flexible representative democracy: an introduction with ... · flexible representative democracy:...

Flexible Representative Democracy: An Introduction with Binary Issues

Ben Abramowitz1 and Nicholas Mattei21Rensselaer Polytechnic Institute, Troy, NY, USA

2Tulane University, New Orleans, LA, [email protected], [email protected]

AbstractWe introduce Flexible Representative Democracy(FRD), a novel hybrid of Representative Democ-racy (RD) and Direct Democracy (DD), in whichvoters can alter the issue-dependent weights of aset of elected representatives. In line with the liter-ature on Interactive Democracy, our model allowsthe voters to actively determine the degree to whichthe system is direct versus representative. How-ever, unlike Liquid Democracy, FRD uses strictlynon-transitive delegations, making delegation cy-cles impossible, preserving privacy and anonymity,and maintaining a fixed set of accountable electedrepresentatives. We present FRD and analyze it us-ing a computational approach with issues that areindependent, binary, and symmetric; we comparethe outcomes of various democratic systems us-ing Direct Democracy with majority voting and fullparticipation as an ideal baseline. We find throughtheoretical and empirical analysis that FRD canyield significant improvements over RD for emu-lating DD with full participation.

1 IntroductionSince the Athenian Ecclesia in 595 BCE Direct Democ-racy (DD) as an ideal collective decision making scheme hasloomed large in the western imagination [Dunn, 1995]. WhileDD may be desirable it becomes impractical at scale as itplaces too much burden on individual decisions makers: ev-eryone must be well-informed on every issue and available tovote [Green-Armytage, 2015]. In addition to the attention re-quirements, voters are also required to know and be able to ar-ticulate their preferences at the time of every vote. While pref-erences and preference learning are large research areas inAI [Domshlak et al., 2011] every voter may not have enoughknowledge, information, time, energy, or incentive to partici-pate, particularly when issues are numerous or complex.

Given the prohibitive costs of implementing a large-scaleDD in both human and agent societies, we often resort to rep-resentation, relying upon a set of proxies to decide on thevoters’ behalf. Countries have parliaments, companies haveelected boards, and groups of agents select leaders to rep-resent them [Yu et al., 2010]. Sets of representatives have

been used in many contexts and disciplines to reduce thecomputation and communication burden of decision makers.The Computational Social Choice (COMSOC) community[Brandt et al., 2016] has produced a large body of researchon how to select and weight representatives. Indeed, usingmulti-winner voting [Skowron et al., 2016], we can view thewinners as a set of exemplars that may be used to decide somedownstream application.Often it is beneficial to elect fixedcommittees which meet certain axiomatic criteria. For exam-ple, committees should be proportional and have justified rep-resentation of the voters [Aziz et al., 2017]. Intuitively, thesedifficulties in electing committees carry through to the set-ting of Representative Democracy (RD) where the commit-tee makes decisions in the interest of the voters/agents whoelect them. This setting was studied by Skowron [2015] whoproved that when we want to optimize for the sum of voterswho are represented on each issue the k-Median rule optimal.

Since DD can be impractical and RD comes with inher-ent tradeoffs and limitations, hybridizations of the two havearisen under the umbrella of Interactive Democracy. Coupledwith modern communication technologies, a large number ofproposed democratic decision making systems have been pro-posed, and Interactive Democracy has become an importantarea of research and application for AI [Brill, 2018]. Perhapsthe most popular version of this today is Liquid Democracy(LD); which has received significant attention in the politicalscience [Green-Armytage, 2015], AI [Kahng et al., 2018] andagents communities [Brill and Talmon, 2018], and has beenimplemented in both corporate [Hardt and Lopes, 2015] andpolitical settings [Blum and Zuber, 2016].

In contrast to existing proposals, our model of FlexibleRepresentative Democracy (FRD) maintains a set of expertrepresentatives while allowing voters to guarantee their ownrepresentation without raising the voters’ minimum requiredburden. In an FRD voters elect a set of representatives toserve a term during which they decide the outcomes over aset of issues. Each voter, by default, allocates a fraction oftheir voting power to each member of the committee. If thisallocation is uniform and we stop here, we are left with thetraditional model of RD where each representative has equalpower. However, for each issue under consideration in FRD,the voters may deviate from this default by delegating theirvoting power to any subset of the committee. If all voters usetheir option to delegate on each issue, as long as there is at

Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence (IJCAI-19)

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least one representative who agrees with each voter’s view,the outcome perfectly recovers DD. Voters have both the elec-tion and the flexible delegation option as tools for achievingrepresentation and holding representatives accountable.

