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Fairness in the Multi-Secretary Problem

Georgios Papasotiropoulos, Zein Pishbin

TL;DR

This work addresses fairness in online multi-winner selection with cardinal ballots by formalizing the online multi-secretary/election model and proving that Extended Justified Representation (EJR) cannot be satisfied online. It then develops and analyzes online adaptations of voting rules, notably Online Method of Equal Shares (with BOS), Greedy Budgeting, and Online Nash, providing probabilistic guarantees and empirical validation. The theoretical results emphasize impossibility of strong proportionality and justify risk-tolerant online strategies, while experiments show BOS often yields the strongest fairness signals and robust performance across diverse datasets. The findings offer practical mechanisms for fair online decision-making in settings such as hiring, participatory budgeting, and online funding platforms.

Abstract

This paper bridges two perspectives: it studies the multi-secretary problem through the fairness lens of social choice, and examines multi-winner elections from the viewpoint of online decision making. After identifying the limitations of the prominent proportionality notion of Extended Justified Representation (EJR) in the online domain, the work proposes a set of mechanisms that merge techniques from online algorithms with rules from social choice -- such as the Method of Equal Shares and the Nash Rule -- and supports them through both theoretical analysis and extensive experimental evaluation.

Fairness in the Multi-Secretary Problem

TL;DR

This work addresses fairness in online multi-winner selection with cardinal ballots by formalizing the online multi-secretary/election model and proving that Extended Justified Representation (EJR) cannot be satisfied online. It then develops and analyzes online adaptations of voting rules, notably Online Method of Equal Shares (with BOS), Greedy Budgeting, and Online Nash, providing probabilistic guarantees and empirical validation. The theoretical results emphasize impossibility of strong proportionality and justify risk-tolerant online strategies, while experiments show BOS often yields the strongest fairness signals and robust performance across diverse datasets. The findings offer practical mechanisms for fair online decision-making in settings such as hiring, participatory budgeting, and online funding platforms.

Abstract

This paper bridges two perspectives: it studies the multi-secretary problem through the fairness lens of social choice, and examines multi-winner elections from the viewpoint of online decision making. After identifying the limitations of the prominent proportionality notion of Extended Justified Representation (EJR) in the online domain, the work proposes a set of mechanisms that merge techniques from online algorithms with rules from social choice -- such as the Method of Equal Shares and the Nash Rule -- and supports them through both theoretical analysis and extensive experimental evaluation.

Paper Structure

This paper contains 16 sections, 7 theorems, 2 equations, 4 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fairness Considerations from Social Choice
  4. Proportional Selection with Certainty
  5. Satisfying and Approximating Extended Justified Representation
  6. Satisfying Weakenings of Extended Justified Representation
  7. Proportional Selection with Risk
  8. Online Method of Equal Shares and Beyond
  9. A Framework for Bringing Voting Rules Online & the Case of MES with Bounded Overspending
  10. Online Nash Product Rule
  11. Empirical Evaluation
  12. Experiment 1: Pabulib Instances
  13. Experiment 2: Sushi and MovieLens Datasets
  14. Experiment 3: Sampled Datasets
  15. Experiment 4: Polarized Instances
  16. ...and 1 more sections

Key Result

Theorem 1

There is no online voting rule that satisfies $\beta$-EJR, for any finite positive value of $\beta$, even for range ballots.

Figures (4)

  • Figure 1: Gini coefficient against the number of voters (top-left), average satisfaction (top-right) and 25th percentile (bottom-left) relative to Offline MES against the number of voters, and Exclusion ratio against the approval probability $p$ (bottom-right) in the IC model (Experiment 3).
  • Figure 2: Gini coefficient against the number of voters (top-left), average satisfaction (top-right) and 25th percentile (bottom-left) relative to Offline MES against the number of voters, and 25th percentile relative to Offline MES against the dispersion parameter $\phi$ (bottom-right) in the Mallows model (Experiment 3).
  • Figure 3: Gini coefficient against the number of voters (top-left), average satisfaction (top-right) and 25th percentile (bottom-left) relative to Offline MES against the number of voters, and 25th percentile relative to Offline MES against the dispersion parameter $\phi$ (bottom-right) in the Normalized Mallows model (Experiment 3).
  • Figure :

Theorems & Definitions (19)

  • Example 1
  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Definition 3
  • ...and 9 more