aa r X i v : . [ c s . M A ] N ov Electing the Executive Branch
Ehud ShapiroWeizmann [email protected] Nimrod TalmonBen-Gurion [email protected]
Abstract
The executive branch, or government, is typically not elected directlyby the people, but rather formed by another elected body or person such asthe parliament or the president. As a result, its members are not directlyaccountable to the people, individually or as a group. We consider ascenario in which the members of the government are elected directly bythe people, and wish to achieve proportionality while doing so.We propose a formal model consisting of k offices, each with its owndisjoint set of candidates, and a set of voters who provide approval bal-lots for all offices. We wish to identify good aggregation rules that assignone candidate to each office. As using a simple majority vote for each of-fice independently might result in disregarding minority preferences alto-gether, here we consider an adaptation of the sequential variant of Propor-tional Approval Voting (SPAV) to our setting, and demonstrate—throughcomputer-based simulations—how voting for all offices together using thisrule overcomes this weakness. We note that the approach is applicablealso to a party that employs direct democracy, where party members electthe party’s representatives in a coalition government. Consider a scenario in which a government in some country has to be populated;i.e., there should be selected the minister of health, the minister of education,etc. Usually this assignment process is done via a non-participatory process.Here we describe a way of doing this process in a participatory way. In essence,we would like each of the citizens of the country to describe her preferencesregarding the assignment of alternatives to each office.While the above setting is indeed quite imaginary, as this is not the way itis usually done in practice, our particular motivation for this work comes frompopulating governments in coalitional systems; indeed, in coalitional systems,following coalition negotiations, each party in the coalition is being allocatedsome set of offices that, in turn, has to be populated with ministers. Specifically,we are interested in the process in which a party that got allocated some officesthrough such a negotiation, shall decide internally—via democratic vote of itsmembers–how to assign ministers to each of her allocated offices.1e view this process as a social choice setting and first observe that onenatural and simple way to approach it is to view it as k independent elections,where k is the number of offices allocated to the party. For example, eachvoter can select a set of alternatives for each of the offices and, for each officeindependently, we can select the alternative that got the highest number ofvotes.Observe, however, that such a process might disregard the preferences ofminorities; in particular, if a strict majority vote for some alternatives for eachof the offices, then only the alternatives voted by the majority would be selected,and none of the alternatives of the minority would be, even if the minority consistof 50 − ǫ percent of the votes.To overcome this weakness, we view the setting as a whole, in which we haveone election whose output would be the complete assignment of alternatives tothe full set of k offices, and we aim to find a process that would guarantee somesort of proportionality.To this end, we offer an adaptation of the sequential variant of ProportionalApproval Voting (in short, SPAV) to this setting and show – via computer-basedsimulations – that indeed, in many cases, it guarantees proportional represen-tation to minorities. To the best of our knowledge, the specific setting we are considering in thispaper is new. However, the model suggested by Aziz and Lee [2], while using adifferent jargon, generalizes our setting, mainly by considering selecting severalcandidates from each office. Furthermore, the model studied by Conitzer etal. [4] study a different generalization of our model, using general cardinal utili-ties and not approval ballots; moreover, they study different axioms and differentrules, concentrating on fairness notions adapted from the fair division literature.We also mention the literature on voting on combinatorial domains [7] that for-mally captures our setting as well; that literature, however, concentrates oninterconnections and logical relations between the different “domains” (in ourjargon – the different offices).Further apart, below we mention some more related models studied in thesocial choice literature. First, we mention the work of Boehmer et al. [3] thatconsiders an assignment social choice problem, but differs from our model inthat voters provide numerical utilities and alternatives can run for few officesin parallel (so the output decision shall take into account the suitability ofalternatives to offices, while we derive the suitability directly for the votes).Generally speaking, our social choice task is of selecting a committee, andthus is related to the extensive work on committee selection and multiwinnerelections [5]. In our setting, however, we do not aim simply at selecting k alternatives, but at selecting an assignment to k offices.Related, the line of work dealing with committee selection with diversityconstraints (e.g., see [6, 1]) has some relation to our work, in particular, as one2an choose a quota of “at most one health minister”, “at most one educationminister”, and so on. We model our situation as follows: We have a set of k offices and k correspondingdisjoint sets of alternatives, A j , j ∈ [ k ] so that each candidate runs for at mostone office). Let us denote the set of all alternatives by A := ∪ i ∈ [ k ] A j . Here weconsider the approval setting, so, in particular, we have a set V = { v , . . . , v n } of n votes such that v ⊆ A , v ∈ V . Then, an aggregation method for our settingtakes as input such an instance ( A, V ) and outputs one alternatives a j ∈ A j foreach j ∈ [ k ]. Example 1.
