1 Introduction
Since the Athenian Ecclesia in 595 BCE Direct Democracy (DD) as an ideal collective decision making scheme has loomed large in the western imagination [18]. While DD may be desirable it becomes impractical at scale because it places too much burden on individual decisions makers: everyone must be wellinformed on every issue and always available to vote [25]. In addition to the attention requirements, voters are also required to know and be able to articulate their preferences at the time of every vote. While preferences and preference learning are large research areas in AI [17, 23], every voter may not have enough knowledge, information, time, or energy to participate, particularly when issues are complex.
Given the prohibitive costs of implementing a largescale Direct Democracy in both human and agent societies, we often resort to forms of representation, relying upon a set of proxies to decide on the voters’ behalf. Countries have parliaments, companies have shareholders, and even groups of agents select leaders to represent them [46]. Sets of representatives have been used in many contexts and disciplines to reduce the computation and communication burden of decision makers. Within computer science, many applications face the task of selecting representatives for downstream decision making. In portfolio selection a particular set of algorithms and hyperparameters are selected from a large pool of candidates and then used as representatives for later problems [31], in multiagent systems the role assignment problem uses distributed voting to decide on tasks for agents [47], and in group recommendation settings this can correspond to picking a set of experts to later make decisions. The COMSOC community [11] has produced a large body of research on how to select and weight representatives. Indeed, using multiwinner voting [40], we can view the winners as a set of exemplars that may be used to decide some downstream application – e.g., we select a set of points in space and then aggregate these points (votes) over the set.
Often it is beneficial to elect fixed committees which meet certain axiomatic criteria. For example, committees should be proportional and have justified representation of the voters [4]. Intuitively, these difficulties in electing committees carry through to the setting of Representative Democracy (RD) where the committee makes decisions in the interest of the voters/agents who elect them [41].
Since DD is impractical and RD comes with inherent tradeoffs and limitations, hybridizations of the two have arisen under the umbrella of interactive democracy. This idea, coupled with modern communication technologies, has spawned a large number of proposed democratic decision making systems, and interactive democracy has become an important area of research and application for AI [12]. Perhaps the most popular version of this today is Liquid Democracy. Liquid Democracy has received significant attention in the political science [25], AI [29] and agents communities [13], and has been implemented in both corporate [28] and political settings [10, 7].
In contrast to existing interactive democracy proposals, our model of Flexible Representative Democracy (FRD) maintains a set of expert representatives while allowing voters to guarantee their own representation without raising the minimum required burden on them. In an FRD voters elect a set of representatives to serve a term during which they decide the outcomes over a set of issues. Each voter, by default, allocates a fraction of their voting power to each member of the committee. If this allocation is uniform and we stop here, we are left with a traditional model of RD where each representative has equal power. However, for each issue under consideration in FRD, the voters can deviate from this default by delegating their voting power to any subset the committee. If all voters use their option to delegate on each issue, as long as there is at least one representative who agrees with each voter’s view, the outcome can become exactly that of DD. Voters have both the election and the flexible delegation option as tools for achieving representation and holding representatives accountable. The degree to which the ideal of Direct Democracy is achieved under FRD is determine by the granularity with which the voters choose to express their preferences.
Contributions
We introduce Flexible Representative Democracy (FRD), a new model of Interactive Democracy which smoothly transitions, at the discretion of the voters, between Representative Democracy and Direct Democracy. Our proposal for FRD solves standing issues in the literature on interactive democracy including maintaining a fixed, elected committee to generate legislation and making delegation cycles impossible, all while increasing voter representation. We analyze our model in decision making scenarios involving binary, symmetric issues and (1) show that electing an optimal set of representatives is hard for any largescale Representative Democracy that uses a multiwinner voting rule, (2) investigate the performance of various polynomialtime multiwinner voting rules to select committees, (3) demonstrate the theoretical ability of issuespecific delegations under FRD to overcome the limitations of RD, and (4) provide empirical results demonstrating that FRD outperforms both RD and Proxy Voting for representing the will of the voters.
2 The Model
A Flexible Representative Democracy (FRD) is a multiagent decision system over a set of issues and is characterized by: a set of issues , a set of voters , a set of candidates , an election rule , a default distribution, a weighting function, and a decision rule .
