An Analysis Framework for Metric Voting based on LP Duality

11/17/2019 ∙ by David Kempe, et al. ∙ 0

Distortion-based analysis has established itself as a fruitful framework for comparing voting mechanisms. m voters and n candidates are jointly embedded in an (unknown) metric space, and the voters submit rankings of candidates by non-decreasing distance from themselves. Based on the submitted rankings, the social choice rule chooses a winning candidate; the quality of the winner is the sum of the (unknown) distances to the voters. The rule's choice will in general be suboptimal, and the worst-case ratio between the cost of its chosen candidate and the optimal candidate is called the rule's distortion. It was shown in prior work that every deterministic rule has distortion at least 3, while the Copeland rule and related rules guarantee worst-case distortion at most 5; a very recent result gave a rule with distortion 2+√(5)≈ 4.236. We provide a framework based on LP-duality and flow interpretations of the dual which provides a simpler and more unified way for proving upper bounds on the distortion of social choice rules. We illustrate the utility of this approach with three examples. First, we show that the Ranked Pairs and Schulze rules have distortion Θ(√(()n)). Second, we give a fairly simple proof of a strong generalization of the upper bound of 5 on the distortion of Copeland, to social choice rules with short paths from the winning candidate to the optimal candidate in generalized weak preference graphs. A special case of this result recovers the recent 2+√(5) guarantee. Finally, our framework naturally suggests a combinatorial rule that is a strong candidate for achieving distortion 3, which had also been proposed in recent work. We prove that the distortion bound of 3 would follow from any of three combinatorial conjectures we formulate.

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1 Introduction

Voting is an important and widespread way for a group to choose one out of multiple available candidate options.111In this submission, we do not consider the equally important and widely studied problem of a group ranking all of the available options. The group could be a country, academic department, or other organization, and the candidate options they choose from could be courses of action or human candidates. Typically, each voter submits a total order of all options, called a ranking or preference order. Based on all the submitted rankings, a social choice rule (or mechanism) determines the winning option.

Different mechanisms will have different desirable and undesirable properties, and it is important to articulate and analyze these properties to guide an organization’s choice of mechanisms. The axiomatic approach, dating back at least several centuries [17, 18], articulates natural axioms about the properties that the mapping from rankings to a winner should satisfy, and has led to extensive work (see, e.g., [12] for an overview). Unfortunately, many of the key results are impossibility results, in particular the famous Gibbard-Satterthwaite Theorem [21, 31] showing that there is no truthful mechanism satisfying very minimal additional properties.

An alternative that has gained much recent popularity, in particular in the computer science community, is to view social choice through the lens of optimization and approximation. In this line of work (e.g., [10, 14, 29, 30]), it is assumed that one can quantify the utility (or cost) that a voter derives from a candidate. These individual utilities or costs can then be aggregated into a social welfare or cost, e.g., by taking the average or median. The social welfare/cost captures how good of a choice a candidate is for the voter population overall.

The problem with this approach, articulated clearly in [11, 3], is that voting mechanisms typically allow voters only to communicate a ranking

of candidates, but not the actual utilities/costs; furthermore, even if a mechanism provided a way to communicate numerical scores, it is not clear that voters could compute or estimate them accurately. In other words, “one can quantify” is more of an abstract statement than one referring to any decision maker involved in the process. Thus, even though the voting mechanism must optimize a

cardinal objective function, it only receives ordinal information as input, namely, for each voter, whether her222 For ease of presentation, we use female pronouns for voters and male pronouns for candidates throughout. utility/cost for candidate is larger or smaller than that for candidate .

As a result, mechanisms must optimize the social welfare robustly, choosing a candidate that has high welfare regardless of what the actual cardinal objective values are — so long as they are consistent with the reported ordinal rankings. The distortion of a mechanism is the worst-case ratio between the welfare/cost of the mechanism’s selected (based only on ordinal information) candidate and the optimum (with full knowledge of the cardinal values) candidate, over all possible inputs. (Formal definitions of this concept and all other terms can be found in Section 2.)