Consider the k sets of alternatives being A = { a, b } , A = { c, d } ,and A = { e, f, g } , and the set of votes being v = { a, c, e } , v = { a, c, f } ,and v = { a, d, f } . An output of an aggregation method might be { a, c, g } ,corresponding to alternative a being selected for the first office, alternative c being selected for the second office, and alternative g being selected for thethird office. Perhaps the most natural and simple solution would be to view the settingas running k independent elections; for example, selecting to each office thealternative that got the highest number of approvals. This, however, would beproblematic; in particular it would not be proportional. Example 2.
Consider a society with strict majority voting for a j ∈ A j for each j ∈ [ k ]. Now, disregarding how the other voters vote, a j ∈ A j , j ∈ [ k ] would beselected. In particular, even a minority of 49% would not be represented in thegovernment. To overcome the difficulty highlighted above, here we aim at identifying a votingrule for our setting that does not completely disregard minorities.To this end, here we adapt SPAV. SPAV is used for multiwinner electionsand is known to be proportional for that setting. It works as follows: Initially,each voter has a weight of 1; the rule works in k iterations (as the task instandard multiwinner elections is to select a set of k alternatives), where ineach iteration one alternative will be added to the initially-empty committee.In particular, in each iteration, the alternative with the highest total weight fromvoters approving it is selected, and then the weight of all voters who approvethis alternative is reduced; the reduction follows the harmonic series, so thata voter whose weight is reduced i times will have a weight of 1 / ( i + 1) (e.g.,3nitially the weight is 1; then, a voter reduced once would have a weight of 1 / /
3, and so on).In the proposed adaptation of SPAV to our setting of electing an executivebranch, in each iteration, we again select the alternative with the highest weightfrom approving voters; say this is some a j ∈ A j . Now, we fix the j th office to bepopulated by a j ; then, as it is fixed, we remove all other a i ∈ A j from furtherconsideration (as the j th office is already populated) and reweight approvingvoters as described above (in the description of SPAV for the standard settingof multiwinner elections). Example 3.
Consider again the election described in Example 1, consisting of k sets of alternatives: A = { a, b } , A = { c, d } , and A = { e, f, g } ; 3 voters: v = { a, c, e } , v = { a, c, f } , and v = { a, d, f } .In the first iteration of SPAV, we will select alternative a to populate thefirst office; then we reweight all votes to be 1 / a ). In thenext iteration we will select either c or f (as both has total weight of 1); saythat our tie-breaking selects c . Then, we reweight v and v to be both 1 / e has 1 / f has 1 / / f . Thus, SPAV assigns a to the first office, c to thesecond office, and f to the third office. Our main aim is to achieve some sort of proportionality, in that minorities wouldnot be completely disregarded when populating the offices. Consider first thefollowing example.
Example 4.
Consider a toy society instantiating the general situation presentedin Example 2, with 3 offices and 3 voters, v , v , and v , voting as follows: Foreach office i , i ∈ { , , } , v votes for a i , while v and v vote for b i . Note that,indeed v and v are a cohesive majority, however SPAV might select b for thefirst office, resulting in reducing the weight of v and v by half, thus makingsure that at least a or a would be selected, thus the minority would also betaken into account.We ran computer-based simulations to evaluate the extent of such minorityrepresentation. To this end, we generated artificial voter profiles and checked,for each of them, whether the following property holds, namely: whether it isthe case that, for each group V ′ of n/k voters for which, for each office, thereis at least one alternative approved by all voters of V ′ , it holds that there isat least one office for which the selected alternative is approved by at least onevoter from V ′ ; we view satisfying this property to mean that the voting rule doesnot completely disregard the preferences of big enough and cohesive minorities.We first generated completely random profiles, in particular, profiles in whicheach voter has some probability to approve each of the alternatives. For such we omit discussion on tie breaking as it technically clutters the presentation; say that wedo it arbitarily following some predefined order over all alternatives. p ∈ { . , . , . } , where p is theprobability of a voter to approve an alternative, after generating 1000 such ran-dom profiles, in each of them the property mentioned above is indeed satisfied.We then generated profiles differently, in particular, we pick a group of n/k voters, set all of them to approve some randomly-chosen alternative for eachoffice, and generate all remaining n − n/k voters randomly as described