There are four main components to FRD: the election, the default distribution of voting power, the delegation phase, and the weighted decision rule. Initially, the set of voters elect a set of representatives from a set of candidates via an election rule (multiwinner voting rule) [48, 19]. These representatives serve a term during which they will publicly debate and vote upon the set of issues. For each issue, every voter has one divisible unit of voting power. By default after the election, the voting power of each voter on each issue is distributed among the representatives. For each issue, the representatives have a deadline to cast their ballots, after which voters have a suitable time period to further alter how their voting power is distributed to the representatives before the decision rule is applied. The outcome of each issue is decided based on the weighted votes of the representatives, whose weights are a function of the issuedependent voting power assigned to them through delegation and/or default.
In an FRD, voters have great flexibility in determining how they are represented and the public votes of the representatives guarantee that an attentive voter can be fully informed about how their voting power will be and was used. For example, the day after the election, an inattentive voter might choose a few elected representatives they trust and delegate the power of their vote to these few for all the issues and pay no attention until the next election. A more attentive voter might alter their power delegations on an issuebyissue basis as the issues arise, reacting to the votes of the representatives. In general, voters determine the granularity with which they privately express their preferences over issues through representatives and may change at any time. For ease of exposition, our analysis will assume that the voters who delegate on an issue only do so after becoming informed of the representatives’ fixed votes and these delegating voters are rational.
Below we provide formal definitions for the various components of our model for this initial analysis. We make several concrete assumptions, i.e, binary issues, that can easily be relaxed. In Section 7 we will discuss alternative formulations and extensions for many of these notions.
Voters, Candidates, Issues, and Outcomes.
In our specification of FRD there is a fixed set of binary issues where each issue results in an outcome . These values represent dichotomous outcomes like {yes, no}, {pass, fail}, {accept, reject}, {hire, don’t hire}. Let be a set of voters and be a disjoint set of candidates. Each voter and candidate has a preferred outcome for every issue . The preferences of each voter are denoted by where represents the preferred outcome of voter on issue . Similarly, we let where represents the preferred outcome of candidate on issue . Without loss of generality, we label the outcome preferred by the weak majority of voters as 1 and the other by 0 so that . We break ties randomly (when is even).
Preferences Over Candidates and Committee Selection
In FRD we assume that voters’ preferred outcomes are private, while candidates’ declare their preferred outcomes before the election (i.e. election campaigns). We assume voters’ preferences over candidates are consistent with the proportion of the issues they agree upon. That is, the agreement between voter and candidate is where is the Hamming Distance.
We consider three ways voter preferences over candidates may be derived from agreement: approvals, total orderings, and normalized weights. We say that voter approves of candidate if , meaning they have the same preferred outcome on a strict majority of the issues. When voters report their preferences over candidates as total orderings , they order all candidates so that implies where ties are broken privately (e.g. randomly) by the voter. When voters report their preferences as normalized weights, .
We consider only anonymous election rules where the voters’ preferences remain private and voters are otherwise identical, motivating adherence to “one person, one vote”. We consider rules with a fixed, odd committee size
so that the set of elected representatives is where . In our simulations, all rules considered are polytime and deterministic other than randomized tiebreaking. In the setting where voters submit approval ballots, we consider Approval Voting and Reweighted Approval Voting (AV, RAV). When voters submit their preferences over candidates as total orderings, we consider SingleTransferrable Vote (STV) and Borda. When voters submit their ballots as normalized weights over the candidates, we consider the rule which selects the candidates who receive the largest total weight. Lastly, we compare these rules to selecting representatives uniformly at random from the set of candidates. Formal definitions of AV, RAV, STV, and Borda can be found in the book chapter by [48].Properties of Sets of Representatives.
Given a set of elected representatives where we want to evaluate the capability of this set to represent the will of the voters, i.e., recover the outcome of a Direct Democracy with majority voting. To this end we introduce the notions of coverage, full coverage, and agreement. Let and represent the number of representatives who prefer 1 and 0 on issue respectively, such that .
 Coverage.

Issue is covered if at least one representative agrees with the voter majority ().
 Full Coverage.

Issue is fully covered if the representatives are not unanimous ().
 Majority Agreement.

There is majority agreement on issue if the majority of representatives agree with the majority of voters ().
