters of the group shouldget some representation in the committee W . In this paper, we generalizedichotomous preferences to trichotomous preferences which allows an extensionin expressiveness to the voters by allowing them not only to elicit approval ordisapproval but also indifference over the candidates in the candidate set. Werepresent the position of a candidate c for a voter i such that pos c ( i ) ∈ { , , − } if the candidate lies in the approval, indifference and disapproval set respectivelyof voter i . We define the positional score of a candidate c as the sum of positionsof the candidate in the preference ballots of all voters in the electorate i.e.Σ i ∈ V pos c ( i ). Moreover, we use [ k ] as an abbreviation for the set { , , . . . k } . Remark 1.
A dichotomous approval voter profile A dic = ( A , A , . . . A n ) isessentially a trichotomous voter profile A tri = (( A +1 , A − ) , . . . ( A + n , A − n )) with A + i ∪ A − i = C ∀ i ∈ V , or A i = ∅ ∀ i ∈ V which means that a trichotomous profilewith every voter casting an empty indifference ballot is essentially a dichotomousprofile. The basic unit of voters whom a proportionally representative committee mightproffer representation to would be a group of voters who have the proclivityto have similarly aligned preferences. Ideally, this calls for the pursuit of pro-viding representation to every large enough and seemingly “cohesive” group ofvoters [9], who essentially form a solid coalition amongst themselves [25].We present two different classes of axioms for proportional representationin trichotomous preference domains the fundamental difference amongst whomis the definition of ‘cohesiveness’ of a voter group. Essentially, Class I axiomsprovide a more liberal definition of cohesive representation while Class II axiomsassert a stricter definition. As a result of this, the number of voter groups to beserved representation in Class I axioms is higher than that in Class II axioms.We further propose new polynomial time executable voting rules and show theirtendency to satisfy our proposed axioms through simulations.
Our first stride in eliciting axioms for proportional representation in trichoto-mous voting domains is given next.
Remark 2.
We say that a group of voters V (cid:48) is worthy of justified represen-tation if its size is at least the quantity that reflects uniform distribution of k seats amongst n voters i.e. nk , while the voters in the group are at least as pref-erentially aligned as to support a set of candidates each of which is approved y at least one voter, but none is disapproved by any. In effect, the set of vot-ers is said to be worthy and preferentially aligned if its size is at least nk and | ∪ i ∈ V (cid:48) A + i \ ∪ i ∈ V (cid:48) A − i | (cid:54) = ∅ . Axiom 1 (Strong Preliminary Representation (SPR)) . A committee is said tosatisfy
Strong Preliminary Representation if all subset of voters V (cid:48) ⊆ V of size | V (cid:48) | ≥ nk satisfy | ( ∪ i ∈ V (cid:48) A + i ) \ ( ∪ i ∈ V (cid:48) A − i ) | ≥ ∀ l ∈ [ k ] ,the committee contains at least one candidate approved by at least one voterin V (cid:48) whilst not containing any unanimously disapproved candidate by the votersin the group. That is, ∃ i ∈ V (cid:48) : | W ∩ A + i | ≥ and | W ∩ ( ∩ j ∈ V (cid:48) A − i ) | = 0 . The definition above captures the idea that a large enough group with simi-larly aligned preferences should have at least one member who gets her favorablecandidate in the committee but at the same time the unanimously disliked can-didate should not feature in the committee. The committees satisfying thisaxiom take an all encompassing approach to the satisfaction of voter approvalas well as voter resentment towards the committee.
Example 3.1.
Suppose there are voters with the following voter preferencesover the set of candidates a, b, c, d and k = 2 . { d, a, b } (cid:31) c (cid:31) e { a, b } (cid:31) c (cid:31) { d, e } a (cid:31) { b, c } (cid:31) { d, e } { b, c } (cid:31) a (cid:31) { d, e } The committee { a, b } forms a committee adherent to the above axiom sincethere is at least one voter in each group who has a favorite candidate in thecommittee and there is no such group of voters which is large enough and findtheir commonly disliked candidate in the committee. It is useful to note thatwhile Strong Preliminary Representation seems to provide appropriate and insome ways a seemingly balanced allocation of candidates amongst the voters, itis indeed a strong notion and is unfortunately, not guaranteed to exist.
Example 3.2.