Our discussion so far has been in terms of general utilities/cost. While some positive results can be obtained for fairly general classes of utility functions (e.g., [10, 14, 29, 30]), stronger results are achievable when the functions take more specific forms. A particularly natural way of defining costs is in terms of a joint metric space defined on candidates and voters, where the distance between voter and candidate captures their difference in opinion, and hence the cost. Voters then rank candidates by non-decreasing distance from themselves.333Such distance-based rankings had been considered in a large body of earlier work, e.g., [8, 9, 19, 26, 25, 7, 6], though most of the listed papers studied such rankings specifically when the metric is the line; such preference orders are often called single-peaked. The approach of using the distances explicitly as the cost objective for optimization was proposed in [3]; [2] is an expanded/improved journal version, and [1] provides a broader overview of the area and its results. While [2] consider both the average and median of all voters’ costs as the overall objective, here, we focus solely on the average/total cost.

The main result of [3, 2] is that under the model of metric costs, many widely used voting rules (including Plurality, Veto, Borda count, and others) have distortion linear in the number of candidates or worse. Furthermore, even with just 2 candidates and a 1-dimensional metric space, every deterministic voting mechanism has distortion at least 3. On the positive side, [3, 2] show that any rule which always outputs a candidate from the uncovered set of candidates has distortion at most 5, for all metric spaces and numbers of candidates. Uncovered sets are defined in terms of the tournament graph on candidates in which the directed edge is present iff a (weak) majority of voters prefer to . The uncovered set is the set of candidates that have a directed path of length at most 2 in to every other candidate (see [27]). Very recently, Munagala and Wang [28] gave a voting rule based on uncovered sets in a weighted tournament graph which improves the upper bound from 5 to .

There is an obvious gap between the lower bound of 3 for the distortion of every mechanism, and the upper bound of . In the original version of [3], it was conjectured that a mechanism called Ranked Pairs (defined in Section 2) achieves a distortion of 3. This conjecture was disproved by [22], who showed a lower bound of 5 on the distortion of Ranked Pairs (and the Schulze rule, also defined in Section 2).

The proof of the upper bound of 5, the recent upper bound of , and many other proofs in the literature are based on reasoning about all metric spaces that are consistent with assumed rankings. They often involve intricate case distinctions and rather ad hoc arguments. So far, a more solid foundation and framework for distortion proofs has been missing from the literature.

1.1 Our Contribution

Our main contribution, presented in Section 3

, is an analysis framework based on LP duality and flows for proving upper bounds on the metric distortion of voting mechanisms. Our point of departure is a well-known linear program for the following problem: given the rankings of all voters, a winning candidate (presumably selected by a mechanism) and an “optimum” candidate, find a metric space maximizing the distortion of this choice; that is, find a metric that makes the selected winner as expensive as possible, subject to the “optimum” candidate having cost 1.

444This approach can of course immediately be leveraged into an optimal polynomial-time voting mechanism; we discuss this more in Section 3.1. We show that the dual of the cost minimization LP can be interpreted as a flow problem with an unusual objective function. Using this framework, in order to show an upper bound on the metric distortion of a particular mechanism, rather than having to explicitly consider all possible metric spaces, it is enough to exhibit a flow of small cost meeting certain demands. We illustrate the power of this analysis framework with three applications.

First, in Section 4, we resolve the distortion of the Ranked Pairs and Schulze rules (defined in Section 2): we show that both have distortion . The upper bound is a clean application of the duality framework, while the lower bound is obtained with a generalization of the example which [22] used to lower-bound the distortion of both rules by 5. The distortion of both rules is thus significantly higher than the distortions of 5 and achieved by the uncovered set mechanisms. Understanding the distortion of the Schulze rule in particular is of importance because it is widely used in practice.

Then, in Section 5, we give a strong generalization of the key lemmas from [3] (Theorem 7) and [28] (Lemma 3.7), used to prove distortions of 5 and for the respective mechanisms under consideration. The common idea of both is that when a large enough fraction of voters prefer to , and a large enough fraction prefer to , then the cost of can be bounded in terms of the cost of . Theorem 7 of [3] is the special case where both fractions are , while Lemma 3.7 of [28] is the case when the first fraction is , and the second is . These bounds immediately imply the upper bounds on the distortion for any candidate in the uncovered set of a suitably defined tournament graph. We give a generalization to arbitrary chains of preferences, and upper-bound the cost of in terms of the cost of when a fraction of voters prefers over , for each . For the specific case when all , the bound can be stated very cleanly: the cost of is at most times that of if is even, and at most times that of if

is odd. Our results fully recover and generalize the bounds of

[3] and [28]. The generalization to longer path lengths can be useful in analyzing voting mechanisms that are missing information. This can happen if the environment restricts the communication between voters and the mechanism, so that parts of the ranking remain unknown, as in [24]. In fact, the results of Section 5 can be used to significantly improve the upper bounds on the performance of “Copeland-like” mechanisms with missing information, compared to the bounds in [24].