In an FRD, voters have great flexibility in determining howthey are represented and the mandated disclosure of represen-tatives’ votes guarantees that an attentive voter can be fullyinformed about how their voting power will be and was used.For example, the day after the election an inattentive votermight choose a few elected representatives they trust, appor-tion the power of their vote to these few for all future issuesand pay no attention until the next election. A more attentivevoter might alter their allocations on an issue-by-issue basisas issues arise, reacting to representatives’ votes. In general,voters determine the granularity with which they privately ex-press their preferences over issues via the representatives. Inaddition, in an FRD voters may or may not be permitted tovote directly on the issues, depending upon the application.Contributions. We introduce Flexible RepresentativeDemocracy (FRD), a new model of Interactive Democracywhich transitions, at the discretion of the voters, betweenRD and DD. FRD solves standing issues in the literatureon Interactive Democracy including maintaining a fixed,elected committee to generate legislation, making delegationcycles impossible, and preserving voter anonymity. Weanalyze our model theoretically using independent, binary,symmetric issues. We show that electing an optimal set ofrepresentatives is hard for any large-scale RD no matter thevoting rule, thus motivating the use of flexible delegations.Thus, we demonstrate the theoretical ability of delegationsunder FRD to overcome the limitations of RD, providingempirical results demonstrating that FRD outperforms bothRD and Proxy Voting for representing the majority will.

2 Model and PreliminariesWe primarily consider three democratic decision systems: Di-rect Democracy (DD), Representative Democracy (RD), andour model: Flexible Representative Democracy (FRD). Givena set of voters V with preferences over the alternatives foreach issue in a set of issues S , we represent their collectivepreferences by a preference profile PVS . In DD, a decisionrule RS takes the PVS as input maps them to a set of out-comes over the issues ODD;RS(PVS)→ ODD. 1

By contrast, in RD voters’ preferences on the issues maynever be directly elicited. Rather, voters report their prefer-ences over a set of candidates C. We denote the collectivepreferences of the voters over the candidates by the elec-toral profile PVC . An election rule RE (i.e. multi-winnervoting rule) is then used to aggregate these preferences andselect a subset of candidates D to serve as representatives,RE(PVC) → D ⊆ C. In a standard RD, a decision rule isthen applied to the public preferences of the representativesover the issues PDS to determine the set of outcomes on allissues ORD; RS(PDS) → ORD. Clearly, RD may producedifferent outcomes than DD, and may leave accessible infor-mation about voter preferences unsolicited and unused.

1More generally, different decision rules could be used on differ-ent issues within a single DD, RD, or FRD.

In FRD, as with RD, the voters elect a set of representa-tives RE(PVC) → D ⊆ C. However, for every issue, a divis-ible unit of voting power is given to each voter rather than toeach representative. Automatically after the election, the vot-ers’ issue-specific votes are distributed among the representa-tives according to some default distribution mechanism. Sub-sequently, every voter has the option to alter how their votingpower is assigned to the representatives and may change thisfor any subset issues at once or on an issue-by-issue basis. Werefer to this process of deviating from the default and activelyallocating voting power to representatives as delegation. Del-egations are not permanent and may be altered before an issueis decided. We let W i

jl represent the voting power allocatedby voter vj ∈ V to candidate cl ∈ C on issue si ∈ S , yield-ing a collective matrix of weights W . In FRD, a decision ruleRS is then applied to the representatives’ preferences takingthese weights into accountRS(W,PDS)→ OFRD. If votershave the option to vote directly on issues rather than havingtheir voting power only distributed to representatives (e.g.,more similar to LD), W and PDS can be augmented to allowvoters to “delegate” to themselves.

2.1 Model SpecificationWe restrict our attention to a simple type of FRD. Our ob-jective is to compare the extent to which RD and FRD canemulate DD, which we hold as “optimal.” We consider a set-ting with symmetric, binary issues so each issue si ∈ S hastwo possible outcomes Oi ∈ {0, 1}, and there are 2r possi-ble outcome vectors over |S| = r of the form O ∈ {0, 1}r.We assume all issues are independent, this is a simplifyingassumption that circumvents issues raised in judgment aggre-gation [List and Puppe, 2009], though an important directionfor future work. Without loss of generality, we label each ofthe alternatives preferred by the (weak) majority of voters 1and the other 0, breaking ties randomly (when N is even).Thus, the ideal majoritarian outcome over the issues is {1}r.

Each voter in the set of voters V = {v1, . . . , vN} has apreferred alternative vij ∈ {0, 1} for every issue si in the setof issues S = {s1, . . . , sr}. We let vector ~vj = {v1j , . . . , vrj}represent the preferred outcome of voter vj over the issues,resulting in the collective profile PVS = {~vj : vj ∈ V}. Sim-ilarly, when we have representatives, we represent the prefer-ence profile of the candidates as PCS = {~cl : cl ∈ C} wherecandidate cl has preferred outcome ~cl ∈ {0, 1}r.