Suppose there are two voters and two candidates { v , v } and { c , c } respectively and the voter profile is as follows: c (cid:31) {} (cid:31) c c (cid:31) {} (cid:31) c Suppose that the committee size is k = 2 , which means that there is only onecommittee possible which is { c , c } . This committee would not satisfy the axiomsince there is at least one candidate in the committee that is despised by eachvoter, who in this case individually form a ‘deserving’ and ‘cohesive’ group ofvoters.
6n order to mitigate the non-existence guarantees of the above axiom, wepropose its weakened version. In effect, we relax the mandatory debarring ofthe unanimously despised candidate from the committee. We assert that a largeenough and cohesive group of voters is granted representation on the committeeif there are at least some candidates in the committee that are approved by atleast some voters in the voter group.
Axiom 2 (Weak Trichotomous Justified Representation (WTJR)) . A commit-tee W is said to follow Weak Trichotomous Justified Representation if for allsets of voters V (cid:48) of size | V (cid:48) | ≥ nk having | ( ∪ i ∈ V (cid:48) A + i ) \ ( ∪ i ∈ V (cid:48) A − i ) | ≥ thefollowing is satisfied: | ( ∪ i ∈ V (cid:48) A + i ) ∩ W | ≥ Axiom 3 (Weak Trichotomous Proportional Justified Representation (WT-PJR)) . A committee W is said to follow Weak Trichotomous Proportional Jus-tified Representation if for all sets of voters V (cid:48) of size | V (cid:48) | ≥ l nk for all l ∈ [ k ] having | ( ∪ i ∈ V (cid:48) A + i ) \ ( ∪ i ∈ V (cid:48) A − i ) | ≥ l the following is satisfied: | ( ∪ i ∈ V (cid:48) A + i ) ∩ W | ≥ l In effect, Weak Trichotomous Justified Representation is nothing but a spe-cial case for Weak Trichotomous Proportional Justified Representation with l = 1. As an instance, consider example 3.1 where the committee { a, b } satisfiesboth the above axioms since for any large enough voter group, the set differencebetween the unions of approval and disapproval sets of voters contain c as wellas c . Remark 3.
A committee satisfying even
Weak Trichotomous Justified Repre-sentation is not guaranteed to exist for all voter profiles. For instance, if thereare five voters and three candidates { a, b, c } and k = 2 with the following voterprofile: {} (cid:31) { a, b, c } (cid:31) {} {} (cid:31) { a, b, c } (cid:31) {} a (cid:31) {} (cid:31) { b, c } b (cid:31) {} (cid:31) { a, c } c (cid:31) {} (cid:31) { a, b } In this case, whatever the size committee is, a voter group of voters , andany one of , , is unsatisfied. On the contrary, we find that there always exists a committee that satis-fies
Weak Trichotomous Proportional Justified Representation if there is somestructure to the preferences of voters in the electorate.
Definition 3.1.
Decisive Electorate
An electorate is said to be decisive if noneof the voters is indifferent about the available alternatives. Formally, A i = ∅∀ i ∈ V . 7 roposition 1. A committee W satisfies Proportional Justified Representationin dichotomous preference domains if and only if it satisfies Weak TrichotomousProportional Justified Representation in trichotomous preference domains in adecisive electorate.Proof. Since the committee W satisfies PJR, | ∩ i ∈ V (cid:48) A i | ≥ l ∀ l ∈ [ k ] and forall subsets of voters V (cid:48) ⊆ V such that | V (cid:48) | ≥ l nk . Now since the electorate isdecisive, A i = ∅ ∀ i ∈ V (cid:48) for all subsets of voters V (cid:48) ⊆ V : | V (cid:48) | ≥ l nk . Thismeans that for all such subsets of voters in the dichotomous domain, | ∩ i ∈ V (cid:48) A i | = | ∪ i ∈ V (cid:48) A i \ ( C \ ∪ i ∈ V (cid:48) A i ) | ≥ l , in effect, the intersection of approvalsets of all voters in a deserving and cohesive group V (cid:48) is equal to the unionof the approval sets minus the union of disapproval sets of all voters in thegroup. When projected in trichotomous domain, since this electorate is decisive, A i = A + i and