Home /  Social Choice Seminar: "Resolving the Optimal Metric Distortion Conjecture" & "Breaking the Metric Voting Distortion Barrier"

# Seminar

Social Choice Seminar: "Resolving the Optimal Metric Distortion Conjecture" & "Breaking the Metric Voting Distortion Barrier" November 27, 2023 (11:00 AM PST - 12:00 PM PST)
Parent Program: Algorithms, Fairness, and Equity SLMath: Baker Board Room
Speaker(s) Daniel Halpern (Harvard University), Kangning Wang (Stanford University)
Description No Description
Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
Video
Abstract/Media

In this session, we will have two spotlight talks about recent advances in developing voting rules with low metric distortion, by Daniel Halpern (Harvard) and Kangning Wang (Stanford).

"Resolving the Optimal Metric Distortion Conjecture" - Daniel Halpern

Abstract: We study the following metric distortion problem: there are two finite sets of points, V and C, that lie in the same metric space, and our goal is to choose a point in C whose total distance from the points in V is as small as possible. However, rather than having access to the underlying distance metric, we only know, for each point in V, a ranking of its distances to the points in C. We propose algorithms that choose a point in C using only these rankings as input and we provide bounds on their distortion (worst-case approximation ratio). A prominent motivation for this problem comes from voting theory, where V represents a set of voters, C represents a set of candidates, and the rankings correspond to ordinal preferences of the voters.

A major conjecture in this framework is that the optimal deterministic algorithm has distortion 3. We resolve this conjecture by providing a polynomial-time algorithm that achieves distortion 3, matching a known lower bound. We do so by proving a novel lemma about matching voters to candidates, which we refer to as the ranking-matching lemma. This lemma induces a family of novel algorithms, which may be of independent interest, and we show that a special algorithm in this family achieves distortion 3. We also provide more refined, parameterized, bounds using the notion of α-decisiveness, which quantifies the extent to which a voter may prefer her top choice relative to all others. Finally, we introduce a new randomized algorithm with improved distortion compared to known results, and also provide improved lower bounds on the distortion of all deterministic and randomized algorithms.

"Breaking the Metric Voting Distortion Barrier" - Kangning Wang

Abstract: We consider the following well studied problem of metric distortion in social choice. Suppose we have an election with nvoters and m candidates who lie in a shared metric space. We would like to design a voting rule that chooses a candidate whose average distance to the voters is small. However, instead of having direct access to the distances in the metric space, each voter gives us a ranked list of the candidates in order of distance. Can we design a rule that regardless of the election instance and underlying metric space, chooses a candidate whose cost differs from the true optimum by only a small factor (known as the distortion)?

A long line of work culminated in finding deterministic voting rules with metric distortion 3, which is the best possible for deterministic rules and many other classes of voting rules. However, without any restrictions, there is still a significant gap in our understanding: Even though the best lower bound is substantially lower at 2.112, the best upper bound is still 3, which is attained even by simple rules such as Random Dictatorship. Finding a rule that guarantees distortion 3−ε for some constant ε has been a major challenge in computational social choice.

In this work, we give a rule that guarantees distortion less than 2.753. To do so we study a handful of voting rules that are new to the problem. One is Maximal Lotteries, a rule based on the Nash equilibrium of a natural zero-sum game which dates back to the 60’s. The others are novel rules that can be thought of as hybrids of Random Dictatorship and the Copeland rule. Though none of these rules can beat distortion 3 alone, a careful randomization between Maximal Lotteries and any of the novel rules can.