We define the agreement between any two outcome vectorsO1,O2 of length r as L(O1,O2) = 1 − 1

r

∑ri=1 |Oi1 − Oi2|,

i.e., the fraction of issues for which the outcomes are thesame. When comparing to an ideal set of outcomes (i.e.ODD), this is the number of issues decided “correctly.” Wewill often refer to the agreement between a voter and candi-date L(~vj ,~cl) as well as the agreement between the outcomesof different democratic systems, i.e. L(ODD, ORD).

We consider three possible ways voters might express theirpreferences over the candidates: approvals, total orderings,and normalized weights. We make an assumption about thesepreferences to give RD the greatest chance of maximizingL(ODD,ORD): we assume each voters’ preferences over thecandidates are induced by their agreement. When voters sub-


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mit approval ballots, we assume vj approves of cl if and onlyif L(~vj ,~cl) > 1/2. When voters report total orderings (�j),we assume they order all candidates so that cl �j ch onlyif L(vj , cl) ≥ L(vj , ch) where ties are broken privately (e.g.randomly). When voters report their preferences as normal-ized weights, wlj = L(vj ,cl)/

∑ch∈C L(vj ,ch).

For DD, we only consider the simple majority rule asour decision rule RS as our issues are binary and sym-metric. For our representative systems we compare severalcommon, anonymous election rules RE with a fixed, oddcommittee size k, i.e., D ⊆ C where |D| = k. All rulesconsidered in our simulations are deterministic other thanrandomized tie-breaking. In the setting where voters sub-mit approval ballots, we consider Approval Voting and Re-weighted Approval Voting (AV, RAV). When voters submittheir preferences over candidates as total orderings, we con-sider Single-Transferrable Vote (STV), Borda, k-Median, andChamberlin-Courant (CC). When voters submit their ballotsas normalized weights over the candidates, we consider therule which selects the k candidates who receive the largesttotal weight. Lastly, we compare these rules to selecting krepresentatives uniformly at random from the candidates. Werefer the reader to Brandt et al. [2016] and Skowron [2015]for complete definitions. We assume that the preferences ofcandidates do not change before or after they are elected asrepresentatives. Without loss of generality, let the winningcandidates be the lowest k indexed {c1, . . . , ck} such that rep-resentative dl ∈ D is candidate cl and ~dl = ~cl.

Given a set D of k representatives, we want to evaluatethe capability of this set to represent the majority will of thevoters, i.e., recover the outcome of DD with full voter turnout.Hence, we define coverage and majority agreement as metricsto evaluate these systems. Let ki1 and ki0 represent the numberof representatives who prefer alternatives 1 and 0 on issue sirespectively, such that ki1 + ki0 = k.• The majority agreement on a set of issues is the frac-

tion of issues on which the majority of representativesagree with the majority of voters (ki1 >

k2 > ki0). This

quantity is 1r

∑i∈{1,...,r} L(OiRD,OiDD).

• The coverage of a set of issues is the fraction of issuescovered on which at least one representative agrees withthe voter majority (0 < ki1).

We allocate each voter one divisible vote for each inde-pendent issue, maintaining the principle of “one person, onevote.” By default, each voter’s unit of voting power is dis-tributed uniformly among the representatives on each issueso W i

l = N/k initially for each of the k candidates and r is-sues (exactly as in RD). However, various distributions fromthe literature on voting power [Shapley and Shubik, 1954;Banzhaf III, 1964] and Proxy Voting [Alger, 2006] are worthconsideration in the future. We do not consider abstentions byrepresentatives nor voter, i.e., a voter assigns less than a fullvote across representatives, and we do not permit voters tovote directly on the issues. Hence, the total voting power heldby the representatives remains N collectively for all issues.

For our purposes, the total weight W il assigned to repre-

sentative dl on issue si is the sum of the voting power theyreceive from default and delegation,

∑jW

ijl. Consequently,

the total weight assigned to representatives who agree withthe voter majority is Xi

1 =∑dil=1W

il . In this paper our

decision rule for FRD is weighted majority with random tiebreaking: Oi = 1 if Xi

1 > N/2, Oi = 0 if Xi1 < N/2, and

Oi = 1 with probability 1/2 if Xi1 = N/2.