Recall that in practice the number of representatives on either side of the issue is known, but it is not typically known which side corresponds to the voter majority. Therefore the majority agreement of any set of representatives for an issue may be unknown, but it is always known on which issues they achieve full coverage. Full coverage implies coverage, but otherwise the status of coverage is unknown.
Default, Delegation, and Decision.
Each voter receives one divisible vote for each independent issue, maintaining the principle of “one person, one vote”. Once the set of representatives has been elected, each voter’s unit of voting power is distributed among the representatives on each issue. In this paper we distribute this power uniformly by default, so the total voting power initially assigned to each of the representatives is for each issue. In general, various distributions from the literature on voting power [38, 6] and Proxy Voting [1] are worth consideration. We do not consider abstentions by representatives nor voter abstentions, whereby a voter assigns less than a full vote to the representatives as a whole. The total voting power held by the representatives remains collectively for all issues. A voter may choose to override the default distribution of their voting power on any subset of issues by using their delegation option before the decision rule is applied. Otherwise, they may take no further action and leave their voting power distributed by default.
Let be the subset of candidates who are elected as representatives. We refer to the fraction of voters who use their delegation option as the delegation rate , and the delegations rates for the majority and minority () may differ. For our purposes, the weight assigned to representative on issue is simply the sum of the voting power they receive from default and delegation. Consequently, the total weight assigned to representatives who agree with the voter majority is . In this paper the committee uses the weighted majority decision rule with random tie breaking. That is, if , if , and
with probability
if .In our theoretical analysis we assume that voters who delegate do so optimally after the representatives have voted. A voter delegates optimally if they only delegate voting power to representatives who agree with their preferred outcome on an issue. This is equivalent to restricting voters to only delegate to a single representative, or to vote directly on the issue (if there is full coverage). We relax the assumption of optimal delegations in our simulations and will consider voters who delegate only to their most preferred candidate(s) or divide their delegation evenly across their approved set. Note that if an issue is not fully covered because the representatives are unanimous, the outcome is already determined, since opposing voters have no one to whom they can delegate. We discuss this further in Section 7.
Example 1.
Consider an FRD instance with issues and , three voters, and three representatives. Below, the solid arrows from voter to representative indicate delegations, and any voter without an arrow defaults on that issue. The voter and representative preferences are given in the tables above and below the agents. Notice that both delegations are optimal.
On issue , the representative majority agrees with the voter majority, so RD would yield as desired. However, since only the voter in the minority () delegates, the weighted majority of representatives now decides the outcome in favor of the voter majority (). This can occur if the number of voters in the minority is large enough, the number of representatives who agree with the voter minority is large enough, and the voters in the minority delegate at a substantially higher rate than the voters in the majority. Fortunately, if any two or all voters delegate optimally on , the outcome will always agree with the voter majority.
On issue the representative majority disagrees with the voter majority so the RD outcome (without delegations) would be, regrettably, . Looking again at the figure we see the delegations flip the result to what would be achieved by Direct Democracy (). Hence, FRD can improve the outcomes over RD as measured against DD.
3 Related Work
[33], inspired by [44] and shareholder proxy voting, suggested an interactive democratic system for legislation which could take place at scale using computers. Miller lamented the lack of flexibility in traditional Representative Democracy and sought to remedy this using a dynamic system of proxies, although admitted this was not conducive to creating legislation.
Soon after, [39] warned that electronic systems may accelerate the legislative process in undesirable ways and suggested holding every referendum twice to guarantee time for sufficient public deliberation. Our use of a fixed, elected set of representatives answers Miller’s question of how to produce legislation, and rather than holding redundant referenda we give the voters sufficient time to continue deliberation and alter their delegations after the representatives vote.
Just shy of the dawn of the internet, [45] revisited the ideas in a proposal that motivates the default distribution and delegation mechanism in FRD [44]. The notion of the default distribution is also similar to that proposed by [1], which suggests that the weights of representatives be based on the preferences of voters expressed in the election, but these weights are fixed during their term. By contrast, in FRD the weight of each representative on each issue is not strictly determined by the election, yielding expectedly different outcomes. [16] took an analytical approach to studying a Proxy Voting model very close to that of [1] for decision making with no election, infinite voters, spatial preferences, and assuming agents lie in a metric space.