C \ A i = A − i and A i = ∅ . This means that all sets of voters thatare assured representation in PJR are also guaranteed to have representation inWTPJR. Since the approval sets of voters under trichotomous and dichotomousdomains are the same in a decisive electorate, the amount of representation thatthey get is also the same, i.e. | ∪ i ∈ V (cid:48) A i ∩ W | = | ∪ i ∈ V (cid:48) A + i ∩ W | ≥ l . Sincewe have established the fact that the approval and disapproval sets of voters indichotomous and trichotomous preference domains are equal, the sets of votersof a decisive electorate that get representation in the committee due to WTPJRare the same as those subsets of voters that get representation due to PJR andrespectively get the same representation. Therefore, in a decisive electorate, acommittee that satisfies WTPJR also satisfies PJR.We define a still weaker axiom with the objective that it still provides atmost as much representation to large enough voter groups as the previouslygiven stronger axioms give. Axiom 4 (Weak Ambivalent Representation (WAR)) . A committee W satisfies WAR if for every group of voters V (cid:48) of size V (cid:48) ≥ l nk for some l ∈ [ k ] the followingcondition is satisfied: | ( ∪ i ∈ V (cid:48) A + i ) \ ( ∪ i ∈ V (cid:48) A − i ) | ≥ l = ⇒ | (( ∪ i ∈ V (cid:48) A + i ) ∪ ( ∩ i ∈ V (cid:48) A i )) ∩ W | ≥ l Intuitively, the definition says that the committee should have at least l candidates from the set of candidates defined by the union of all the approvedcandidates and the unanimously ‘not cared about’ candidates by the voters. Ina way, the definition further degrades the utility that the voters in the votergroups are entitled to in the previous axioms.Although the above definition is weaker than the previous axioms, there are stillprofiles for which there does not exist a committee such that it follows WAR.For instance, there is no committee that satisfies WAR if the preference profileis the same as mentioned in remark 3 i.e. the voter group formed by the firsttwo voters and any one of the other three voters is always dis-satisfied with anycommittee formed out of the three candidates.8 xiom 5 (Weakest Axiom (WA)) . A committee W satisfies weakest axiom iffor every subset of voters V (cid:48) ⊆ V of size | V | ≥ l nk for all l ∈ [ k ] , the followingimplication stands true: | (( ∪ i ∈ V (cid:48) A + i ) \ ( ∪ i ∈ V (cid:48) A − i )) | ≥ l = ⇒ | (( ∪ i ∈ V (cid:48) A + i ) ∪ ( ∪ i ∈ V (cid:48) A i )) ∩ W | ≥ l This is the most liberal axioms that we propose in Class I axioms. Intu-itively, a committee satisfies this axiom if at least some every large enoughgroup (same as defined before) get some candidates from their approval as wellas indifference classes. Unfortunately, even committees satisfying this axiom arenot guaranteed to exist, but as we present in the following sections, there is ahigh probability for a committee computed by some voting rules to satisfy thisaxiom.
All the Class I axioms proposed have been in the spirit that the voters formcohesive groups with other voters even if there is a candidate about whom oneof them is in approval while the other has neutral feelings as a result of whichthere are a high number of deserving and cohesive voter groups formed leadingto an obvious difficulty in accommodating every voter’s choice. In the definitionsdescribed in this section, we do away with this definition of cohesiveness to aless accommodating version but strengthen the amount of representation thatthe voters in the voter groups get in the committee.
Axiom 6 (New Cohesiveness Representation (NCR)) . A committee satisfies
New Cohesiveness Representation if for no set of voters of size | V (cid:48) | ≥ l nk for all l ∈ [1 , k ] , if | ∩ i ∈ V (cid:48) A + i | ≥ l then | ( ∩ i ∈ V (cid:48) A + i ) ∩ W | < l . Proposition 2.
For a given committee size k, a committee satisfying
NewCohesiveness Representation always exists.Proof.