We assume that all voters are incisive on all issues. A voteris incisive if they only delegate voting power to representa-tives who agree with their preferred alternative. Similarly, werefer to delegations as being incisive if they exhibit this prop-erty on an issue. We relax the assumption of incisive votersin our simulations and consider voters who delegate only totheir most preferred candidate(s) or divide their delegationevenly across their approved set. Relaxing these assumptionsare important directions for future work discussed in the fullversion of this paper [Abramowitz and Mattei, 2018].Example 1. Consider an FRD instance with issues s1 ands2, three voters, and three representatives. Below, the solidarrows from voter to representative indicate delegations, andany voter without an arrow defaults on that issue. The voterand representative preferences are given in the tables aboveand below the agents; both delegations are incisive.

v1 v2 v3

1 1 0

d1 d2 d3

1 1 02/3 2/3 5/3

vijvj

dl

dilW il

Issue s1

X11 = 4/3 < N/2

v1 v2 v3

1 1 0

d1 d2 d3

1 0 05/3 2/3 2/3

Issue s2

X21 = 5/3 > N/2

On issue s1, the representative majority agrees with the votermajority, so RD would yield O1

RD = 1 as desired. How-ever, since only the voter in the minority (v13 = 0) delegates,the weighted majority of representatives now decides the out-come in favor of the voter minority (X1

1 < N/2). Hence, FRDcan make the outcomes worse than RD as measured againstDD in some cases. This can occur if the number of voters inthe minority is large, the number of representatives who agreewith the voter minority is large, and the voters in the minoritydelegate at a higher rate than the voters in the majority.

On issue s2 the representative majority disagrees withthe voter majority so the RD outcome (without delegations)would be O1

RD = 0. Looking again at the figure we see thedelegations flip the result to what would be achieved by DD(X2

1 > N/2). Hence, FRD can improve the outcomes over RDas measured against DD. Fortunately, for both issues, if anytwo or all of the voters delegate incisively, the outcome willalways agree with the voter majority.

3 Related WorkMiller [1969], inspired by Tullock [1967] and the idea ofshareholder proxy voting, suggested an interactive demo-cratic system for legislation which could take place at scaleusing computers. Miller lamented the lack of flexibility in tra-ditional Representative Democracy and sought to remedy this


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using a dynamic system of proxies, although he admitted thiswas not conducive to creating legislation. Soon after, [Shu-bik, 1970] warned that electronic systems may accelerate thelegislative process in undesirable ways and suggested holdingevery referendum twice to guarantee time for sufficient publicdeliberation. Our use of a fixed, elected set of representativesanswers Miller’s question of how to produce legislation, andrather than holding redundant referenda we make representa-tives’ votes public and then give the voters sufficient time todeliberate and alter their delegations.

Before the dawn of the internet, [Tullock, 1992] revisitedhis ideas from [Tullock, 1967] in a proposal that motivatesthe default and delegation mechanisms in FRD. The notionof the default distribution is similar to that proposed by Al-ger [2006], which suggests that the weights of representa-tives be based on the preferences of voters expressed in theelection, but these weights are fixed during their term. Bycontrast, in FRD the weight of each representative on eachissue is not strictly determined by the election. Cohensius etal. [2017] took an analytical approach to studying a ProxyVoting model close to that of Alger [2006] for decision mak-ing with no election, infinite voters, spatial preferences, andthat agents lie in a metric space.

The hallmark of interactive democracies like FRD and Liq-uid Democracy is that rather than adjudicating whether a di-rect or representative system is better for achieving some ob-jective and asserting it by fiat, the extent to which the systemis direct or representative is itself a function of the “will of thevoters”. Currently, Liquid Democracy is the most well-knownand well-studied form of Interactive Democracy, and has beenstudied from an algorithmic perspective as a decision-makingprocess in the AI and COMSOC literature [Brill and Talmon,2018; Kahng et al., 2018; Bloembergen et al., 2018; Christoffand Grossi, 2017] and elsewhere [Green-Armytage, 2015;Blum and Zuber, 2016; Brill, 2018; Hardt and Lopes, 2015].2Unlike Liquid Democracy, FRD does not allow transitive del-egations nor delegations to another voter, thereby violatingthe second axiom proposed by Green-Armytage [2015]. Frac-tional delegations in FRD serve a similar function to that ofthe virtual committees proposed by Green-Armytage [2015],although in theory FRD could incorporate virtual committeesas well as other mechanisms for delegating voting power.

The design of FRD is largely based on work in proba-bilistic voting, binary aggregation, statistical decision the-ory, and computational social choice. In particular, workon the optimal weighting of experts [Baharad et al., 2012;Nitzan and Paroush, 2017; Grofman and Feld, 1983; Nitzanand Paroush, 1982; Ben-Yashar and Nitzan, 1997], the Con-dorcet Jury Theorem [Grofman et al., 1983], variable elec-torates [Feld and Grofman, 1984; Smith, 1973; Paroush andKarotkin, 1989], and optimal committee sizes [Auriol andGary-Bobo, 2012; Karotkin and Paroush, 2003; Magdon-Ismail and Xia, 2018]. In FRD, one can view the voter del-egations as a pseudo-tie breaking mechanism for the rep-resentatives or, conversely, see the default distribution as away to dampen the variance in the outcome in that may oc-

2Also see B. Ford, Delegative Democracy at http://brynosaurus.com/deleg/deleg.pdf.

cur DD when the set of participating voters is small or bi-ased. Another view is that electing representatives is analo-gous to a compression algorithm [Rodriguez and Steinbock,2004], which is the algorithmic version of John Adams’s al-leged intuition that the representatives should be a micro-cosm of the population (taken from [Alger, 2006]). In thisview, the delegations in FRD are analogous to a decompres-sion mechanism where a higher delegation rate reduces the“loss” of representation. Our evaluations are similar to thoseof Skowron [2015], however, in their model the quality of thecommittee is measured as the sum of the voter proportion be-ing represented for each issue, while we focus only on thetotal number of issues correct according to DD.