In our view, the hallmark of interactive democracy is that rather than predetermining whether a direct or representative system is better for expressing the will of the voters, the degree to which the system is direct or representative is itself a function of the will of the voters. Currently, the most wellknown and wellstudied form of Interactive Democracy is Liquid Democracy, which has been studied from an algorithmic perspective as a decisionmaking process in the AI and COMSOC literature [13, 29, 9, 14] and elsewhere [25, 22, 10, 12, 28]. Unlike Liquid Democracy, FRD does not allow transitive delegations nor delegations to another voter, thereby violating the second axiom proposed by [25]. However, as we discuss in Section 7, the notion of voluntary representatives can be maintained if desired. Fractional delegations in FRD serve a similar function to that of the virtual committees proposed by [25], although in theory FRD could incorporate virtual committees as well as many other mechanisms for delegating voting power.
The design of FRD is also largely based on work in probabilistic voting, binary aggregation, statistical decision theory, and computational social choice. In particular, work on the optimal weighting of experts [5, 35, 26, 34, 8], the Condorcet Jury Theorem [27], variable electorates [21, 43, 36], and optimal committee sizes [3, 30, 32]. Relatedly, [42] approached the question of choosing a suitable election rule based on the decision rule the elected representatives will use with probabilistic voting, and considering pairs of such rules. In this model, the median OWA rule is best when the representatives decide using simple majority vote. Unfortunately, the outcome of the median OWA rule is NPHard to compute, and so it does not scale to large instances.
Consequently, one can view the voter delegations as a pseudotie breaking mechanism for the representatives or, conversely, see the default distribution as a way to dampen the variance in the outcome which occurs in Direct Democracy when the sample of participating voters is small or biased. At the same time, one can view the process of voters weighting the representatives as a decentralized approximation for the optimal weights. Another view is that electing representatives is analogous to a compression algorithm
[37], which is the algorithmic version of John Adams’s alleged intuition that the representatives should be a microcosm of the population (taken from [1]). In this view, the delegations in FRD are a decompression mechanism where a higher delegation rate reduces the “loss” of representation.4 Difficulties of Representative Democracy
Electing good committees is hard. Even if we knew the view of the voter majority on every issue, electing a set of representatives which maximizes majority agreement on binary issues is NPHard. But suppose we wanted to solve an easier problem: maximizing coverage. Even maximizing coverage is NPHard, as is maximizing full coverage. Worse yet, even for small instances where the problem is computationally tractable, there are pathological examples for which truthful voters whose derived preferences over the candidates are perfectly consistent with their preferences over the issues will elect horrible representatives. We refer to the problems of selecting representatives to maximize coverage, full coverage, and majority agreement as Max kCoverage, Max kFull Coverage, and Max kMajority Agreement, respectively. Below we provide complexity results and pathologies, followed by simulated results to show how well our various polynomial time multiwinner voting rules perform in terms of majority agreement when voter and candidate preferences are generated uniformly at random.
Complexity and Pathologies.
Even if the outcome preferred by the majority of voters is known for every issue, electing a subset of candidates that maximizes coverage, full coverage or agreement is NPHard.
Theorem 2.
Selecting the set of representatives which maximizes coverage is NPHard.
More formally, the theorem can be stated as: Consider a set of binary issues and a set of candidates where each candidate has preference on each issue , the problem of selecting the subset of candidates that maximizes the number of issues on which is NPHard.
Proof.
Our proof of the hardness of Max kCoverage is a reduction from the NPHard problem of Max kcover [24, 20]. The input to max kcover is a set of points, a collection of subsets of , and an integer . The objective of max kcover is to select subsets from such that their union has maximum cardinality. Given an instance of max kcover we create an instance of Max kCoverage as follows. For every point create an issue and for every subset create a candidate . For all points and subsets , if then let , otherwise let . And let be the number of representatives we will elect. There is a onetoone correspondence between the number of issues covered by our representatives and the cardinality of the corresponding subsets in the original max kcover instance. Therefore, any set of candidates that maximizes coverage corresponds exactly to a collection of subsets in our max kcover instance whose union has maximum cardinality∎
Note that if the majority view of the voters were known, coverage could be approximated deterministically in polynomial time within a factor of by a greedy algorithm, and this bound is tight [20]. Therefore, none of our deterministic polynomialtime election rules can provide a better guarantee than this, although they may provide decent approximations in expectation. The proofs for the two theorems below follow directly, although nontrivially, from our proof for Max kCoverage.
Theorem 3.