As a third application, the flow interpretation naturally suggests a candidate mechanism that might achieve distortion 3, which we present in Section 6. The analysis points to a sufficient condition for distortion 3: that for every given preference profile of the voters, there be a candidate such that for all other candidates , a certain bipartite graph on the voters have a perfect matching. In fact, the mechanism itself can be phrased in this terminology, leading to a purely combinatorial polynomial-time mechanism.

This mechanism was independently discovered and presented in [28]. In [28], it is also shown — again with a case distinction proof over metric spaces — that if such a candidate exists, the mechanism guarantees distortion 3. Our duality framework gives a cleaner and simpler proof of this fact. The main question is then whether the desired candidate always exists.

Munagala and Wang [28] conjecture — as do we — that it does. They phrase a conjecture which is essentially a restatement of the fact that the algorithm succeeds in finding a candidate . In Section 6, we present a slight rephrasing of this conjecture, along with two more very different-looking (in fact, much more self-contained) conjectures, each of which would resolve the question positively, i.e., establish a distortion of 3. One of the two new conjectures is phrased in terms of certain preferences between candidates and sets under randomly drawn preference orders, while another talks about cycles in certain induced subsets of a type of graph we define. The fact that they are sufficient to establish distortion 3 is based on Hall’s Marriage Theorem for bipartite graphs. We have verified the conjecture by hand for candidates, and using exhaustive computer search for . Resolving any of the three conjectures positively would answer the key open question of the field of metric voting, closing the gap between the upper bound of on the best distortion of any deterministic mechanism, and the lower bound of 3.

1.2 Additional Related Work

The observation that mechanisms may have to optimize a cardinal objective function while only given ordinal information (i.e., rankings) extends beyond just voting mechanisms, to more general problems. See, e.g., [5, 1] for results on other optimization problems under ordinal information.

The lower bound of 3 on the distortion of any mechanism is based on worst-case input instances. Better bounds can be obtained when additional assumptions are placed on the instances. As one example, [4, 23] show that when instances are decisive, in the sense that each voter has a candidate she strongly prefers over all others, better upper bounds on the distortion are obtained. As another example, when the candidates are drawn i.i.d. from the set of all voters, [15] gives improved constant distortion bounds in the case of two candidates, while [16] shows that many position-based scoring rules now achieve constant distortion (instead of linear).

The lower bound of 3 on the distortion of voting mechanisms only applies to deterministic mechanisms. Randomization can lead to lower distortion [4]. For example, it is known that the Randomized Dictatorship mechanism, which outputs the first choice of a uniformly random voter, has distortion strictly smaller than 3.

Our work ignores the issue of incentives, i.e., whether voters truthfully report their preferences. The connection between strategy proofness and distortion in metric voting is studied in [20].

The use of LP duality for analyzing the performance of optimization algorithms has a long history, e.g., in approximation algorithms (see [34]). Another more recent example is the duality framework of Cai, Devanur, and Weinberg [13] (see also references in [13] to prior, less general, work) for analyzing the revenue of Bayesian Incentive Compatible mechanisms. In their case as well, dual solutions can be interpreted as flows, and Cai et al. obtain performance guarantees by exhibiting particular types of “canonical” flows that can be interpreted as witnesses for the revenue guarantees. While this work and ours have the use of duality, and the interpretation as flows, in common, the specific technical details are very different.

2 Preliminaries

2.1 Voters, Candidates, and Social Choice Rules

An instance consists of a set of candidates , and the voters’ preferences among these candidates. Candidates will always be denoted by lowercase letters (and their variations), with specifically reserved for a candidate chosen as winner by a mechanism (which will be clear from the context). Sets of candidates are denoted by uppercase letters . The voters are denoted by and variants thereof, and the set of all voters is .