4 Difficulties of Representative DemocracyElecting good committees is hard. In fact, electing a set ofrepresentatives which maximizes majority agreement on bi-nary issues is NP-Hard even if we know the view of everyvoter on every issue. The easier problem of maximizing cov-erage is also NP-Hard. We refer to the problems of selectingk representatives to maximize coverage and majority agree-ment as Max k-Coverage and Max k-Majority Agreement, re-spectively. Note that if the majority view of the voters wereknown, coverage could be approximated deterministically inpolynomial time within a factor of 1 − 1/e by a greedy al-gorithm, and this bound is tight [Feige, 1998]. The proofs forour complexity results can be found in the full version of thispaper [Abramowitz and Mattei, 2018].Theorem 2. Max k-Coverage: If the candidates’ preferencesand the outcome preferred by the majority of voters are knownfor every issue, selecting the subset of candidates that maxi-mizes the number of issues covered is NP-hard.

Proof. We show a reduction from the NP-Hard problem ofMAX K-COVER [Garey and Johnson, 1979; Feige, 1998]to Max k-Coverage. In MAX K-COVER, given a set U ={x1, . . . , xr} of r points, a collection Z = {z1, . . . , zm} ofsubsets of U , and an integer k we must select k subsets fromZ such that their union has maximum cardinality. Given aninstance (U,Z, k) of MAX K-COVER we create an instance ofMax k-Coverage as follows. For every point xi ∈ U create anissue si and for every subset zl ∈ Z create a candidate cl. Forall points xi and subsets zl, if xi ∈ zl then let cil = 1, oth-erwise let cil = 0. Let k be the number of representatives wewill elect. There is a one-to-one correspondence between thenumber of issues covered by our k representatives and the car-dinality of the corresponding subsets in the original instance;any set of k candidates that maximizes coverage correspondsexactly to a collection of k subsets in our MAX K-COVER in-stance whose union has maximum cardinality.

The hardness of achieving coverage even in the unrealisticcase where the voter majority is known for every issue sup-ports the idea that voters should be able to vote directly whenthe issues are binary to ensure the majority will is always re-coverable. Or, at the very least, a dummy candidate with nodefault power should be created automatically on any binaryissue for which the representatives are unanimous. Note thatadding such a candidate automatically can only improve the


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outcome, as it can only change the outcome of the majorityof voters delegate to the dummy.Theorem 3. Max k-Agreement: If the candidates’ prefer-ences and the outcome preferred by the majority of voters areknown for every issue, selecting the subset of k candidateswhose majority agrees with the voter majority on the greatestnumber of issues is NP-hard.

Proof. We prove the hardness of Max k-Majority Agree-ment by polynomial-time reduction from our problem ofMax k-Coverage. Given an instance of Max k-Coveragewith input (S = {s1, . . . , sr}, C = {c1, . . . , cm},PCS ={~c1, . . . , ~cm}, k) we construct an instance of Max k-MajorityAgreement with input (S, C, ˜PCS , k) as follows. Create a setof binary issues S = {s1, . . . , s2r+1} and a set of candi-dates C = {c1, . . . , cm+k+1}. Let k = 2k + 1. For l ≤ m,cil = cil for i ≤ r and cil = 0 for r < i ≤ 2r + 1. Form < l < m+ k+1, cil = 1 for all i. And for l = m+ k+1,cil = 0 for i ≤ r and cil = 1 for r < i ≤ 2r + 1.D agrees with the voter majority on issues

{sr+1, . . . , s2r+1} iff D contains {cm+1, . . . , cm+k+1},because this is the only way at least k + 1 out of the2k + 1 representatives can agree with the voter majorityon any of these issues. Therefore any set D ⊆ C of 2k + 1representatives which maximizes majority agreement mustcontain {cm+1, . . . , cm+k+1}.

Selecting candidates {cm+1, . . . , cm+k+1} provides ex-actly k representatives who agree with the voter majority onissues {s1, . . . , sr}. Since we are selecting 2k + 1 represen-tatives in total, on any of these first r issues we need only1 more representative who agrees with the voter majorityon each issue to achieve majority agreement. Therefore, se-lecting k additional representatives from {c1, . . . , cm} whichmaximize coverage over issues {s1, . . . , sr}, maximizes themajority agreement of the 2k + 1 representatives over S .These k representatives are a one-to-one correspondence tothe k representatives in the solution to our original Max k-Coverage problem.