Selecting the set of representatives which maximizes full coverage is NPHard.
Proof.
We now prove the hardness of Max kFull Coverage by polynomialtime reduction from Max kCoverage. To do this we construct an instance of Max kFull Coverage by adding an additional candidate , adding additional issues to the original issues, and desire a set of candidates. We show that in this new instance of Max kFull Coverage the additional candidate must be selected in any optimal solution because they are uniquely required to cover the added issues, and the remaining candidates in the solution set correspond exactly to the optimal candidates in the solution to our original Max kCoverage instance.
Given an instance of Max kCoverage we construct an instance of Max kFull Coverage as follows. Create a set of binary issues and augment it with additional binary issues so that . Create a set of candidates where for all , for all and for all issues . Let for all for issues and let for all issues . Our objective is to select a set of candidates from which maximizes full coverage. We will now prove that for all solutions to our new Max kFull Coverage problem, where is a set of candidates whose corresponding counterparts maximize coverage over issues in our original Max kCoverage instance.
Lemma 4.
must contain
Proof.
Clearly, the set achieves full coverage for issues for any , and any set which does not contain cannot fully cover (or cover) . Since comprises more than half the issues, any set of candidates for which maximizes the number of issues fully covered, must contain . ∎
Given that for all issues , the set of candidates , which maximizes full coverage for issues is the set of candidates which maximizes coverage over issues in . Therefore, the candidates corresponding to are the solution to our original instance of Max kCoverage and given the solution to Max kCoverage we simply add to find . ∎
Theorem 5.
Selecting the set of representatives which maximizes majority agreement is NPHard.
Proof.
We now prove the hardness of Max kMajority Agreement by polynomialtime reduction from our problem of Max kCoverage. Similar to our proof for Max kFull Coverage, we replicate the instance of Max kCoverage and add issues to the original issues such that . However, we now augment the candidate set with additional candidates who must be included in any committee which maximizes majority agreement. The objective is to select the candidates which maximize majority agreement. The additional candidates must be in the solution set for Max kMajority Agreement, and the remaining candidates selected will correspond exactly to the candidates in the solution to our original instance of Max kCoverage.
Given an instance of Max kCoverage with input we construct an instance of Max kMajority Agreement with input as follows. Create a set of binary issues and a set of candidates . We can think of as being made up of three sets of candidates based on how we will construct their preferences over , that is, . The first set has identical preferences to the candidates in the original problem over the first issues, and prefers on the rest. The second set unanimously prefer the outcome of on all issues. The last candidate prefers on the first issues, and on the remaining issues. Formally, for , for and for . For , for all . And for , for and for .
Lemma 6.
Any set of representatives which maximizes majority agreement must contain .
Proof.
Clearly, agrees with the voter majority on issues if and only if contains , because this is the only way at least out of the representatives can agree with the voter majority on any of these issues. This directly implies there is agreement on more than half the issues if and only if . ∎
Selecting candidates provides exactly representatives who agree with the voter majority on issues . Since we are selecting representatives in total, on any of these first issues we need only more representative who agrees with the voter majority on each issue to achieve majority agreement. Therefore, selecting additional representatives from which maximize coverage over issues , maximizes the majority agreement of the representatives over . Clearly, these representatives are a onetoone correspondence to the representatives in the solution to our original Max kCoverage problem. Likewise, given the solution to the original Max kCoverage problem, taking the corresponding candidates and adding maximizes majority agreement. ∎
In our model, we do not know the view of the voter majority on any issue before the election, so even if the parameters are small enough for the fullinformation problem to be computationally tractable we must work with partial information. Thus our election rule uses the preferences of voters over the candidates, induced by their underlying preferences over the issues. A fairly dire example can be found in [2] with 11 voters and 11 issues. In this example, when voters only approve of candidates who agree with them on the majority of issues, the worst imaginable candidate would be approved by 7 of the 11 voters, while the best conceivable candidate would only receive 4 approvals. On top of the pathological cases and the computational complexity of maximizing majority agreement, we show that our polynomial time election rules do not perform particularly well in approximating maximum majority agreement in expectation.