Each voter has a total order (or preference order or ranking — we use the three terms interchangeably) over the candidates. We write to denote that (strictly) prefers over , and to denote that weakly prefers over ; the difference is that the latter allows . We extend this notation to sets, writing, for instance, to denote that (strictly) prefers all candidates in over all candidates in . We write for the set of voters who rank strictly ahead of all candidates in , and for the set of voters who rank strictly behind all candidates in .

A vote profile

is the vector of the rankings of all voters

. A social choice rule (we use the term mechanism interchangeably) is given the rankings of all voters, i.e., , and deterministically produces as output one winning candidate .

2.2 (Pseudo-)Metric Space and Distortion

The voter preferences are assumed to be derived from distances between voters and candidates. The distance between voter and candidate captures how similar their positions on key issues are. The distances form a pseudo-metric, i.e., they are non-negative and satisfy the triangle inequality555Distances between pairs of voters, or between pairs of candidates, could be defined using shortest-path distances; however, they are irrelevant for our analysis. Symmetry, another defining property of a pseudo-metric, would arise automatically when using this definition. for all voters and candidates .

A vote profile is consistent with the pseudo-metric if and only if each voter ranks the candidates by non-decreasing distance from herself; that is, if whenever . When is consistent with , we write . If there are ties among distances, several vote profiles will be consistent with .

[Social Cost, Distortion]

  1. The social cost of candidate is the sum of distances from to all voters: .

  2. A candidate is an optimum candidate iff he minimizes666There could be multiple optimum candidates — for our analysis, it will never matter which of them is designated as “the” optimum. the social cost: .

  3. The distortion of a mechanism is the largest possible ratio between the cost of the candidate chosen by , and the optimal (with respect to the pseudo-metric , which does not know) candidate :

The main cause for (large) distortion is that while the social choice rule knows the voter preferences , it does not know the pseudo-metric . We can think of the pseudo-metric as being chosen adversarially, based on the winning candidate = chosen by the mechanism. However, the adversary is constrained by having to ensure that is consistent with .

2.3 Ranked Pairs and the Schulze Rule

In Section 4, we will characterize the distortion of the Ranked Pairs and Schulze Rules. We briefly review these rules here. Both are based on a weighted directed graph on the set of candidates . The weight of the edge from candidate to is the fraction of voters who have . As a result, for all .

In Ranked Pairs [33]

, the (ordered) pairs

are considered in non-increasing order of . When the pair is considered, the directed edge is inserted into the graph if and only if doing so creates no cycle. When the insertion process terminates, the graph has a unique source node, which is returned as the winner.

In the Schulze Method [32], a directed weighted graph is created in which each ordered pair has an edge with weight . Then, for each pair , let be the width of the widest path from to , that is, the largest such that there is a path from to on which all edges have . It has been shown [32] that there is a candidate node such that for all other candidates . Any such candidate is returned as the winner.

For the purposes of our analysis, the only property of these methods that matters is captured by the following lemma, which is well known. (We prove it only for completeness.)

Let be the candidate selected by the rule (either Ranked Pairs or Schulze), and any other candidate. Then, there exists a and a sequence of (distinct) candidates with the property that at least a fraction of voters prefer over (for each ), and at most a fraction of voters prefer over .

Proof.

For the Ranked Pairs rule, because was selected, it has no incoming edges in the DAG that is constructed. In particular, this means that Ranked Pairs did not insert the edge , so when it was considered for insertion, it would have caused a cycle, meaning that there was a path from to all of whose edges had been inserted earlier. By the definition of the Ranked Pairs insertion order, this means that all edges on this path had a higher fraction of voters agreeing with them, giving us the path claimed above.

For the Schulze rule, recall that the winner has the property that for all candidates . Let . Then, there is a path from to in which each edge corresponds to a preference by at least a fraction of voters. On the other hand, because is a path from to , at most an fraction of voters can prefer to . ∎

3 The LP Duality Approach and Flows

In this section, we develop the key tool for our analysis: the dual linear program for distortion in metric voting.

The voters’ preferences are given. Let be a candidate that the mechanism is considering as a potential winner, and the optimal candidate. Following [2, 22], we phrase the adversary’s problem of finding the distortion-maximizing metric as a linear program whose variables denote distances between voters and candidates . These distances must be non-negative, obey the triangle inequality, and be consistent with the reported preferences of the voters. The objective is to maximize the distortion, i.e., the ratio between the cost of and the cost of .

