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A Unified Framework of Multi-Stage Multi-Winner Voting: An Axiomatic Exploration

Shengjie Gong, Lingxiao Huang, Shuangping Huang, Yuyi Wang, Zhiqi Wang, Tao Xiao, Xiang Yan, Chunxue Yang

TL;DR

Multi-winner elections face manipulation and strategic voting; this paper proposes a unified multi-stage framework $\mathcal{R}=(R_1,\ldots,R_t)$ to mitigate manipulation costs by successive shortlists. It develops a formal score-based rule class using parameters $(\beta,\gamma)$ and shows how classic rules such as SNTV, Bloc, Borda, and CC fit into the framework, with STV and Baldwin's rule as multi-stage instances. The authors prove that Solid Coalition is preserved across stages when stage rules satisfy it, but demonstrate that Committee Monotonicity, Candidate Monotonicity, and Consistency generally fail in the multi-stage setting under rational rules. They provide guidance on rule selection, suggesting using SC-preserving stage rules like SNTV and noting that approval-based Thiele rules exhibit stronger multi-stage monotonicity properties. The work lays a theoretical foundation for multi-stage voting design and highlights directions for future axiomatic analyses and manipulation-resistance investigations.

Abstract

Multi-winner voting plays a crucial role in selecting representative committees based on voter preferences. Previous research has predominantly focused on single-stage voting rules, which are susceptible to manipulation during preference collection. In order to mitigate manipulation and increase the cost associated with it, we propose the introduction of multiple stages in the voting procedure, leading to the development of a unified framework of multi-stage multi-winner voting rules. To shed light on this framework of voting methods, we conduct an axiomatic study, establishing provable conditions for achieving desired axioms within our model. Our theoretical findings can serve as a guide for the selection of appropriate multi-stage multi-winner voting rules.

A Unified Framework of Multi-Stage Multi-Winner Voting: An Axiomatic Exploration

TL;DR

Multi-winner elections face manipulation and strategic voting; this paper proposes a unified multi-stage framework to mitigate manipulation costs by successive shortlists. It develops a formal score-based rule class using parameters and shows how classic rules such as SNTV, Bloc, Borda, and CC fit into the framework, with STV and Baldwin's rule as multi-stage instances. The authors prove that Solid Coalition is preserved across stages when stage rules satisfy it, but demonstrate that Committee Monotonicity, Candidate Monotonicity, and Consistency generally fail in the multi-stage setting under rational rules. They provide guidance on rule selection, suggesting using SC-preserving stage rules like SNTV and noting that approval-based Thiele rules exhibit stronger multi-stage monotonicity properties. The work lays a theoretical foundation for multi-stage voting design and highlights directions for future axiomatic analyses and manipulation-resistance investigations.

Abstract

Multi-winner voting plays a crucial role in selecting representative committees based on voter preferences. Previous research has predominantly focused on single-stage voting rules, which are susceptible to manipulation during preference collection. In order to mitigate manipulation and increase the cost associated with it, we propose the introduction of multiple stages in the voting procedure, leading to the development of a unified framework of multi-stage multi-winner voting rules. To shed light on this framework of voting methods, we conduct an axiomatic study, establishing provable conditions for achieving desired axioms within our model. Our theoretical findings can serve as a guide for the selection of appropriate multi-stage multi-winner voting rules.
Paper Structure (26 sections, 10 theorems, 10 equations, 2 figures)

This paper contains 26 sections, 10 theorems, 10 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Technique Contributions
  3. Other Related Work
  4. Unified Framework for Multi-Stage Multi-Winner Voting
  5. Score-Based Rules
  6. Examples of score-based rules.
  7. Axioms Related to Score-Based Rules
  8. Remark on Axiom Selection.
  9. Axiomatic Study for Multi-Stage Score-Based Rules
  10. Preserving Axioms in Multi-Stages
  11. Violating Axioms in Multi-Stages
  12. Discussion on Voting Rules Selection
  13. Axiomatic Study for Multi-Stage Approval-Based Rules
  14. Approval-Based Rules and Related Axioms
  15. Axioms Related to Approval-Based Rules.
  16. ...and 11 more sections

Key Result

theorem thmcountertheorem

Let $t\geq 1$ be an integer and $\mathcal{R} = (R_1, R_2, \ldots, R_t)$ be a $t$-stage multi-winner voting rule. If $R_r$ satisfies Solid Coalition for each $r\in [t]$, $\mathcal{R}$ also satisfies Solid Coalition.

Figures (2)

  • Figure 1: Score of the winning committee under two-stage voting. The blue line represents the score, and the green shade represents the standard deviation of the score.
  • Figure 2: Gini index of the winning committee under two-stage voting. The blue line represents the Gini index, and the green shade represents the standard deviation of the Gini index.

Theorems & Definitions (34)

  • definition thmcounterdefinition: Multi-Winner Voting Rules
  • definition thmcounterdefinition: $t$-Stage Multi-Winner Voting Rules
  • definition thmcounterdefinition: Score-based rules; $(\beta, \hbox{\boldmath$\gamma$})$-rule
  • definition thmcounterdefinition: Solid Coalition
  • definition thmcounterdefinition: Committee Monotonicity
  • definition thmcounterdefinition: Candidate Monotonicity
  • definition thmcounterdefinition: Consistency
  • theorem thmcountertheorem: Solid Coalition preserves in multi-stage voting
  • proof
  • definition thmcounterdefinition: Rationality of $(\beta, \hbox{\boldmath$\gamma$})$-rules
  • ...and 24 more