Worse yet, even for small instances where the problemis computationally tractable, there are pathological examplesfor which truthful voters whose derived preferences over thecandidates are perfectly consistent with their preferences overthe issues will elect horrible representatives.Theorem 4. No Condorcet-consistent election rule using ap-provals or total orderings can approximate Max k-MajorityAgreement.

Our proof for Theorem 4 can be found in the full version ofthis paper [Abramowitz and Mattei, 2018] and is derived froman example found in [Anscombe, 1976] with 11 voters and 11issues. This example is particularly pathological, because theworst candidate gets elected over the best candidate.

The difficulty of achieving majority agreement between thevoters and candidates using the election alone - reinforcedby our simulations below - and the existence of pathologicalcases even at small scales motivate the use of flexible delega-tions. However, as we saw from Example 1, delegations are

not guaranteed to improve outcomes. We explore the poten-tial benefits of such delegations in Section 5.

Simulation ResultsWe investigate the properties of coverage and majority agree-ment as a function of the numbers of candidates, issues, andcommittee size. In all our simulations, for all issues si we letvij = 1 and cil = 1 with probability 1

2 for all voters and can-didates. This means that all candidates and voters come fromthe same populations, i.e., that they are both drawn from thesame distribution; relaxing this assumption is an interestingdirection for future work. In all of our runs, coverage was1.0 for all combinations, hence we omit it from the graphsin Figure 1. For all simulations we perform 50 iterations ateach datapoint and plot the mean (σ2 ≤ 0.002). For a first setof simulations we included rules that have NP-hard winnerdetermination problems: CC and k-Median [Skowron, 2015].We implemented these rules in Gurobi 8.1 and used a serverwith 16 cores and 32GB of memory; taking nearly 24 hours togenerate results for a smaller setting (|C| up to 17). We foundthat both CC and k-Median are outperformed by Weights,STV, and AV at all settings and hence, we drop these rulesin our larger analysis.

For our larger simulations we hold |V| = 501 fixed as wedid not observe a strong dependence on the number of votersas long as it was sufficiently larger than the number of candi-dates. Turning first to Figure 1a we hold |C| = 60, |k| = 21and vary |S| ∈ {15, . . . , 150} in steps of 15. We see thatfor a small number of issues the AV, RAV, and the weightedvoting rule can be expected to select a committee with veryhigh majority agreement (0.8). However, as we add issues tothe docket, the voting rules seem to converge around 0.6. InFigure 1b we fix |k| = 21, |S| = 150 and vary the numberof candidates between |C| ∈ {21, . . . , 100} in steps of 5. Weobserve again that AV, RAV, and weighted voting are the bestfollowed closely by STV. As we increase the number of can-didates it is possible for rules to select committees with highermajority agreement, but this number does not climb above65%. Finally, in Figure 1c we hold |C| = 100, |S| = 150and vary |k|. These simulations reinforce the idea that elect-ing an ideal committee, i.e., one that obtains perfect majorityagreement, is a hard problem. In the next section we will ex-plore how FRD can outperform RD and its dependence on thecomparative use of delegations by the majority and minority.

5 Benefits of FlexibilityThe flexibility of issue-specific delegations is the motivatingfeature of FRD. We look at basic features of FRD in a deter-ministic setting before considering probabilistic delegations.

5.1 Deterministic DelegationA voter’s delegation is incisive if the voter only delegates torepresentatives who agree with them on that issue. Observethat if the representatives are unanimous with no direct vot-ing or dummy candidates to guarantee coverage only one out-come is possible, but as long as there is some dissent in thecommittee FRD can take advantage and return decision mak-ing power to the voters.


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0 25 50 75 100 125 150Number of Issues, jSj

0.4

0.5

0.6

0.7

0.8

0.9

1.0M

ajor

ity A

gree

men

tAV AgreementBorda AgreementSTV AgreementRAV AgreementRandom AgreementWeights Agreement

(a) Vary |S|; fix k = 21, |C| = 60.

20 40 60 80 100Number of Candidates, jCj

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Maj

ority

Agr

eem

ent

AV AgreementBorda AgreementSTV AgreementRAV AgreementRandom AgreementWeights Agreement

(b) Vary |C|; fix k = 21, |S| = 150.

0 20 40 60 80 100Committee Size, k

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Maj

ority

Agr

eem

ent

AV AgreementBorda AgreementSTV AgreementRAV AgreementRandom AgreementWeights Agreement

(c) Vary k; fix |C| = 100, |S| = 150.