We present a constructive proof, inspired by the proof of proposition 3.7presented in [26], which we specialize to the settings of multi-winner voting.Let’s say, the algorithm iterates over l (cid:48) with its initial value k and it terminatesas soon as the value of l (cid:48) reaches 0. Initially A (cid:48) = V , where A (cid:48) acts as a containerset for the voters who still remain to be served representation in the committeeand let W = ∅ . The algorithm greedily satisfies voter groups by providing themwith representation in the committee and once they have been represented, theyare removed from consideration to provide representation. At the beginning ofevery iteration, the condition | W | + l (cid:48) ≤ k in order to ensure the committeesize remains bounded by k ; if this condition fails in any iteration, reduce l (cid:48) by1 and continue to the next iteration. Then, let C ∗ denote the set of subsets ofcandidates of size l (cid:48) . C ∗ = { C (cid:48) ⊆ C : | C (cid:48) | = l (cid:48) } If C ∗ is empty, this means that there are no subsets of candidates of size exactly l (cid:48) and we cannot satisfy any subset of voters of size at least l (cid:48) nk at this juncture9f the algorithm, so we reduce l (cid:48) by 1 and continue to the next iteration. If notso, for each subset of candidates C (cid:48) ∈ C ∗ , define A + ( C (cid:48) ) as follows: A + ( C (cid:48) ) = { i ∈ A (cid:48) : C (cid:48) ⊆ A + i } select the voter set A + ( C (cid:48) ) of maximal size, and check if A + ( C (cid:48) ) ≥ l nk ; if notso, decrease l (cid:48) by 1 and continue to the next iteration, since the consequenceof this step is that there is no subset of unrepresented voters which approves l (cid:48) candidates and has a size of at least l (cid:48) nk . But, if the condition is indeed true,set W → W ∪ C (cid:48) and A (cid:48) → A (cid:48) \ A + ( C (cid:48) ). Preserving the value of l (cid:48) as isin pursuit of finding another subset of candidates of size l (cid:48) which is supportedby correspondingly large group of unsatisfied voters and continue to the nextiteration. If the algorithm reaches the point where l (cid:48) = 0 but | W | < k , it meansthat there are no subsets of voters that are large and cohesive enough to have acandidate in the committee. In that case, we arbitrarily add candidates in thecommittee in order to fill it to its size, k .Now we prove the correctness of the algorithm. Suppose that a committeethat is computed by the above algorithm W does not satisfy New CohesivenessRepresentation , which means that there is at least one set of voters V (cid:48) such thatits size is V (cid:48) ≥ l (cid:48) nk and |∩ i ∈ V (cid:48) A + i | ≥ l (cid:48) but |∩ i ∈ V (cid:48) A + i ∩ W | < l (cid:48) . Every candidatein the committee W represents a group of at least nk voters, each of which oncegranted representation is not entertained further. This means that the numberof voters getting representation in this committee is | W | · ( nk ) = n thus, there does not exist any l (cid:48) such that a set of voters V (cid:48) not have l (cid:48) ≤ | V (cid:48) | k/n candidates to represent it in the committee. Thus, the contradiction.The above axiom promises representation to cohesive groups of voters fromexactly that set of voters about which they mutually agree. We further presenta weaker version of the Weaker New Cohesiveness Representation whereby weweaken the amount of representation provided to the voters in a cohesive anddeserving group. In some ways, this axiom mimics Proportional Justified Rep-resentation in dichotomous preferences.
Axiom 7 (Weaker New Cohesiveness Representation (WNCR)) . A committeesatisfies
New Cohesiveness Representation if for no set of voters of size | V (cid:48) | ≥ l nk for all l ∈ [1 , k ] , if | ∩ i ∈ V (cid:48) A + i | ≥ l then | ( ∪ i ∈ V (cid:48) A + i ) ∩ W | < l . Proposition 3.
There exists a polynomial time algorithm that determines acommittee that satisfies
Weaker New Cohesiveness Representation . The proof of this proposition can be argued to by asserting the equivalencebetween Weaker New Cohesiveness Representation in trichotomous preferencesand Proportional Justified Representation in dichotomous preferences. Thisaxiom mimics Proportional Justified Representation since the merging of dis-approval and indifference sets of voters in the electorate would neither change10igure 1: Relationship between the proposed Class I and Class II axiomsthe nature of the large enough voting groups who deserve representation norwould it change the nature of representation that they receive in the committeeotherwise. Since we know that well known polynomial time rules like SequentialPhragmen’s Rule satisfy Proportional Justified Representation [27], we concludethat committees satisfying Weaker New Cohesiveness Representation always ex-ist.
In this section we present trichotomous versions of popular scoring rules espe-cially with the objective of finding the most suitable rule providing proportionalrepresentation. While some voting rules have been taken from results cited cor-respondingly, we propose the approximate variants of trichotomous versions ofChamberlin Courant and PAV respectively and to the best of our knowledge,use Droop-STV in trichotomous settings.