[eqn:primal-lp]Maximize∑_v + + & for all x, y, v, v’ (Triangle Inequality)
& for all x, y, v such that (consistency)
∑_v = 1 & (normalization)
∑_v ≥1 & for all (optimality of )
≥0 & for all x, v.

As is standard in the use of LPs for optimizing a ratio, the normalization side-steps the issue of having to write a ratio: for any worst-case metric, one could simply rescale all distances by a constant so that the normalization holds — this does not change any ratios, and thus also not the distortion.

3.1 An Efficient Optimal Mechanism

As already observed in [2, 22], the LP (LABEL:eqn:primal-lp) can be leveraged to immediately yield an instance-optimal polynomial-time mechanism for minimizing distortion, as follows. Given the voter preferences , let denote the maximum LP objective of the LP (LABEL:eqn:primal-lp) for the winner and putative optimum . The distortion for as a winner is then . The mechanism returns as winner any candidate in .

Because the algorithm only involves solving linear programs, it runs in polynomial time. By definition (and correctness of the LP (LABEL:eqn:primal-lp)), the distortion for a given vote profile and winner is ; thus, the mechanism does indeed minimize distortion. Unfortunately, as also observed in [2], it is not immediate from the mechanism and the LP formulation how to bound the distortion for all vote profiles; though [22] conjecture that the LP-based algorithm guarantees distortion at most 3.

The dual program provides a very useful tool towards making the LP-based algorithm combinatorial, and for reducing an analysis of its distortion to simpler combinatorial conjectures. More generally (and perhaps importantly), the dual program provides a general approach for bounding the metric distortion of other voting rules, too.

3.2 The Dual Linear Program

Rearranging the primal LP into normal form, taking the dual, and switching the signs of the variables (for clarity) yields the following dual LP (LABEL:eqn:dual-lp). In this LP, the are the dual variables for the triangle inequality constraints, are the dual variables for the consistency constraints, and the are the dual variables for the normalization/optimality constraints.

[eqn:dual-lp]Minimize∑_x + ∑_y: - ∑_y: + ∑_y, v’ ( - - - )
  ≥  {1 if x = 0 if x ≠ & for all v, x
≥0 & for all v, v’, x, y
≥0 & for all v, x, y
≤0 & for all x ≠.

Notice that is in fact unconstrained, due to the equality constraint in the normalization.

The advantage of studying the dual linear program instead of the primal (or reasoning about the distortion directly) is that it omits any reference to any metric space. Rather than having to reason about all candidate metric spaces consistent with given voting patterns, by weak duality, we only have to exhibit one setting of the dual variables that yields a small dual objective value. Thus, our goal in analyzing a mechanism will be to show that for any voter preferences , with a suitably chosen winner , there is a setting of dual variables giving a small objective value.

3.3 Using the Dual by Exhibiting Flows

The LP (LABEL:eqn:dual-lp) looks rather unwieldy, mostly due to the terms involving the variables. However, by making some specific choices for these variables, it can be interpreted as a flow777Some sources use the word “flow” only when there is a single source and a single sink; here, we will have multiple sources and sinks. We will still use the word “flow” in a generic sense. problem on a suitably defined graph, with a somewhat unusual objective function. This is captured by the following lemma:

Let be a directed graph with vertex set , and edges defined as follows:

  • Whenever , contains the directed edge . We call such edges preference edges.

  • For all and , contains the directed edge . We call such edges sideways edges.

Let be a flow on such that exactly one unit of flow originates at the node for each voter , and flow is only absorbed at nodes for voters . Define the cost of at voter to be .

Then, .

The graph has two types of edges. For any fixed voter , the preference edges (over all candidate pairs ) exactly correspond to ’s preference order. For any fixed candidate , the sideways edges (over all voter pairs ) form a complete directed graph.

The flow’s cost function has two terms for each voter . The first is fairly standard in the study of multi-commodity flows: the capacity required at the sink node to be able to absorb all of the flow. The second one is rather non-standard: for each voter , there is an additional penalty term for all incoming and outgoing flows of nodes for along sideways edges. In other words, using preference edges is much less costly than using sideways edges: the former just route flow, while the latter route the flow, but also incur a cost penalty at both endpoints.

Proof.