above.For this distribution of profiles, again, we get 100% satisfaction of the propertydescribed above for 1000 profiles, each with 9 offices with 9 alternatives in each,and 27 voters, for p ∈ { . , . , . } .While, indeed, there might be profiles for which SPAV does not satisfy theproportionality property described above, we view this preliminary experimentalevaluation as demonstrating that in practice there is reason to believe that SPAVwould indeed not violate the property, thus providing sufficient representationto big-enough cohesive minorities. We briefly discuss how to deal with candidates who are selected to an office butdecline to serve in the office: In particular, assume that, for a given instance,there is a candidate c that is selected to some office A j as a winner; however,when the day comes, c refuses to populate the j th office (say, e.g., that c acceptsa different career).A simple solution would be to simply run the aggregation method again, afterremoving c from the election. However, when using SPAV, it might be the casethat, as a result, other offices will get different candidates as winners. As thismight be unacceptable (as it means, e.g., that if the foreign minister declines toaccept then we change the environmental minister), we offer a different option,as follows.In particular, we can keep the other k − c fromthe election, and run a single further iteration of SPAV, resulting in a differentcandidate to be selected for the office that c was originally elected for. Thisensures that the other winners are kept as they were, while the weights of allvoters are calculated properly, and a different candidate is being selected forthat office. An essential property of a voting rule is that it can be easily explained tothe voter. Another important property is that its realization does not requiretrusting external elements (e.g., hardware and software). Fortunately, SPAV wasinvented before computers and hence must have been realized initially withoutthem.For completeness, clarity, and ease of implementation, we describe here a5encil-and-paper realization of our SPAV protocol for electing the executivebranch, including for determining replacements for declined candidates.The basic process is as follows:1. Before the vote commences, there is a finite list of candidates and a finitelist of voters. Each candidate name is associated with one office.2. During the vote, every voter writes a list of names on a note, places thenote in an envelope and then in the ballot box.3. All envelopes are collected and opened. If there is a limit on the numberof names a voter can vote for, then all excess names on a note, as well asnames of non-candidates, are stricken with X’s. If any name in the noteis stricken with a line, then it is stricken again with X’s.4. The weight of a name in a note is 1 / ( k + 1), where k is the number ofnames stricken with a line in the note, if the name appears in the note,and zero otherwise.5. Before vote counting commenced, all offices are vacant and no name onany note is stricken with a line.6. Counting proceeds in rounds until all offices are occupied (or no vacantoffice has a candidate named in a note) as follows: In each round, thecombined weight of each name in all notes is computed. The highest-weighted name for a vacant office is elected, occupies its office, and thename is stricken with a line from each note it appears in.This completes the description of the basic voting process. In case an electedminister cannot fill her office, a replacement is needed (as described in Section 6.To realize the concept described there (in Section 6), the following simple pro-cedure is followed: The name of the declining minister-elect is stricken from allnotes with X’s, the highest-weight candidate for the vacated office is elected,and her name is stricken with a line from all notes. We have described the setting of selecting the executive branch via direct democ-racy. For this setting we suggest the use of an adaptation of SPAV and show,via computer-based simulations, that it indeed does not disregard minorities inmany cases.
References [1] Haris Aziz. A rule for committee selection with soft diversity constraints.
Group Decis. Negot. , 28(6):1193–1200, 2019.62] Haris Aziz and Barton E Lee. Sub-committee approval voting and general-ized justified representation axioms. In
Proceedings of the 2018 AAAI/ACMConference on AI, Ethics, and Society , pages 3–9, 2018.[3] Niclas Boehmer, Robert Bredereck, Piotr Faliszewski, Andrzej Kaczmar-czyk, and Rolf Niedermeier. Line-up elections: Parallel voting with sharedcandidate pool. In
In proceedings of SAGT ’21 , 2021. To appear.[4] Vincent Conitzer, Rupert Freeman, and Nisarg Shah. Fair public decisionmaking. In
Proceedings of EC ’17 , pages 629–646, 2017.[5] P. Faliszewski, P. Skowron, A. Slinko, and N. Talmon. Multiwinner voting:A new challenge for social choice theory. In U. Endriss, editor,
Trends inComputational Social Choice . AI Access Foundation, 2017.[6] Rani Izsak. Working together: Committee selection and the supermodulardegree. In