The α -CC (Chamberlin Courant) rule is computed as follows. A satisfactionfunction for a committee W ⊆ C such that | W | ≥ k is defined as follows: sat α − CC ( v, W ) = (cid:40) | A + v ∩ W | − | A − v ∩ W | < α otherwise α -CC finds a committee that maximizes Σ v sat α − CC ( v, W ), that is, thecommittee in which the highest number of voters have difference between thenumber of approved and disapproved candidates at least α in it. Since this ruleis nothing but a slight variant of dichotomous α -CC, computing a committeeusing this rule is NP-Hard [12] [23]. The TPAV score of a voter for a committee W is calculated as follows; declaresatisfaction and dis-satisfaction functions as follows: sat T P AV ( v, W ) = (cid:40) | A + v ∩ W | = 0Σ | A + v ∩ W | p =1 1 p otherwisedissat T P AV ( v, W ) = (cid:40) | A − v ∩ W | = 0Σ | A − v ∩ W | p =1 1 p otherwise The committee that maximizes Σ v ( sat T P AV ( v, W ) − dissat T P AV ( v, W )) is theone that is selected by the rule as the winner. From remark 1, since TPAV is atleast as hard as dichotomous PAV, winner determination in TPAV is NP-Hard.[23] The sequential variants of these rules can be generalized as follows; we startwith an empty committee W = ∅ and iteratively add a candidate c in everyiteration till | W | < k such that Σ v sat α − CC ( v, W ∪ c ) and Σ v ( sat T P AV ( v, W ∪ c ) − dissat T P AV ( v, W ∪ c )) are maximized for Sequential Trichotomous α -CCand Sequential Trichotomous PAV respectively. Note that at every iteration,the candidate added in the committee is deleted from the set of available can-didates to be added in the committee. Generally, a candidate c is added to thecommittee W for a rule R , where R ∈ { Sequential TCC, Sequential TPAV } ifΣ v ( sat R ( v, W ∪ c ) − dissat R ( v, W ∪ c )) is maximized, where dissat T CC ( v, W ) = 0 ∀ W ∈ V k . There are several relevant approximation algorithms for the Monroe rule [8][28]. Here, we adapt Algorithm A proposed in [28] to trichotomous settings. Weproceed in k steps, greedily building the committee W by adding a candidate c at every iteration. Define a satisfaction function Γ( c ) : C → V (cid:100) nk (cid:101) such that fora candidate c not yet added in the committee and | Γ( c ) | = (cid:100) nk (cid:101) . Explicitly, Γ( c )returns the set of voters such that ∀ i ∈ Γ( c ) and j ∈ V \ Γ( c ) , pos c ( i ) > pos c ( j )i.e. we choose the top (cid:100) nk (cid:101) voters who have the highest positional score for thecandidate c . Induct the candidate c in the committee if the sum of positionalscores of voters in Γ( c ) is the highest and remove the voters in Γ( c ) from the set12f unsatisfied voters and c from the set of available candidates. In cases wherethe number of unrepresented voters becomes ∅ , we randomly add candidatesinto the committee W until | W | < k . The approximation ratio is guaranteedto be 1 − k − m − − H k k where m is the number of candidates and H k is the k th harmonic number [28]. We adapt this rule from the family of STV rules mentioned in [25]. We greedilyconstruct a committee W in the following manner; find the candidate c withmaximum plurality score (the important step here is to break ties randomly, notlexicographically) and compare it with the Droop quota i.e. (cid:98) nk +1 (cid:99) + 1 and addthe candidate to the committee if its score is greater than the quota, removing itfrom the set of candidates available for induction to the committee. Otherwiseif the maximum score is less than the quota, we again break ties randomly,which is an important step for winner determination and remove that candidatefrom the list of available candidates. At the end of each iteration, we delete thecandidate in focus, c from the preference order of every voter and move on tothe next iteration if | W | < k . We conduct a series of experiments, each of which in itself has been conducted anumber of times and hence we present the average results scaled so that optimumrule gives out a probability at most 1. We conduct these experiments in orderto quantitatively show which voting rule is the best suited for trichotomoussettings. In order to do this, we consider 4 of the well known polynomial timeexecutable voting rules mentioned above and find the probability that the outputof each of these voting rule satisfies our proposed axioms by generating 10 , C . Additionally, in each of the voter profiles, the number of voters andthe number of candidates are picked randomly ranging from 4 to 20 and 1 to 15respectively.We use the Impartial Culture of Voting since it is a standard method of ran-domized profile generation to study voting mechanisms. As mentioned earlier,we divide the axioms in two different classes i.e. Class I and Class II. For everyaxiom presented, we find the probability that our chosen voting rules select acommittee that satisfies it. 13oting Rules Class I Class IIWA WAR WTPJR WNCR NCRSequential Monroe 0.9996 0.992 0.9878 0.996 0.769Sequential α -Chamberlin Courant 0.9995 0.984 0.9696 0.9874 0.788Multi-winner STV 0.9998 0.993 0.99 0.9974 0.789Sequential PAV 0.9997 0.9934 0.9854 0.9928 0.8058Table 1: Probabilities of voting rules satisfying Class I and Class II axioms over10,000 randomly generated profiles with | V | ∈ [4 ,
20] and | C | ∈ [1 ,
15] between1 and 15 and k ∈ [1 , | C | − We present our results in Table 1. We present a characterization of both ClassI and Class II rules and further draw a contrast between the two and reasonabout the kind of results mentioned for both the classes correspondingly.
