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Metric Distortion in Peer Selection

Javier Cembrano, Golnoosh Shahkarami

TL;DR

This work initiates metric distortion analysis for peer selection, where voters and candidates coincide, focusing on a line metric and four objective-cost combinations. It develops the Median Alternation rule and a two-location reduction technique to obtain tight distortion bounds, revealing that hardness persists even on the line while enabling better constants in some cases. Key contributions include near-tight bounds for utilitarian additive cost (distortion in $[1,2]$ with a precise closed form), sharp results for utilitarian $q$-cost (unbounded for $q\le\frac{k}{2}$ and tight for $k=q=2$), and optimal, parity-aware results for egalitarian additive cost (via $k$-Extremes) and egalitarian $q$-cost (constant distortion for $q>\frac{k}{3}$). The findings advance understanding of how ordinal information constrains representation quality in peer committees and point to future work on broader metric spaces and strategyproof mechanisms.

Abstract

In the metric distortion problem, a set of voters and candidates lie in a common metric space, and a committee of $k$ candidates must be elected. The objective is to minimize a social cost, defined as a function of the distances between voters and their chosen representatives, while the voting rule only has access to ordinal preferences. The distortion of a rule is the worst-case ratio between the social cost of its outcome and that of the optimal committee, taken over all consistent preferences and metrics. We initiate the study of metric distortion in peer selection, where voters and candidates coincide. We consider four objectives, obtained by combining two aggregation rules with two types of social cost. Under additive aggregation, an individual's cost is the sum of their distances to all committee members; under $q$-cost, it is their distance to the $q$th closest member. The overall social cost is either utilitarian, given by the sum of all individual costs, or egalitarian, given by the maximum individual cost. Surprisingly, we find that even on the line metric, peer selection retains much of the hardness of the general case: Lower bounds remain strictly larger than one for all objectives, and cases where bounded distortion is impossible in general remain so here as well. On a positive note, cases with bounded distortion in the general setting achieve better constants in peer selection. For utilitarian cost, selecting the $k$ middle agents achieves a distortion between $1$ and $2$ under additive aggregation. Under $q$-cost, we show positive results for $q=k=2$, but impossibility results largely carry over. For egalitarian cost, selecting the extremes yields an optimal distortion of $2$ under additive aggregation and for $q$-cost with $q>k/3$. Thus, while peer selection on the line metric allows better constants, fundamental hardness barriers persist.

Metric Distortion in Peer Selection

TL;DR

This work initiates metric distortion analysis for peer selection, where voters and candidates coincide, focusing on a line metric and four objective-cost combinations. It develops the Median Alternation rule and a two-location reduction technique to obtain tight distortion bounds, revealing that hardness persists even on the line while enabling better constants in some cases. Key contributions include near-tight bounds for utilitarian additive cost (distortion in with a precise closed form), sharp results for utilitarian -cost (unbounded for and tight for ), and optimal, parity-aware results for egalitarian additive cost (via -Extremes) and egalitarian -cost (constant distortion for ). The findings advance understanding of how ordinal information constrains representation quality in peer committees and point to future work on broader metric spaces and strategyproof mechanisms.

Abstract

In the metric distortion problem, a set of voters and candidates lie in a common metric space, and a committee of candidates must be elected. The objective is to minimize a social cost, defined as a function of the distances between voters and their chosen representatives, while the voting rule only has access to ordinal preferences. The distortion of a rule is the worst-case ratio between the social cost of its outcome and that of the optimal committee, taken over all consistent preferences and metrics. We initiate the study of metric distortion in peer selection, where voters and candidates coincide. We consider four objectives, obtained by combining two aggregation rules with two types of social cost. Under additive aggregation, an individual's cost is the sum of their distances to all committee members; under -cost, it is their distance to the th closest member. The overall social cost is either utilitarian, given by the sum of all individual costs, or egalitarian, given by the maximum individual cost. Surprisingly, we find that even on the line metric, peer selection retains much of the hardness of the general case: Lower bounds remain strictly larger than one for all objectives, and cases where bounded distortion is impossible in general remain so here as well. On a positive note, cases with bounded distortion in the general setting achieve better constants in peer selection. For utilitarian cost, selecting the middle agents achieves a distortion between and under additive aggregation. Under -cost, we show positive results for , but impossibility results largely carry over. For egalitarian cost, selecting the extremes yields an optimal distortion of under additive aggregation and for -cost with . Thus, while peer selection on the line metric allows better constants, fundamental hardness barriers persist.

Paper Structure

This paper contains 29 sections, 16 theorems, 118 equations, 10 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Our Contributions and Techniques
  3. Utilitarian Additive Cost.
  4. Utilitarian $q$-Cost.
  5. Egalitarian Additive Cost.
  6. Egalitarian $q$-Cost.
  7. Further Related Work
  8. Single-Winner Voting.
  9. Multi-Winner Voting.
  10. Restricted Voting Settings.
  11. Mechanism Design in Committee Selection.
  12. Preliminaries
  13. Election.
  14. Line metric.
  15. Social cost.
  16. ...and 14 more sections

Key Result

Lemma 1

For every election $\mathcal{E}=([n],k,\succ)$, we can compute a permutation $\pi\colon [n]\to[n]$ of the agents such that, for any consistent position vector $x\in (-\infty,\infty)^n$ with $x\ \rhd \succ$, we have either $x_{\pi(1)} \leq x_{\pi(2)} \leq \dots \leq x_{\pi(n)}$ or $x_{\pi(n)} \leq x_

Figures (10)

  • Figure 1: Distortion of Median Alternation stated in \ref{['thm:Util-Add']} for $n=100$ and $k\in \{2,\ldots,99\}$.
  • Figure 2: Metrics considered in the proof of \ref{['thm:utilitarian-add-lb']}. In this and all similar figures throughout the paper, the (sets of) agents are represented by circles, with the identity of the agents or sets below them, and the distances between them are written on top of the corresponding line segments. All figures consider indistinguishable metrics for a certain preference profile of the agents and thus any voting rule must select the same subsets for any of these metrics.
  • Figure 3: Metrics considered in the proof of \ref{['thm:utilitarian-1cost']}. In this and all similar figures throughout the paper, the (sets of) agents are represented by circles, with the identity of the agents or sets below them, and the distances between them are written on top of the corresponding line segments. All figures consider indistinguishable metrics for a certain preference profile of the agents and thus any voting rule must select the same subsets for any of these metrics.
  • Figure 4: Lower bound on the distortion of any rule for utilitarian $q$-cost stated in \ref{['thm:utilitarian-lowerbound-qcost']}, and metrics used to prove it.
  • Figure 5: Stair diagrams for 9 and 8 agents. The common cost incurred by any committee is shown in gray; the additional cost of two specific committees is shown in red and green.
  • ...and 5 more figures

Theorems & Definitions (33)

  • Lemma 1: elkind2014recognizing, babashah2024distortionmultiwinnerelectionsline
  • Lemma 2
  • proof
  • Theorem 1
  • Lemma 3
  • proof : Proof of \ref{['lem:agent-move-distortion']}
  • Claim 3.1
  • proof
  • Lemma 4
  • proof
  • ...and 23 more