Let be a flow with one unit of flow originating at each node , such that flow is only absorbed at nodes . We define dual variables, and show that these dual variables are feasible. Then, we will obtain the statement of the lemma by weak LP duality.

For each triple , we set . For each triple , we set ; notice that we carefully choose as the additional candidate for the dual variable. Finally, we set . All other dual variables are set to 0.

We now verify that this assignment satisfies all dual constraints. First, because and whenever , the dual constraints for all are exactly circulation constraints; that is, they require that (at least) one unit of flow originate with , and that flow be conserved (or appear) at each node . Thus, all of these constraints are satisfied for the given dual variable assignment.

For pairs , the left-hand side of the dual constraint totals the flow into along any edges (these are the variables and the variables), as well as all the and variables, for all . By definition of the dual variables, this is exactly the flow into , plus the flow into and out of all nodes for along edges of the form and . Thus, it is exactly . Because we set , the dual constraints for all pairs are also satisfied.

Since we have a dual feasible solution of objective value , by weak duality, for every metric, the cost of the primal is at most . This completes the proof. ∎

4 Distortion of Ranked Pairs and Schulze

As a first application of Lemma 3.3, we pin down the distortion of the Ranked Pairs and Schulze rules to within constant factors.

Both the Ranked Pairs mechanism and the Schulze rule asymptotically have distortion at most and at least .

Proof.

We begin by proving the upper bounds. Let be the candidate selected by the rule, and the optimum candidate. By Lemma 2.3, applied with , there exists a and a sequence of distinct candidates with the property that for each , at least a fraction of voters prefer over , and at most a fraction of voters prefer over . The existence of with these properties is all that we need from the specific voting rules. The rest of the proof will be completely generic, and would thus also apply to any other voting rule satisfying Lemma 2.3.

We consider two cases, depending on the value of . The case is easy. In this case, at least a fraction of voters prefer over . Lemma 6 from [2] states that if at least a fraction of voters prefer over , then . Applying this lemma to and , the distortion of is at most .

When , we use Lemma 3.3, and define a flow. Let .888To ensure that , we may assume that . For smaller , it is easy to see that the distortion of both rules is at most a constant, which of course can be absorbed in the term. Let . Because , we get that . Consider the candidates for , and . For each , let be the set of voters who prefer candidate to . Because for each , at least a fraction of voters prefer to , a union bound over the candidates shows that for each , at least a fraction of voters prefer over ; that is, .

We are now ready to construct the flow, which we will do by increasing . Initially, each node has one unit of incoming flow. Each node with distributes its unit of flow evenly over the nodes with . Then, for each and each voter , the node routes all its flow to the node along the preference edge ; this preference edge exists because . Subsequently, the flow into gets distributed uniformly to nodes with along sideways edges. This concludes the description of the flow.

We now analyze the flow’s cost, according to the cost metric of Lemma 3.3. Because for all , no node ever has more than units of flow. Now focus on any voter . Cost for is incurred by incoming flow into nodes along sideways edges, outgoing flow from nodes along sideways edges, and flow into . Each of these cost terms is bounded by by the preceding observation. Thus, the total cost for node associated with one particular (for ) is at most , while the cost associated with is at most . Adding these terms of all gives an upper bound of . Since this holds for each , Lemma 3.3 implies that the distortion is at most .

We now turn to a lower bound. Our lower-bound construction is a straightforward generalization of the construction that [22] used to show a lower bound of 5 on the distortion of the two rules. Let be given (assumed even), and set (with foresight999The choice gives the tightest lower bound for this type of construction. Other choices work as well; for instance, gives a lower bound of instead of .) . Our construction has voters and candidates . Voters have for all . We call this the default order. To define the order (and later: distances) for voters , we define the following blocks of consecutive (according to the default order) candidates. Block for consists of the candidates . Block consists of the candidates . (Notice that .) Finally, block consists of the candidates . Voter ranks the blocks in reverse order ; within each block, ranks the candidates by the default order. An example of the block structure and ordering is shown in Figure 1.

Default order:
Voter 4’s ranking:
Figure 1: An illustration of the blocks and the preference ranking of voter 4. In this example, .

Intuitively, this means that on a “global” scale, voters completely disagree with the default order, but locally, it looks like they agree. More formally, each voter disagrees with the default order only for pairs when . In particular, this means that for every pair , only one voter disagrees with the default order. For every pair with , at least the voters have , so the edges have highest weight. In particular, this means that they are inserted first by Ranked Pairs, and hence Ranked Pairs chooses . Similarly, has a path of width to each , exceeding the width for any other node. Thus, the Schulze method also chooses .

