1. Introduction
Collective decision making and social choice are the cornerstones of modern democracy. Mathematically, a social choice rule is a function that maps ordinal preferences of voters for alternatives (or candidates) to a single winning alternative. One way to characterize social choice rules is by adopting the axiomatic approach. Exemplary works include (Gibbard, 1973) and (Satterthwaite, 1975), where several natural desirable properties of social choice rules are proposed, and many of them are shown to be incompatible with each other.
An alternative approach is to adopt the utilitarian view, where agents have latent cardinal preferences underlying the ordinal preferences (Procaccia and Rosenschein, 2006; Caragiannis and Procaccia, 2011; Boutilier et al., 2015). The goal is to optimize certain social objectives, the most common being the sum of utilities of the voters. The term distortion of a social choice rule refers to the worstcase ratio between the objective value that the rule achieves and the optimal one, where the worst case is over all cardinal preferences that are consistent with the given ordinal preferences. However, it has been established that many common rules have unbounded distortion (Procaccia and Rosenschein, 2006) and even randomized social choice rules have distortion of at least , where is the number of candidates (Boutilier et al., 2015).
These impossibility results necessitate imposing additional structure on the underlying cardinal preferences. One common approach is the spatial model, where voters and candidates are located in a certain space (e.g. the Euclidean space) and the distance between a voter and a candidate is part of the cost function of the voter (Davis and Hinich, 1966; Enelow and Hinich, 1984, 1990; Merrill III et al., 1999; Schofield, 2007). For instance, for the social issue of building a facility serving a neighborhood, it is natural to assume the cost of an individual is the distance between the her and the facility. As another example, consider a government in the process of making a public policy. If people are viewed to be on the leftright political spectrum, the cost of an individual can be modeled as the distance between her and the policy on the spectrum. In both cases, the voters and candidates are located in a metric space and the costs are the corresponding distances.
When the cardinal costs are assumed to be distances in an underlying metric space, it is possible to achieve constant distortion. The work of (Anshelevich et al., 2018) shows that selecting any candidate in the uncovered set (Moulin, 1986) — for instance, the Copeland voting rule — guarantees distortion of . (Please refer to Section 2.4 for description of several common voting rules.) They also show a lower bound of for the distortion for any (deterministic) social choice rule. The exact value of the optimal distortion remains an open question — neither the upper bound of nor the lower bound of has been improved. The work of (Skowron and Elkind, 2017) shows that the well known voting schemes of using scoring rules, as well as the popular single transferable vote, have superconstant distortion. The work of (Goel et al., 2017) shows the ranked pairs rule and the Schulze method, both of which work on the weighted tournament graph over candidates, have distortion of at least contrary to previous belief that they could improve the bound of . The weighted tournament graph places a directed edge between two candidates with weight equal to the portion of voters who prefer the first candidate over the second. The work of (Goel et al., 2017) then conjectured any social choice rule that only looks at the weighted tournament graph over candidates cannot beat .
1.1. Our Results
Our first result in Section 3 disproves the conjecture that weighted tournament rules cannot improve the distortion bound below . We improve the distortion upper bound to using a weighted tournament rule. We achieve the bound by generalizing the notion of uncovered sets to a class of weighted rules. We show is indeed the best distortion achievable by this new class of social choice rules.
This still leaves a gap between the best lower bound of and the upper bound of for deterministic social choice rules. In Section 4, we propose another social choice rule based on bipartite matchings that generalizes uncovered sets in a different way. In Theorem 4.4, we show that if a candidate exists that satisfies our criterion, then choosing this candidate yields distortion of . In Section 4.3. we present a combinatorial conjecture that implies such a candidate always exists, and verify this conjecture for small numbers of voters and candidates (at most in all). As another advantage of using this framework, we show a new result in Section 4.4 for certain types of preference profiles: If the weighted tournament graph on the instance is cyclically symmetric, then selecting any candidate yields distortion of at most for that profile.
1.2. Related Work
Distortion of randomized social choice rules
In addition to deterministic social choices rules, randomized rules have been studied in the metric distortion setting. It is possible that they can achieve better distortion than deterministic ones: Randomly selecting a voter to be the dictator achieves distortion of (Feldman et al., 2016; Anshelevich and Postl, 2017). On the other hand, no randomized rule can get distortion better than (Feldman et al., 2016; Anshelevich and Postl, 2017), no truthful randomized rule can beat (Feldman et al., 2016) and no weighted tournament rule can beat (Goel et al., 2017). An almost distortionoptimal rule when the number of candidates is not large is shown in (Gross et al., 2017). Randomized rules are unsatisfying for practical implementation and interpretability, and much literature on social choice has therefore focused on deterministic voting rules.
Median distortion and fairness properties
In our paper, we define the social cost to be the sum of the cost of each individual voter. There are other measures of how well a social choice rule performs. The median distortion setting, where distortion is defined using the median cost instead of the sum is considered in (Anshelevich et al., 2018). They show the Copeland rule achieves distortion of in the setting and is optimal. Fairness properties of social choice rules generalizes distortion in the sense that for any , it considers the sum of the largest costs (Goel et al., 2017). For this setting, the Copeland rule achieves a fairness ratio of . Further, fairness ratio and distortion can only differ by at most additively (Goel et al., 2018). We show in Section 3.2 that though our voting rule improves distortion for the sum of cost, it does not improve distortion bounds for fairness properties.
Other Problems in the Metric Distortion Setting
The metric distortion framework is powerful enough to be applied to settings beyond that considered in this work. When candidates are independently drawn from the population of voters, the distortion can be better than (Cheng et al., 2017) and the class of scoring rules that have constant expected distortion has a clean characterization (Cheng et al., 2018). Lowdistortion algorithms for the maximum bipartite matching problem in metric spaces are proposed in (Anshelevich and Zhu, 2017). Social choice and facility location problems in the distortion framework are studied in (Anshelevich and Zhu, 2018).
2. Preliminaries
2.1. Social Choice Rules
Let be a set of candidates and be a set of voters. Let be the set of entities in the voting system. For each voter , she has a strict preference ordering on the candidates in . We call the voting profile. If prefers to , we write or . Let be the family of strict orderings on elements. A social choice rule is a function
that takes orderings on candidates and outputs one of the candidates. The output of is the winner of the social choice rule.
Throughout the paper, we reserve uppercase letters to denote candidates. is the set of voters that prefer to . is the set of voters that both prefer to and prefer to . For any set , we use to denote the size of . E.g., is the number of voters who prefer to .
We will need the following definition later.
Definition 2.1 ().
For any voter and any candidate , is the set of candidates that likes at least as much as . Similarly, is the set of candidates that likes at most as much as .
2.2. Metric Distortion
We assume the candidates and the voters are located in the same metric space , where is the metric (distance function). As a metric, satisfies the following conditions:

Positive definiteness: .

Symmetry: .

Triangle inequality: .
We say a metric is consistent with a voting profile if
Here we allow a voter to have equal distances to different candidates. This is only to eliminate the need of infinitesimal distances in later analysis. Similar results can be derived if we require and to be different in the above definition.
We define the social cost of a candidate with respective to a metric as:
and the distortion of a social choice rule as:
2.3. Weighted Tournament Graphs
We can construct a tournament graph on the candidates by looking at their pairwise relationships: Let be the set of vertices. For each pair of (), draw an edge from to if pairwise beats : (break ties arbitrarily). The result is a directed graph where there is exactly one directed edge between any pair of vertices. If a social choice rule makes decision only based on the tournament graph, it is called a tournament rule, or a rule (Fishburn, 1977).
One way to generalize the class of tournament rules is to further consider the margin of each winning. For each pair of (), draw an edge from to with weight . In the resulted weighted tournament graph, there are two directed edges between any pair of vertices. A social choice rule making decision only based on the weighted tournament graph is called a weighted tournament rule, or a rule^{1}^{1}1To be precise, in (Fishburn, 1977), a social choice rule is only if it is not . (Fishburn, 1977). We also consider a subclass of weighted tournament graphs that satisfy cyclic symmetry.
Definition 2.2 ().
We say a graph is cyclically symmetric, if there exists a cyclic permutation on the vertices, so that any edge has the same weight as .
Example 2.3 ().
Let . Figure 1 shows a cyclically symmetric weighted tournament graph. For each pair of vertices, only one edge is shown for visibility. The cyclic permutation is
We note that cyclically symmetric weighted tournament graph is not necessarily induced by a cyclically symmetric voting profile.
2.4. Examples of Social Choice Rules
Now we present some social choice rules from existing literature, along with their distortion bounds.
Uncovered
The uncovered set of a tournament graph consists of those candidates such that for any other candidate , either there is an edge from to , or there is another candidate so that there are an edge from to and an edge from to (Moulin, 1986). The uncovered set is always nonempty for any tournament graph. Uncovered is a rule that selects the alphabetically smallest candidate from the uncovered set. It has distortion of (Anshelevich et al., 2018).
Copeland
Copeland selects the candidate that has the most pairwise wins, i.e., the candidate with the largest outdegree in the tournament graph (break ties alphabetically). Copeland is a rule as well, and it can be seen as Uncovered with a specific tiebreaking method. It does not improve the distortion of Uncovered (Anshelevich et al., 2018).
RankedPairs
RankedPairs is a rule proposed in (Tideman, 1987). In RankedPairs, the edges of the weighted tournament graph are sorted to have decreasing weights. We start from an empty graph and add one edge at a time (in the order of decreasing weights) if it does not create a cycle. A directed acyclic graph (DAG) is resulted in the end, and the source of the DAG is selected as the winning candidate. It achieves distortion of when the tournament graph has a circumference (size of the largest cycle) of at most (Anshelevich et al., 2018). However, its distortion is at least in general (Goel et al., 2017).
Schulze
Schulze is a rule proposed in (Schulze, 2011). For two candidates and in the weighted tournament graph, define to be the maximum so that there is a path from to with each edge having weight of at least . It can be shown that there is a candidate so that for any other . This is selected as the winning candidate by Schulze. Its distortion is at least (Goel et al., 2017).
OptimalLP
In (Goel et al., 2017), the authors show that the optimal distortion can be achieved by OptimalLP
involving solving linear programs. The distortion of selecting candidate
comparing to selecting in an instance is given by the following linear program:OptimalLP selects the candidate . Its distortion was known to be between and (inclusive). Our work implies OptimalLP has distortion of at most .
3. WeightedUncoveredSet Voting Rule
In this section, we present the WeightedUncovered rule that has distortion of (Theorem 3.4), improving the best existing upper bound of for any deterministic social choice rule. WeightedUncovered is based on the weighted uncovered set (Definition 3.1), a generalization of the uncovered set (Moulin, 1986). We show our analysis is tight by giving a matching lower bound (Theorem 3.10). We also give a lower bound for the fairness ratio (Theorem 3.12).
We start by introducing weighted uncovered sets.
Definition 3.1 ().
Let be a constant. The weighted uncovered set is the set of candidates such that for any candidate , we have:

either ,

or there is another node so that and .
Now we present our key lemma showing the weighted uncovered set is a meaningful concept.
Lemma 3.2 ().
For any , the weighted uncovered set is nonempty.
Proof.
We prove by contradiction. Suppose there is no such node . Then there must be nodes in the graph. If we set and (), then both of the conditions in the lemma are violated. Figure 2 shows such a cycle.
Let . We have and . When goes from down to , has to drop below at some time, so we have while . Thus we have and . This contradicts with the assumption that when and , there is no such satisfying the second condition in the lemma. ∎
Throughout this section, we use to denote the golden ratio .
Definition 3.3 ().
WeightedUncovered is a social choice rule that picks the alphabetically smallest candidate in the weighted uncovered set.
Because of Lemma 3.2, WeightedUncovered is welldefined and always picks a candidate in the weighted uncovered set. We state the main property of WeightedUncovered below in Theorem 3.4 and prove it later.
Theorem 3.4 ().
WeightedUncovered has distortion of at most .
3.1. Main Theorem: Improved Distortion Bound
Recall that is the social cost if is selected and the metric is . First, if , the social cost gap between choosing and can be easily bounded.
Lemma 3.5 ().
If , then for any metric .
Proof.
We have
When showing Copeland has distortion of at most , (Anshelevich et al., 2018) crucially uses the following lemma:
Lemma 3.6 (Theorem 15 in (Anshelevich et al., 2018)).
If and , then for any metric .
Lemma 3.6 itself is tight, but worstcase analysis is deployed when we assume every edge in the unweighted tournament graph has weight of . Actually, if every edge has weight approaching , any social choice rule will have distortion approaching by a similar argument as Lemma 3.5. Instead of having the stronger condition in the uncovered set that , we allow those candidate ’s with to be in the weighted uncovered set. In this way, we can use unbalanced and and still guarantee the weighted uncovered set is nonempty (Lemma 3.2). It turns out that better distortion can be achieved with carefully chosen parameters.
Lemma 3.7 ().
If and , then for any metric .
To prove Lemma 3.7. We need the following technical lemma from (Anshelevich et al., 2018), whose proof is included here for completeness. Recall that is the set of candidates that likes at most as much as .
Lemma 3.8 (Lemma 14 in (Anshelevich et al., 2018)).
If , then for any metric .
Proof.
By definition of and the triangle inequality,
Proof of Lemma 3.7.
We perform different analysis depending on whether or .
Case (2): When :
Notice that for , . Thus,
Because and ,
The second inequality above comes from the conditions that , and the assumption that in this case. Therefore,
where the second inequality is from and . Again by , we apply Lemma 3.8 to get . ∎
3.2. Lower Bounds
An immediate question is whether our upper bound is tight. Also, is it possible to choose a in the weighted uncovered set to improve the distortion? In this section, we show that our choice of and our analysis are indeed tight. Before showing the lower bounds, we complete the definition of weighted uncovered sets for .
Definition 3.9 ().
Let be a constant. The weighted uncovered set is the set of candidates such that for any candidate , we have:

either (different from the case where ),

or there is another node so that and .
We did not define weighted uncovered sets for earlier because for any such , selecting any candidate in the weighted uncovered set can have distortion worse than . Now we show lower bounds for all .
Theorem 3.10 ().
For any , choosing the alphabetically smallest candidate in the weighted uncovered set has distortion of at least .
Proof.
The tight examples are illustrated in Figure 3. In both cases, all points lie on a straight line. In the left case, let portion of voters be of type and portion of voters be of type .^{2}^{2}2Technically this is only doable for , but we can add an arbitrarily small portion of other voters to make in the weighted uncovered set. When , the distortion of choosing is at least . Since is in the weighted uncovered set and weighted uncovered set, we rule out and .
When , consider the right case in Figure 3. Let portion of voters be of type and portion of voters be of type . is in the weighted uncovered set. However, the distortion for choosing comparing to is at least . When , the distortion is greater than . ∎
The fairness ratio of social choice rules was proposed in (Goel et al., 2017):
Definition 3.11 ().
The fairness ratio of a social choice rule is:
The fairness ratio is at least the distortion because it approximates the sum of largest costs for any including . The distortion of Copeland is (Anshelevich et al., 2018) and its fairness ratio is (Goel et al., 2017) as well. Since WeightedUncovered improves the distortion, one may ask whether it improves the fairness ratio as well. We answer this in the negative:
Theorem 3.12 ().
For any , choosing the alphabetically smallest candidate in the weighted uncovered set has a fairness ratio of at least .
Proof.
When or , the uncovered set contains every candidate so the fairness ratio is . Otherwise, consider a setting where and contains multiple copies of and . The distances between pairs of points are given in Table 1. One can verify they satisfy the triangle inequality.
The preference of is and that of is . Let voters be of type and voters be of type.^{3}^{3}3Again, this is only doable for , but we can add an arbitrarily small portion of other voters to make the weighted uncovered set include . Instead of looking at the worstoff voter in the end, we look at the worstoff portion of voters for a small . is in the weighted uncovered set, but the fairness ratio for choosing is at least comparing to , given by the worstoff voter (). ∎
Points  

4. MatchingUncoveredSet Voting Rule
In this section, we present the MatchingUncovered rule, which is based on an alternative generalization of uncovered sets. We introduce matching uncovered sets and the MatchingUncovered rule in Section 4.1, after which we analyze its distortion (Theorem 4.4) in Section 4.2. In Section 4.3, we propose a conjecture (Conjecture 4.8) which implies the distortion of MatchingUncovered is . We use computerassisted search to verify it holds when at most entities are involved (Theorem 4.11). Finally, we give a sufficient condition (Lemma 4.13) for the matching uncovered set to be nonempty in Section 4.4. We use this new condition to show when the weighted tournament graph is cyclically symmetric, every social choice rule has distortion of at most (Theorem 4.12).
4.1. Description
Recall that as in Definition 2.1, is the set of candidates that likes as least as much as and is the set of candidates that likes as most as much as .
Definition 4.1 ().
The matching uncovered set is the set of candidates such that: For any other candidate , there is a perfect matching in the bipartite graph constructed in the following way:

The vertices on the left and vertices on the right are both .

For a vertex on the left and a vertex on the right, we draw an edge between them if .
Example 4.2 ().
Definition 4.3 ().
MatchingUncovered is a social choice rule that picks the alphabetically smallest candidate in the matching uncovered set. It picks the alphabetically smallest candidate if the matching uncovered set is empty.
4.2. Distortion Analysis
The next theorem is the main reason why we introduce matching uncovered sets.
Theorem 4.4 ().
Selecting any candidate from the matching uncovered set guarantees distortion of at most .
Corollary 4.5 ().
If the matching uncovered set is nonempty, MatchingUncovered has distortion of at most .
Before proving Theorem 4.4, we introduce a new notation specifically for proving distortion of . Fixing the preference orderings , define
Lemma 4.6 ().
A social choice rule has distortion of at most 3 if and only if for any voting profile and for any other candidate ,
for any metric that is consistent with .
Proof.
Let . A social choice rule has distortion of at most if and only if for any profile and any metric consistent with , for any other , i.e., . ∎
Lemma 4.7 ().
For a voter and any consistent metric :

If has preference , then .

If has preference , then .

If has preference , then .

If has preference , then .

For any , .
Proof.
By the preference of and triangle inequality:

If has preference , then

If has preference , then

If has preference , then

If has preference , then

For any ,
Proof of Theorem 4.4.
Because of Lemma 4.6, we only need to show if there is a perfect matching in .
We divide the set of voters into two disjoint subsets: (those who prefer to ) and . By (4) in Lemma 4.7, we have
For a voter , we use to denote the vertex on the left representing and the vertex on the right representing . Because there is a perfect matching in the graph , there has to be a matching of size at least within the vertices representing : . We write to denote the boolean variable whether the vertex is matched in this smaller matching. If is true, we write to denote the common candidate that uses in the matching. Note that the common candidate cannot be or between vertices representing . We have
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