Majority Agreement in Representative Democracy
We investigate the properties of coverage and majority agreement as functions of the numbers of candidates, issues, and committee size. In all our simulations, for all issues we let = 1 and = 1 with probability for all voters and candidates. In all of our runs, coverage was 1.0 for all combinations, hence we omit it from the graphs in Figure 1. For all simulations we hold fixed as we did not observe a strong dependence on the number of voters as long as it was sufficiently larger than the number of candidates. For all simulations we perform 50 iterations at each datapoint and plot the mean of these runs. Variance for all points is so our results are robust to noise [15].
Turning first to Figure 0(a) we hold , and vary in steps of 15. We see that for a small number of issues the AV, RAV, and the weighted voting rule can be expected to select a committee in agreement with the majority nearly 80% of the time. However, as we add issues to the docket, the voting rules seem to converge around 60%. In Figure 0(b) we fix and vary the number of candidates between in steps of 5. We observe again that AV, RAV, and weighted voting are the best followed closely by STV. As we increase the number of candidates it is possible for the system to more frequently recover the will of the majority but this number does not climb above 65% across all treatments. Finally, in Figure 0(c) we hold and vary . These simulations reinforce the idea that electing an ideal committee, i.e., one that represents the will of the majority of the voters on every issue, is a hard problem. In the next section we will explore how FRD can out preform RD and its dependence on the constituent delegation rates .
5 Benefits of Flexibility
The flexibility of issuespecific delegations is the motivating feature of FRD. We first look at basic features of FRD in a deterministic setting, and then consider probabilistic delegations.
Deterministic Delegation
A voter’s delegation is optimal if the voter only delegates to representatives who agree with them on an issue. Observe that if the representatives are unanimous only one outcome is possible but as long as there is some dissent in the committee opinions FRD can take advantage and return potential decision making capability entirely to the voters.
We denote by the numbers of voters and by the numbers of candidates who agree and disagree with the voter majority on issue , respectively. We have labeled the majority view of the voters as 1, so
. The overall outcome of any resolute democratic process over this set of issues is a single vector
, and our ideal outcome is . Treating all issues equally and independently, we seek to maximize . Let and be the number of voters who delegate in favor of each outcome, assuming they delegate optimally. We drop the superscript below because we will be talking about a single issue.Theorem 7.
If all delegations are optimal and the issue is fully covered (), the outcome is guaranteed to agree with the voter majority if the number of voters in the majority who delegate is greater than .
Proof.
For the majority to guarantee the outcome in their favor by delegating optimally, this means that even if all voters in the minority delegate optimally, the outcome must still be . Recall that is the total weight assigned by default and delegation to representatives who prefer the outcome 1. Let . If , then . For the outcome to be guaranteed in favor of the voter majority, it must be that . Solving for we find that . Note that this lower bound may be negative. In this case is so large and is so small that the outcome is guaranteed regardless of delegations of the minority, so . ∎
Theorem 8.
If all delegations are optimal and the issue is fully covered , the outcome will favor the minority if .
Proof.
With only optimal delegations, the outcome favors the minority if . The lower bound can be found directly by solving for and substituting . ∎
Theorem 9.
The minimum size a minority must be for it to be possible for them to yield their favored outcome when is (in the case where all voters in the minority delegate optimally and all voters in the majority default).
The reader can verify this theorem without difficulty.
Probabilistic Delegation
Instead of assuming that some fractions of voters delegate we investigate what happens if each voter chooses to delegate with some fixed individual probability. These results gives us an idea of how motivated or attentive voters must be to improve the outcome of FRD over RD. We assume here that all voter and candidate preferences are independent for all issues.
Since all issues are independent, we will consider a single issue. Suppose each voter chooses to delegate (deviate from the default) with independent probability and that all delegations are optimal. Let be the amount of power voter assigns to candidates who agree with the voter majority (), either by delegation or default. If defaults then , if delegates optimally and is in the voter majority () then , and if delegates optimally and is in the voter minority then . Let be the total power assigned to these candidates via both delegation and default. Let be the expected value of the total power assigned to representatives who agree with the voter majority. Then .
Theorem 10.
Consider an FRD with an odd number of voters , odd committee size , only optimal delegations. Suppose each voter delegates with probability on issue such that . Then the probability that the outcome agrees with the voter majority is bounded by .