Figure 1: Majority Agreement of the elected committee as a function of the number of issues, number of candidates, and committee size.Across all treatments the weighted voting, approval voting, and RAV select the best committees as measured by majority agreement.

We denote by N i1 + N i

0 = N the numbers of voters andby ki0 + ki1 = k the numbers of candidates who agree anddisagree with the voter majority on issue si, respectively.We have labeled the majority view of the voters as 1, so∀si ∈ S : N i

1 ≥ N/2 ≥ N i0. The overall outcome of any

resolute democratic process over this set of issues is an out-come vector O = {O1, . . . ,Or} ∈ {0, 1}r, and our idealoutcome is {1}r. Treating all issues equally and indepen-dently, we seek to maximize

∑si∈S Oi. Let λi1 and λi0 be the

number of voters who delegate from the majority and minor-ity respectively on issue si, assuming they delegate incisively.We drop the i superscript below as we have a single issue.Proposition 5. If all delegations are incisive and the repre-sentatives are not unanimous (0 < k1 < k), the outcomeis guaranteed to agree with the voter majority if the num-ber of voters in the majority who delegate (λ1) is greaterthan Nk−2N1k1

2(k−k1) and the outcome will favor the minority if

λ0 >kλ1+(N−λ1)(2k1−k)

2k1.

5.2 Probabilistic DelegationHere, instead of assuming that some fractions (α1, α0) ofvoters delegate we investigate what happens if each voterchooses to delegate with some fixed individual probability.These results gives us an idea of how motivated or attentivevoters must be to improve the outcome of FRD over RD. Weassume that all voter and candidate preferences are indepen-dent for all issues.

As all issues are independent, we consider a single issue.Suppose each voter vj ∈ V chooses to delegate (deviatefrom the default) with independent probability pj and thatall delegations are incisive. Let xj ∈ [0, 1] be the amountof power voter vj assigns to candidates who agree with thevoter majority (cl = 1), either by delegation or default. Ifvj defaults then xj = k1/k, if vj delegates incisively and isin the voter majority (vij = 1) then xj = 1, and if vj dele-gates incisively and is in the voter minority then xj = 0. LetX1 =

∑vj∈V xj be the total power assigned to these can-

didates via both delegation and default. Let µ = E[X1] =∑vj=1(pj + (1 − pj)k1/k) +

∑vj=0(1 − pj)k1/k be the ex-

pected value of the total power assigned to representatives

who agree with the voter majority.Theorem 6. Consider an FRD with an odd number of vot-ers N , odd committee size k, and only incisive delegations.Suppose each voter vj ∈ V delegates with probability pjon each issue such that µ > N/2. Then the probability thatthe outcome agrees with the voter majority is bounded byP (O = 1) ≥ 1− e−(N−2µ)2/4N .

Proof. Recall that xj ∈ [0, 1] is the amount of voting powervoter vj assigns to candidates who agree with the voter major-ity on an issue andX1 =

∑vj∈V xj . Given some tie breaking

rule, we have that P (O = 1) = P (X1 > N/2) + P (O =1|X1 = N/2) · P (X1 = N/2). First we show that P (X1 =N/2) = 0, then we give a lower bound for P (X1 > N/2).

Lemma 1. If N is odd, k is odd, and all delegations are inci-sive, then no ties are possible.

Proof. Let x′j = k · xj where xj ∈ [0, 1] is the amountof weight (voting power) voter vj assigns to candidates whoagree with the voter majority on an issue via default or dele-gation. If vj defaults then x′j = k1, if vj delegates incisivelyand is in the voter majority (vij = 1) then x′j = k, and if vjdelegates incisively and is in the voter minority then x′j = 0.Therefore, ∀j : x′j ∈ {0, k1, k}. Let X ′1 =

∑vj∈V x

′j and

X ′0 =∑vj∈V(k − x′j). Then X ′1, X

′0 are non-negative in-

tegers and X ′1 + X ′0 = kN . Since kN is odd, it must bethat X ′1 and X ′0 have opposite parity and so they cannot beequal. Therefore X1 = X′

1/k 6= X0 = X′0/k, meaning the total

amounts of weight delegated to the representatives on eitherside of the issue cannot be equal, so no ties may occur.