We will now define a metric which is consistent with these rankings. Voters have distance to each candidate.101010To avoid tie breaking issues, one can easily perturb these and other equal distances by small values so that all distances are unique and the desired order is uniquely induced. We avoid doing so to not overload the proof with inessential notation. Each voter has distance from all candidates . First, these distances explicitly ensure consistency with the voters’ rankings. It remains to verify the triangle inequality. Consider two voters and candidates . We need to show that . This is trivial if , because has distance to all candidates. If , then the right-hand side contains two terms of , and one term that is at least 1. Hence, the inequality holds. Finally, if both , then let be the blocks such that . The definition of the block structure ensures that ; in particular, . Because , the triangle inequality again holds.

Finally, we compute the social costs of (the winner in Schulze and Ranked Pairs) and . is in block 0 for all voters , and in block 1 for voter 1. Hence, his combined distance from these voters is . With the added distance of from each of , the total cost of is . On the other hand, is at distance 1 from all voters and at distance from , for a total cost of . The ratio is thus at least . This completes the proof. ∎

The upper bound in Corollary 4 was a direct application of our flow-based framework. While the lower bound did not explicitly use the framework, the counter-example was in fact discovered after failed attempts to improve the upper bound. The failure to find ways to route flow very clearly suggested the types of rankings that were obstacles (i.e., reversed block structures). In turn, the distances were found essentially using the primal linear program.

5 Generalization of Distortion Bounds for Undominated Nodes

As a second corollary of Lemma 3.3, we obtain a strong generalization of Theorem 7 in [3] and Lemma 3.7 of [28] (which are given below for comparison). The most general version can be stated as follows:

Let be (distinct) candidates such that for each , at least a fraction of voters prefer candidate over candidate . Define , and for . Let . (Here, independence of a set of natural numbers means that the set contains no two consecutive numbers.) Then, .

Proof.

We define a flow and analyze its cost. For each , we call the nodes (for all voters ) layer . Let be the set of voters with , with for notational simplicity.

We construct the flow layer by layer; our construction will ensure that each node with has exactly units of flow entering. This holds in the base case , because each node in layer 1 is the source node of one unit of flow.

For the i step of the construction, we first route all the flow within layer using sideways edges, from nodes with to nodes with . We then route it to nodes in layer using preference edges. More specifically, to route the flow within layer , we first consider voters . For those voters, units of flow simply stay at . The node for such will have additional incoming flow from other nodes (if ) or additional outgoing flow to other nodes (if ). The remaining flow is routed arbitrarily using single sideways edges from nodes with to nodes with , of course ensuring that each such node has in total units of flow entering.

After this redistribution within layer , each routes its flow to . Notice that this is always possible, because for all . The construction is illustrated with an example in Figure 2.

1

1

1

4

2

2

2

4/3

2/3

2/3

4/3

4/3

4/3
Figure 2: An illustration of the flow construction. In the example, there are 4 voters and 4 relevant candidates, with voter preferences shown on the left. The preference fractions are . Sideways flows are shown in solid red, while flow along preference edges is shown in dashed lines. The dashed lines into nodes for candidate are shown in blue (instead of black), to emphasize that they contribute to the objective function. The amount of flow is given numerically, and also shown using the width of the lines/arcs. Edges that are not used by the flow are not shown.

We now analyze the cost associated with any fixed voter . The cost has two components: the incoming flow at (shown in blue in Figure 2), and the cost associated with incoming/outgoing flow using sideways edges incident on for (shown in red in Figure 2). We begin with the incoming flow at : if , the incoming flow is ; otherwise, it is 0.

Next, we consider the cost associated with sideways edges. As a general guideline (subtleties will be discussed momentarily), when , the node has units of flow coming in along sideways edges, and the node has the same amount of flow leaving along sideways edges. The associated cost of both together is . Two obvious exceptions are layers and . For , one unit of flow simply originates with , resulting in no cost. For , no sideways edge is used to route outgoing flow; however, this is compensated by the incoming flow at (discussed in the preceding paragraph), which adds the same cost term.

