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Can a Few Decide for Many? The Metric Distortion of Sortition

Ioannis Caragiannis, Evi Micha, Jannik Peters

TL;DR

The paper advances the study of sortition by formalizing metric distortion to compare panel decisions against population welfare in a shared metric space. It shows that uniform random selection achieves near-optimal ex-ante distortion with panel size $k=O(\varepsilon^{-2}\ln(m/\varepsilon))$, and that the Fair Greedy Capture algorithm attains the same ex-ante bound with $k=O(\varepsilon^{-3}\ln(m/\varepsilon))$ while guaranteeing constant ex-post distortion. Theoretical results are complemented by experiments on real datasets (Adult and ESS Austria), where Fair Greedy Capture consistently outperforms uniform selection in both ex-ante and ex-post metrics. The work illuminates the trade-offs between fairness and representativeness in panel-based decision making and outlines directions for closing gaps between lower and upper distortion bounds and extending to multi-suggestion settings.

Abstract

Recent works have studied the design of algorithms for selecting representative sortition panels. However, the most central question remains unaddressed: Do these panels reflect the entire population's opinion? We present a positive answer by adopting the concept of metric distortion from computational social choice, which aims to quantify how much a panel's decision aligns with the ideal decision of the population when preferences and agents lie on a metric space. We show that uniform selection needs only logarithmically many agents in terms of the number of alternatives to achieve almost optimal distortion. We also show that Fair Greedy Capture, a selection algorithm introduced recently by Ebadian & Micha (2024), matches uniform selection's guarantees of almost optimal distortion and also achieves constant ex-post distortion, ensuring a "best of both worlds" performance.

Can a Few Decide for Many? The Metric Distortion of Sortition

TL;DR

The paper advances the study of sortition by formalizing metric distortion to compare panel decisions against population welfare in a shared metric space. It shows that uniform random selection achieves near-optimal ex-ante distortion with panel size , and that the Fair Greedy Capture algorithm attains the same ex-ante bound with while guaranteeing constant ex-post distortion. Theoretical results are complemented by experiments on real datasets (Adult and ESS Austria), where Fair Greedy Capture consistently outperforms uniform selection in both ex-ante and ex-post metrics. The work illuminates the trade-offs between fairness and representativeness in panel-based decision making and outlines directions for closing gaps between lower and upper distortion bounds and extending to multi-suggestion settings.

Abstract

Recent works have studied the design of algorithms for selecting representative sortition panels. However, the most central question remains unaddressed: Do these panels reflect the entire population's opinion? We present a positive answer by adopting the concept of metric distortion from computational social choice, which aims to quantify how much a panel's decision aligns with the ideal decision of the population when preferences and agents lie on a metric space. We show that uniform selection needs only logarithmically many agents in terms of the number of alternatives to achieve almost optimal distortion. We also show that Fair Greedy Capture, a selection algorithm introduced recently by Ebadian & Micha (2024), matches uniform selection's guarantees of almost optimal distortion and also achieves constant ex-post distortion, ensuring a "best of both worlds" performance.
Paper Structure (19 sections, 18 theorems, 57 equations, 3 figures, 1 algorithm)

This paper contains 19 sections, 18 theorems, 57 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our approach and results.
  3. Related Work
  4. Sortition.
  5. Distortion.
  6. Clustering.
  7. Model and Motivation
  8. The Benefit of Fair Selection
  9. Uniform Selection
  10. Distortion Guarantees for Fair Greedy Capture
  11. Experiments
  12. Experimental Setup
  13. Representation Metric Construction.
  14. Our Results
  15. Ex-Post Distortion.
  16. ...and 4 more sections

Key Result

Theorem 1

Let $k,n$ be positive integers with $k\leq n$ and $\mathcal{A}_k$ a deterministic algorithm that selects a panel of size $k$ among $n$ agents. For any $\varepsilon>0$, there exists an instance with $n$ agents and two alternatives in which $\mathcal{A}_k$ has distortion of at least $5 - \frac{12k}{n+

Figures (3)

  • Figure 1: Metric spaces for \ref{['ex:example-1']}. Edges without labels have length $1$. The graph indicates the probability of the different alternatives being the top choice of a uniformly selected panel, for different panel sizes.
  • Figure 2: Average distortion for $k$ from $1$ to $40$. The figures include the 95% confidence interval.
  • Figure 3: Distribution of ex-post distortion values for $k$ from $1$ to $49$.

Theorems & Definitions (33)

  • Example 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5: S74
  • ...and 23 more