Proof Sketch: The probability that the outcome agrees with the voter majority is where is due to some tiebreaking mechanism. First we show that with odd voters, odd representatives and only optimal delegations there can be no ties. Namely, . This proof is due to parity and holds regardless of the delegation rate. Without ties, we simply need to determine . We use a Chernoff inequality to provide a lower bound on this value based on the probability of delegation of all voters. See Section 7 for discussion about tiebreaking ^{1}^{1}1It is an interesting open question how the distribution of these delegation probabilities effects the outcome when voter and representative preferences are correlated or when this distribution changes over time based on the outcomes of previous issues..
Proof.
Recall that is the amount of weight voter assigns to candidates who agree with the voter majority on an issue via default or delegation and . Given some tie breaking rule, we have that . First we show that , then we give upper and lower Chernoff bounds for .
Lemma 11.
If N is odd, k is odd, and all delegations are optimal, then no ties are possible.
Proof.
Let where is the amount of weight (voting power) voter assigns to candidates who agree with the voter majority on an issue via default or delegation. If defaults then , if delegates optimally and is in the voter majority () then , and if delegates optimally and is in the voter minority then . Therefore, . Let and . Then are nonnegative integers and . Since is odd, it must be that and have opposite parity and so they cannot be equal. Therefore , meaning the total amounts of weight delegated to the representatives on either side of the issue cannot be equal, so no ties may occur. ∎
Given that no ties are possible, we have that . Remember that where is the total weight that delegates to representatives who agree with the voter majority. If then , else if then . Let be the expected total weight assigned to representatives who agrees with the voter majority.
We now use the fact that . Let . If , then . This allows us to apply a Chernoff bound to derive our lower bound
∎
This bound depends on the mild condition that . This requires that the delegation rate of the majority cannot be too small relative to the minority . Observe that this condition is always satisfied when and . Furthermore, as an increasing number of voters delegate optimally, we expect regardless of . Naturally, as the delegation rate increases (), we observe our lower bound approach the ideal . To find , the voter majority cannot be too large compared to the voter minority, must be smaller than or somewhat close to , and/or the voters in the majority must be significantly more apathetic towards delegation than voters in the minority.
Tighter bounds may be achieved when the delegation probabilities are assumed to come from a particular distribution. It is an interesting open question to see how the expected outcome is effected when voters have various motivations to delegate which give rise to different delegation probability distributions.
Simulated Delegations
Given our theoretical results on how FRD can improve the outcomes of a decision making process, we investigate the effect of the overall delegation rate on recovering the ideal outcomes according to Direct Democracy. We use the same model to generate candidates and voter preferences as used in Section 4. For our simulated delegations we create instances with , , , and . We vary in increments of and for each setting of we run 50 iterations. We plot the means in Figure 2 and note again that the variance is at every point.
In Figure 2 we can see the agreement of the outcomes of FRD and RD for the weighted voting committee selection rule for several delegation types and delegation rates. For each of the instances we measure against a baseline of the proportion of issues where the outcome is that of DD. A value of means that the outcomes of all issues are the same under the democracy as they would be in a Direct Democracy.
We compare RD with four different delegation schemes: (1) Approve where voters delegate evenly to the representatives in the committee of whom they approve and do not update per issue; Best Rep where voters delegate to their single most preferred representative and do not update per issue; Best3 Rep where voters delegate equally to their three most preferred members of the committee and do not update per issue; and finally Optimal where voters delegate to a single representative with whom they agree per issue.
Our most surprising finding is how little delegations that are not active and optimal help the system. The Approve delegation system is perhaps closest to the proposal of Proxy Voting espoused by [33] but does not improve RD in a meaningful way. Similarly interesting are the 1Best and 3Best delegations, which also do not move the outcome towards the ideal of Direct Democracy. Hence, we can see that the issuespecific flexibility FRD allows can be effectively used to improve outcomes of decision making systems. Another striking result in Figure 2 is how drastically FRD can improve agreement over RD when voters are highly attentive. With as little as 60% of the population delegating we can improve agreement by 10%, and when the delegation rate reaches 80% we see an almost 20% increase, eventually reaching 100% when everyone delegates if the issue is fully covered. It is an interesting open question to see how this behavior changes when voters have specific motivations to delegate such as the makeup of the representatives or outcomes of past issues.