Given that no ties are possible, we have that P (O =1) = P (X1 > N/2). Remember that X1 =

∑vj∈V xj

where xj is the total weight that vj delegates to represen-tatives who agree with the voter majority. If vij = 1 thenE[xj ] = pj + (1 − pj)

k1k , else if vij = 0 then E[xj ] =

(1 − pj)k1k . Let µ = E[X1] be the expected total weight as-signed to representatives who agrees with the voter majority,then µ =

∑vij=1(pj + (1− pj)k1k ) +

∑vij=0(1− pj)

k1k . We


8

0 20 40 60 80 100Delegation Rate (®)

0.5

0.6

0.7

0.8

0.9

1.0

Maj

ority

Agr

eem

ent

FRD: IncisiveFRD: Best RepFRD: Best-3 RepsFRD: ApproveRD

Figure 2: Majority agreement of the various systems with DD whenthe committee is selected using weighted voting.

now use the fact that P (X1 > N/2) = 1 − P (X1 ≤ N/2).Let δ = (2µ − N)/2µ. If µ > N/2, then δ > 0. This al-lows us to apply a Chernoff bound to derive our lower bound,P (X1 > N/2) = 1 − P (X1 ≤ N/2) = 1 − P (X1 ≤(1− δ)µ) ≥ 1− e−δ2µ2/N = 1− e−(2µ−N)2/4N .

This bound depends on the condition that µ > N/2. Thisassumption is only violated in rare cases when the minor-ity is large, the majority delegates sparingly, and the rep-resentatives are near evenly split. As an increasing numberof voters delegate incisively, we expect µ → N1 > N/2regardless of k1. Naturally, as the delegation rate increases(α → 1), we observe our lower bound approach the ideal1 − e−(2µ−N)2/4N → 1. To observe µ ≤ N/2, the votermajority cannot be too large compared to the voter minority,k1 must be smaller than or somewhat close to k0, and/or thevoters in the majority must be considerably more apathetictowards delegation than voters in the minority.

Tighter bounds may be achieved when the delegation prob-abilities are assumed to come from a particular distribution, orthe preferences of voters and candidates are assumed to comefrom different distributions. It is an interesting open questionto see how the expected outcome is effected when voters havedifferent delegation probability distributions.

5.3 Simulated DelegationsWe investigate the effect of the delegation rate α on the abilityof different systems to recover the DD outcome, i.e., majorityagreement. We use the same model to generate candidatesand voter preferences as used in Section 4. For our simulateddelegations we create instances with |V| = 301, |C| = 60,|S| = 150, and k = 21. We vary α ∈ {0, 1.0} in incrementsof 0.01 and for each setting of α we run 50 iterations. We plotmeans in Figure 2 (σ2 ≤ 0.002).

In Figure 2 we can see the majority agreement for theweighted voting committee selection rule for several delega-tion types and delegation rates. Note that a majority agree-ment of 1.0 means that the outcome of the system is iden-tical to DD. We compare RD with four different delegationschemes: (1) in Approve voters delegate to the representa-tives of whom they approve and do not update; in Best Repvoters delegate to their single most preferred representative

and do not update; in Best-3 Rep voters delegate equally totheir three most preferred members of the committee and donot update; and finally in Incisive where voters delegate to asingle representative with whom they agree per issue.

Most surprising is how little delegations that are not ac-tive and incisive help emulate DD. The Approve system isperhaps closest to the proposal of Proxy Voting espoused byMiller [1969] but does not improve RD in a meaningful way.Hence, we can see that the issue-specific flexibility FRD al-lows can be effectively used to achieve DD outcomes. An-other striking result in Figure 2 is how drastically FRD canimprove majority agreement over RD when voters are highlyattentive. However, the high delegation rates required suggestthat FRD may be burdensome to voters; exploring other del-egation models is an important future direction.

6 ConclusionWe introduced a novel system called FRD which transitionssmoothly, at the discretion of the voters, between direct andrepresentative democracy. We have shown theoretically andempirically that FRD has the potential to overcome the short-comings of other systems such as RD, LD, and Proxy Vot-ing. An important point to remember is that in FRD, unlikein LD: delegations are optional, and not an additional burdenimposed on the system or voters; in contrast to LD, voters inFRD have greater certainty about how their vote will be castahead of time; and delegation cycles are not possible. Further-more, FRD maintains a fixed, elected set of accountable rep-resentatives to produce legislation and hold public debates.This committee of representatives does not need to expandto guarantee proportional or justified representation. As Fig-ure 2 demonstrates, the default distribution of voting powerin FRD prevents a small subset of active voters from entirelydetermining the outcome as can happen in DD.

Our analysis makes best-case assumptions for RD: we as-sume all issues are known before the election, all voters par-ticipate in the election, candidate and representative prefer-ences do not change, and voter preferences over candidatesare consistent with their preferences over the issues. Relax-ing any of these assumptions strengthens the argument forenabling flexible, issue-specific delegations. Flexible delega-tion also minimizes the role that election rule plays in theoutcome. Extensions and future work including more real-istic voter models, different candidate models, incorporatingaspects of judgement aggregation including removing the in-dependence between issues, and different committee decisionrules, to name just a few, see Abramowitz and Mattei [2018].

AcknowledgmentsMuch of this work was completed while both authors wereworking at IBM Research AI, Yorktown Heights, NY. Wewould like to thank Nick Dalmasso and the anonymous re-viewers for their guidance and input.

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