However, simply adding up the bounds from the preceding paragraph over all steps with is too pessimistic, because our flow construction avoids routing more flow than necessary when . A tighter bound is captured by the following lemma, proved below:

Let be the set of all indices with . can be partitioned into disjoint intervals of integers (for some ) such that:

  1. For each , there exists an index such that and ; that is, the are monotone non-increasing from to , and monotone non-decreasing from to .

  2. The total cost of flow (both sideways flow and flow into in case ) associated with nodes with is at most .

To apply Lemma 5, the key observation is that the set is independent, i.e., contains no two consecutive integers. If it did — say, and — then both . If , this would contradict the maximality of in the definition of ; on the other hand, if , then by the definition of , so it is impossible that .

Now, summing up the costs for each of the disjoint intervals, we obtain that the total cost of the flow at nodes associated with (both sideways flow and flow into ) is at most ; because the set of is independent, this sum is at most . Using Lemma 3.3, this completes the proof. ∎

Lemma 5 We inductively define satisfying the first property, then show that they also guarantee the second property. For the base case, we set (for convenience) . For the inductive step, focus on any . Define . (The construction terminates when there is no such .) Let . In words, is the largest index such that all indices between and are in , and the values are monotone non-increasing all the way to . Notice that because , we also have . Now, let . In words, is the largest index such that all indices between and (and thus also between and ) are in , and the values are monotone non-decreasing from to . This definition explicitly ensures that each interval is entirely contained in , and satisfies the given monotonicity conditions. We now verify the second property.

We first consider the case , where . The important observation for the proof, also visible in Figure 2, is that when , this eliminates sideways flow to and from nodes associated with . Specifically, none of the nodes for have outgoing flow along sideways edges (since all their flow stays for the next step). The incoming flow at along sideways edges is exactly (with for convenience); the remaining flow at is what is kept from step . Similarly, none of the nodes for have incoming flow along sideways edges, since the node has enough flow to meet all of the needs of ; the outgoing flow at such nodes along sideways edges is exactly (with ). Summing up all the incoming flows for (a telescoping series), and the outgoing flows for (another telescoping series), the total flow on sideways edges for all with is at most . Because , there is no other cost associated with these nodes. Next, we consider the remaining cases .

  1. If , then by construction, ; otherwise, the fact that would rule out setting . Thus, the entire interval is just . Because there is no incoming sideways flow into , the only cost is for one unit of outgoing sideways flow, i.e., the cost is .

  2. If , then we must have and , because always by definition. We can directly apply the general analysis, except that we can subtract one unit of cost, the reason being (as in the case ) that there is no one unit of sideways flow into . Thus, the total cost associated with the interval is at most .

  3. If , then . There is no sideways flow out of (and there are no nodes for to consider). Thus, the total cost of the sideways flows associated with is at most . On the other hand, in this case, there is also a cost of for flow into (the blue flow in Figure 2); however, the total cost is still bounded by .

This shows that the bound holds for all cases of the interval.

5.1 Special Cases

Lemma 3.7 of [28] is the special case of Corollary 5 with , and . Our Corollary 5 then exactly recovers the bound of .

When we have a uniform lower bound on the , Corollary 5 can be simplified significantly. (A direct proof of Corollary 5.1 would also be simpler than the proof of the more general Corollary 5.)

Let be (distinct) candidates such that for each , at least a fraction of the voters prefer candidate over candidate . Then, if is even, ; if is odd, .

Proof.

We substitute for all in Corollary 5; then, we observe that for even , the independent set of integers giving the largest sum is , while for odd , it is . ∎

The asymmetry between even and odd disappears when (i.e., in the case of the majority graph), where the bound simply becomes . The result thus strongly generalizes Theorem 7 in [3], which is the special case of and .

6 A Candidate Algorithm for Distortion 3

As a third application, we derive a purely combinatorial (i.e., not LP-based) voting mechanism, which we conjecture to have distortion 3. We show that this conjecture would follow from any of three different-looking combinatorial conjectures we will formulate.

The point of departure for the derivation of the mechanism is Corollary 6, which simplifies Lemma 3.3, reducing it to a purely combinatorial property of a certain graph. Corollary 6 was proved as Theorem 4.4 in [28], using a significantly more complex proof.

For any two candidates , we consider the following bipartite graph on the node set ; that is, there is one node on the “left” for each voter