6 Conclusion
We have introduced a novel system of Interactive Democracy called Flexible Representative Democracy which transitions smoothly, at the discretion of the voters, between direct and representative democracy. We have shown theoretically and empirically that FRD has clear benefits over other systems such as Representative Democracy and Proxy Voting as FRD has the ability to improve outcomes. An important point to remember is that delegations are optional, and not an additional burden imposed on the system or voters. In contrast to Liquid Democracy, voters in FRD have greater certainly about how their vote will be cast ahead of time and delegation cycles are not possible. Furthermore, FRD maintains a fixed, elected set of accountable representatives to produce legislation and hold public debates. This committee of representatives does not need to expand to guarantee proportional or justified representation, because as long as there is full coverage of an issue voters have the power to guarantee these properties for themselves.
In our analysis, we seek to create a bestcase scenario for traditional RD to achieve high agreement: we assume the full list of issues is known before the election, all voters participate in the election, candidates are truthful and do not change their preferred outcome after election, and that voter preferences over candidates are consistent with their preferences over the issues. Intuitively, relaxing any of these assumptions only strengthens the argument for enabling flexible, issuespecific delegations. Flexible delegation also has the effect of minimizing the role that the choice of election rule plays in the outcome.
7 Extensions
In addition to the model laid out here, there are a number of interesting and important extensions to FRD that one could consider. Nothing in our system prevents moving to issues where there are nonbinary domains and/or different domains for every issue. One can also consider asymmetric issues with decision rules other than simple majority rule, such as quota rules. Additionally, we have modeled the weight given to representatives as a simple sum of the defaults and delegations, but this could be any function and may treat delegations and defaults differently; though such a function should be monotone with respect to the delegated weights. Note that we can easily relax the assumptions in our analysis that there are an odd number of voters, optimal delegations, and no abstentions. Relaxing any of these requires us to account for potential ties in the outcomes of our analysis.
If ties are broken randomly, then we only need to compute the probability of a tie occurring. This probability is computable, but in general we should not expect the values to come from a nice, symmetric, wellbehaved distribution. It is also worth noting that ties can be broken in other ways including based on the observed delegation rate. For example, when the delegation rate is low one might break ties in favor of the representative majority, whereas when the delegation rate is high one might break ties in favor of the outcome with more weight delegated to it.
Lastly, one of the attractive features of Liquid Democracy is the notion of voluntary representatives who need not be formally elected. These voluntary representatives are not beholden to an election cycle and can be local leaders, with personal relationships to voters who support them. This can be incorporated into a Flexible Representative Democracy by allowing any voter or agent to become a voluntary representative on a single issue by casting their vote publicly by the same deadline as the elected representatives. However, voluntary representatives do not receive any voting power by default, may not receive delegated voting power on any issue before their vote is declared, and can only receive delegations on a perissue basis. Since no voluntary representatives receive any default voting power, all our results still hold.
In fact, the addition of representatives who receive no power by default constitutes a Pareto improvement because they effectively serve to guarantee full coverage (and coverage) and if the outcome changes as a result it can only change in the direction of the voter majority (assuming optimal delegations). Alternatively, an FRD can automatically add a contrarian singleissue dummy representative whenever the representatives are unanimous. Keep in mind that while full coverage can be artificially guaranteed in this way, any attempt to guarantee majority agreement would constitute rigging the election.
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8 Appendix
Preference Representation
For clarification, there are two ways voters could report their strict orderings over the candidates. The first is a list where the position in the list denotes the rank and the value in the list denotes the candidate. This is what is typically thought of as an ordering. We use a second, equivalent, representation where the position in the list denotes the candidate and the value denotes their rank. Consider a paper ballot in which you want voters to rank their candidates in order of preference. The first representation corresponds to writing the numbers on the paper and having the voters fill in the names of the candidates next to them. Our second representation is like listing the candidates on the page, and having the voters fill in the numbers next to the names to denote their rank. The choice of representation in practice does not impact our analysis, as they represent the exact same information. Similarly with approvals. Agents can just report the subset of candidates of which they approve, or they could write whether they approve next to each candidate name on a ballot. Again, we use the second representation and the two are equivalent.
Additional Simulated Results
In Figure 3 we can see the agreement of the outcomes of FRD and RD for the weighted voting, AV, and Borda election rules for several delegation types and delegation rates. For each of the instances we measure against a baseline of the proportion of issues where the outcome is that of DD. A value of means that all issues are the same under the democracy as they would be in a Direct Democracy. Overall, this plots show that RD under any of these rules cannot achieve above agreement with Direct Democracy in expectation.