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Boosting Sortition via Proportional Representation

Soroush Ebadian, Evi Micha

TL;DR

The paper addresses proportional representation in sortition by formalizing a core-based notion in a representation metric space and defining a $q$-cost to measure representational closeness. It proves that uniform random sampling is fair and achieves a constant ex ante core approximation, but may fail ex post for many $q$; to overcome this, it introduces FairGreedyCapture, a fair algorithm that yields panels in a constant-approximate ex post core (6-approx for all $q\in [k]$). It also provides an auditing method to quantify core violations and demonstrates empirically that FairGreedyCapture substantially improves ex post core guarantees while maintaining competitive social-cost performance on real datasets (Adult and ESS-UK). The work connects core-based proportionality with practical quotas and shows how to translate core guarantees into quotas over features, offering a path toward transparent, fair, and proportionally representative sortition in real-world settings.

Abstract

Sortition is based on the idea of choosing randomly selected representatives for decision making. The main properties that make sortition particularly appealing are fairness -- all the citizens can be selected with the same probability -- and proportional representation -- a randomly selected panel probably reflects the composition of the whole population. When a population lies on a representation metric, we formally define proportional representation by using a notion called the core. A panel is in the core if no group of individuals is underrepresented proportional to its size. While uniform selection is fair, it does not always return panels that are in the core. Thus, we ask if we can design a selection algorithm that satisfies fairness and ex post core simultaneously. We answer this question affirmatively and present an efficient selection algorithm that is fair and provides a constant-factor approximation to the optimal ex post core. Moreover, we show that uniformly random selection satisfies a constant-factor approximation to the optimal ex ante core. We complement our theoretical results by conducting experiments with real data.

Boosting Sortition via Proportional Representation

TL;DR

The paper addresses proportional representation in sortition by formalizing a core-based notion in a representation metric space and defining a -cost to measure representational closeness. It proves that uniform random sampling is fair and achieves a constant ex ante core approximation, but may fail ex post for many ; to overcome this, it introduces FairGreedyCapture, a fair algorithm that yields panels in a constant-approximate ex post core (6-approx for all ). It also provides an auditing method to quantify core violations and demonstrates empirically that FairGreedyCapture substantially improves ex post core guarantees while maintaining competitive social-cost performance on real datasets (Adult and ESS-UK). The work connects core-based proportionality with practical quotas and shows how to translate core guarantees into quotas over features, offering a path toward transparent, fair, and proportionally representative sortition in real-world settings.

Abstract

Sortition is based on the idea of choosing randomly selected representatives for decision making. The main properties that make sortition particularly appealing are fairness -- all the citizens can be selected with the same probability -- and proportional representation -- a randomly selected panel probably reflects the composition of the whole population. When a population lies on a representation metric, we formally define proportional representation by using a notion called the core. A panel is in the core if no group of individuals is underrepresented proportional to its size. While uniform selection is fair, it does not always return panels that are in the core. Thus, we ask if we can design a selection algorithm that satisfies fairness and ex post core simultaneously. We answer this question affirmatively and present an efficient selection algorithm that is fair and provides a constant-factor approximation to the optimal ex post core. Moreover, we show that uniformly random selection satisfies a constant-factor approximation to the optimal ex ante core. We complement our theoretical results by conducting experiments with real data.
Paper Structure (31 sections, 19 theorems, 36 equations, 3 figures, 5 tables, 3 algorithms)

This paper contains 31 sections, 19 theorems, 36 equations, 3 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our approach
  3. Proportional Representation via Core.
  4. Representation Metric Space.
  5. $q$-Cost.
  6. Our Contribution
  7. Related Work
  8. Preliminaries
  9. Ex Post $q$-Core.
  10. Ex Ante $q$-Core.
  11. Fairness and Ex Post Core
  12. Sampling panels from the $(k/n)$-fractional allocation.
  13. Ex Post Core and Quotas over Features
  14. Uniform Selection and Ex Ante Core
  15. Auditing Ex Post Core
  16. ...and 16 more sections

Key Result

Theorem 1

For any $q \in [k-1]$ and $\left\lfloor n/k \right\rfloor \ge k$, there exists an instance such that uniform selection is not in the ex post $\alpha$-$q$-core for any bounded $\alpha$.

Figures (3)

  • Figure 1: Diagram for Proof of \ref{['lem:US-ex-ante']}
  • Figure 2: Ex post core violation of $\textsc{FairGreedyCapture}$ and Uniform with $k = 40$
  • Figure 3: Approximation to the optimal social cost of $\textsc{FairGreedyCapture}$ and Uniform with with $k = 40$

Theorems & Definitions (39)

  • Theorem 1
  • proof
  • Theorem 2: Birkhoff-von Neumann
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Lemma 3: Chu–Vandermonde identity
  • ...and 29 more