Our class 1 axioms are based on the antecedent of getting representation beingvery strong i.e. there are larger number of large enough groups qualifying forrepresentation as compared to those in the class 2 axioms. In effect, all oursequential voting rules produce every good results with almost all entries in thetable being close to 98%.Amongst the Class I axioms, the strength of axioms increases as follows;WA, WAR and WTPJR. Due to a very weak nature of WA which allows votergroups getting representation from the union of approval and indifference classes,all voting rules almost always produce a committee that satisfies WA, thoughthere are some exceptions in cases where the ratio k/m is high. The datareflects a careful weakening of the stronger axioms WAR and WTPJR where theprobability of finding a committee that satisfies these axioms is marginally lessthan the weakest version and hence these also emerge to be quite suitable axiomsfor evaluation of the quality of a committee determined by these polynomial timecomputable voting algorithms.
Our Class II axioms reveal an approach towards proportional representationthat is commonly taken by Proportional Justified Representation [11]. Hence,our voting rules tweaked for trichotomous ballots determine committees thatsatisfy axioms of this class with lesser probability as compared to the ClassI axioms. NCR bears similarity to trichotomous version of Strong-BPJR [26]presents a very strong notion of representation in the committee and hence doesnot always exist [10]. This is instantiated by the figures in table 1 where the bestrule for this axiom is Sequential PAV while the worst being Sequential Monroe.14nterestingly, the exact order is not repeated in the case when WNCR is takenas the axiom to be tested though what remains common is that Sequential α -CCremains the voting rule which fares the worst. The general trend in the fitness of voting rules for trichotomous settings turnsout to be that Droop-STV produces the best results while Sequential α -ChamberlinCourant rule produces the worst results. This re-asserts the usefulness of STV inelecting a proportionally representative committee in dichotomous preferencesas well [25] [4] and also asserts the fact that Chamberlin Courant is not the bestsuited voting rule for proportional representation however impeccable it is forthe selection of diverse committees [2]. Additionally, sequential Monroe and Se-quential PAV also provide good results, although a little less than Droop-STV.In general, for the proposed Class I the following order of fittest voting rules forrandomized preference ballot generation is found; Droop-STV, Sequential PAV,Sequential Monroe, Sequential α -Chamberlin Courant. Alternatively, for ClassII axioms, which take a little more cognizance of proportional representationaxioms for dichotomous preferences [11] [10], it is correct to say that Sequential α -CC is not the most suitable rule for randomly generated trichotomous voterprofiles and while Droop-STV fares the best in weaker axioms of the class, itdoes not do so for the stronger axioms. Sequential-PAV is a good rule for thosecases when a strong variant of axioms of this is preferred while other presentedrules might not be the as effective as they are in with the weaker axioms. To be able to accommodate negative feelings, we have proposed two broadclasses of axioms for proportional representation in trichotomous preference do-mains, which provide greater flexibility to the voters than approval ballots. Westudy and propose some voting rules for such input formats and show by sim-ulations that these rules adhere to our axioms to a large extent. Therefore, weprovide strong basis for the design of polynomial time computable voting rulestaking input as trichotomous preferences, which select committees that satisfycertain proportionality axioms to this setting. We mention some future researchdirections below.
Different Axioms and Rules
Here we concentrated on adaptations of JR-style axioms to our setting. Naturally, there are other ways to approach pro-portionality, hopefully also giving rise to a richer landscape of voting rules forthis setting.
Participatory Budgeting
One application in which negative feelings arequite prominent is participatory budgeting, in which some projects might beperceived as hurting certain voter groups (e.g., building a sports stadium causes15raffic jams, building a bus station causes pollution, etc.). Currently such nega-tive feelings are not taken into account while selecting project bundles for par-ticipatory budgeting; lifting the results of our paper to this important usecaseis of theoretical as well as practical importance.
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