Communication, Distortion, and Randomness in Metric Voting

Kempe, David

doi:10.1609/aaai.v34i02.5582

Cited by 32 publications

(43 citation statements)

References 13 publications

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“…e distortion of randomized rules has also received significant a ention. For single-winner voting, the best possible distortion under the utilitarian model is known to be Θ( √ m) [Boutilier et al, 2015], while that under the metric model still remains a challenging open question [Anshelevich and Postl, 2017b;Kempe, 2020a;Charikar and Ramakrishnan, 2022]. Caragiannis et al [2017] provide bounds on the best possible distortion of randomized multiwinner voting rules under the utilitarian model, and our work does so under the metric model.…”

Section: Related Workmentioning

confidence: 84%

The Metric Distortion of Multiwinner Voting

Caragiannis¹,

Shah²,

Voudouris³

2022

Preprint

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We extend the recently introduced framework of metric distortion to multiwinner voting. In this framework, n agents and m alternatives are located in an underlying metric space. e exact distances between agents and alternatives are unknown. Instead, each agent provides a ranking of the alternatives, ordered from the closest to the farthest. Typically, the goal is to select a single alternative that approximately minimizes the total distance from the agents, and the worst-case approximation ratio is termed distortion. In the case of multiwinner voting, the goal is to select a commi ee of k alternatives that (approximately) minimizes the total cost to all agents. We consider the scenario where the cost of an agent for a commi ee is her distance from the q-th closest alternative in the commi ee. We reveal a surprising trichotomy on the distortion of multiwinner voting rules in terms of k and q: e distortion is unbounded when q k/3, asymptotically linear in the number of agents when k/3 < q k/2, and constant when q > k/2.

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Section: Related Workmentioning

confidence: 84%

The Metric Distortion of Multiwinner Voting

Caragiannis¹,

Shah²,

Voudouris³

2022

Preprint

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“…Going beyond these conjectures, we believe that the duality-based framework may be useful for bounding the performance of other voting mechanisms, in particular, those that may miss information on parts of voters' ranking. For instance, such a situation can occur in the setting of (Kempe 2020), where voters can only name the candidates in a subset of positions on their ballot, rather than giving a complete ranking. The analysis of a mechanism proposed in (Kempe 2020) becomes much simpler (and tighter) using the techniques developed here.…”

Section: Discussionmentioning

confidence: 99%

“…For instance, such a situation can occur in the setting of (Kempe 2020), where voters can only name the candidates in a subset of positions on their ballot, rather than giving a complete ranking. The analysis of a mechanism proposed in (Kempe 2020) becomes much simpler (and tighter) using the techniques developed here.…”

Section: Discussionmentioning

confidence: 99%

“…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 (Kempe 2020). 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 (Kempe 2020).…”

Section: Our Contributionmentioning

confidence: 99%

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An Analysis Framework for Metric Voting based on LP Duality

Kempe

2020

AAAI

Self Cite

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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 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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“…Discussion and Related Work. Ordinal approximation for the minimum social cost (or maximum social welfare) with underlying utilities/distances between agents and alternatives has been studied in many settings including social choice [4,6,11,13,15,16,19,21,25,28,30,31,33,38,39,41,[43][44][45], matchings [7-9, 12, 18, 22, 29], secretary problems [37], participatory budgeting [10,34], general graph problems [1,7], hierarchical clustering [23], and many other models in recent years. The general assumption of the ordinal setting is that we only have the ordinal preferences of the agents over the alternatives, and the goal is to form a solution that has close to optimal social cost.…”

Section: While Only Ordinal Information About Agent Preferences Is Kn...mentioning

confidence: 99%

Ordinal Approximation for Social Choice, Matching, and Facility Location Problems Given Candidate Positions

Anshelevich

Zhu

2021

ACM Trans. Econ. Comput.

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In this work, we consider general facility location and social choice problems, in which sets of agents A and facilities F are located in a metric space, and our goal is to assign agents to facilities (as well as choose which facilities to open) to optimize the social cost. We form new algorithms to do this in the presence of only ordinal information , i.e., when the true costs or distances of the agents from the facilities are unknown , and only the ordinal preferences of the agents for the facilities are available. The main difference between our work and previous work in this area is that, while we assume that only ordinal information about agent preferences is known, we also know the exact locations of the possible facilities F. Due to this extra information about the facilities, we are able to form powerful algorithms that have small distortion , i.e., perform almost as well as omniscient algorithms (which know the true numerical distances between agents and facilities) but use only ordinal information about agent preferences. For example, we present natural social choice mechanisms for choosing a single facility to open with distortion of at most 3 for minimizing both the total and the median social cost; this factor is provably the best possible. We analyze many general problems including matching, k -center, and k -median, and we present black-box reductions from omniscient approximation algorithms with approximation factor β to ordinal algorithms with approximation factor 1+2β doing this gives new ordinal algorithms for many important problems and establishes a toolkit for analyzing such problems in the future.

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Communication, Distortion, and Randomness in Metric Voting

Cited by 32 publications

References 13 publications

The Metric Distortion of Multiwinner Voting

The Metric Distortion of Multiwinner Voting

An Analysis Framework for Metric Voting based on LP Duality

Ordinal Approximation for Social Choice, Matching, and Facility Location Problems Given Candidate Positions

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