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Reconfiguring Proportional Committees

Chris Dong, Fabian Frank, Jannik Peters, Warut Suksompong

TL;DR

The paper studies reconfiguring proportional committees in approval-based multiwinner voting, focusing on transitions between JR/EJR committees via single-candidate swaps. It proves that the exact space of JR committees can be disconnected and that deciding connectivity is PSPACE-complete, but any two JR committees can be connected through a path of $2$-JR committees, and any two EJR committees through a path of $4$-EJR committees. It further shows that several well-known rules (e.g., $PAV$, MES, seqPhragmén, seqCCAV, CCAV, GreedyEJR) yield committees that lie in the same JR-connected component, and investigates connectivity in restricted domains CI and VI where JR connectivity has stronger guarantees. Collectively, these results illuminate the structure of proportional committees, reveal fundamental limits of exact connectivity, and point to practical pathways via approximate notions and domain restrictions for robust reconfiguration.

Abstract

An important desideratum in approval-based multiwinner voting is proportionality. We study the problem of reconfiguring proportional committees: given two proportional committees, is there a transition path that consists only of proportional committees, where each transition involves replacing one candidate with another candidate? We show that the set of committees satisfying the proportionality axiom of justified representation (JR) is not always connected, and it is PSPACE-complete to decide whether two such committees are connected. On the other hand, we prove that any two JR committees can be connected by committees satisfying a $2$-approximation of JR. We also obtain similar results for the stronger axiom of extended justified representation (EJR). In addition, we demonstrate that the committees produced by several well-known voting rules are connected or at least not isolated, and investigate the reconfiguration problem in restricted preference domains.

Reconfiguring Proportional Committees

TL;DR

The paper studies reconfiguring proportional committees in approval-based multiwinner voting, focusing on transitions between JR/EJR committees via single-candidate swaps. It proves that the exact space of JR committees can be disconnected and that deciding connectivity is PSPACE-complete, but any two JR committees can be connected through a path of -JR committees, and any two EJR committees through a path of -EJR committees. It further shows that several well-known rules (e.g., , MES, seqPhragmén, seqCCAV, CCAV, GreedyEJR) yield committees that lie in the same JR-connected component, and investigates connectivity in restricted domains CI and VI where JR connectivity has stronger guarantees. Collectively, these results illuminate the structure of proportional committees, reveal fundamental limits of exact connectivity, and point to practical pathways via approximate notions and domain restrictions for robust reconfiguration.

Abstract

An important desideratum in approval-based multiwinner voting is proportionality. We study the problem of reconfiguring proportional committees: given two proportional committees, is there a transition path that consists only of proportional committees, where each transition involves replacing one candidate with another candidate? We show that the set of committees satisfying the proportionality axiom of justified representation (JR) is not always connected, and it is PSPACE-complete to decide whether two such committees are connected. On the other hand, we prove that any two JR committees can be connected by committees satisfying a -approximation of JR. We also obtain similar results for the stronger axiom of extended justified representation (EJR). In addition, we demonstrate that the committees produced by several well-known voting rules are connected or at least not isolated, and investigate the reconfiguration problem in restricted preference domains.

Paper Structure

This paper contains 12 sections, 26 theorems, 28 equations, 2 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Related Work
  4. Preliminaries and Notation
  5. Approval-Based Multiwinner Voting
  6. Proportionality Axioms
  7. Common Rules
  8. Distance, Connectedness, and Isolation
  9. (Extended) Justified Representation
  10. Specific Voting Rules
  11. Restricted Preference Domains
  12. Conclusion and Future Directions

Key Result

Theorem 3.1

For any $k \ge 3$, there exists an approval profile $A$ and a committee $W \in \mathop{\mathrm{JR}}\nolimits(A,k)$ that is $(k-2)$-isolated in $\mathop{\mathrm{JR}}\nolimits(A,k)$, that is, every other committee in $\mathop{\mathrm{JR}}\nolimits(A,k)$ has distance at least $k-1$ to $W$.

Figures (2)

  • Figure 1: Each node represents a committee of size two. An edge is drawn between two committees if the distance between them is one.
  • Figure 2: A visualization of Example \ref{['exp:choice_set_disc']}. Voters are depicted as nodes, and row and column candidates as hyperedges around the nodes. The row committee $\{c'_i\mid 1\le i\le 5\}$ is shown in yellow.

Theorems & Definitions (53)

  • Example 1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Lemma 3.3
  • proof
  • ...and 43 more