[Paper Review] Proportionality and Strategyproofness in Multiwinner Elections

Dominik Peters|arXiv (Cornell University)|Apr 17, 2021

Game Theory and Voting Systems36 references34 citations

TL;DR

The paper proves a fundamental incompatibility: no approval-based multiwinner rule can be both proportional (even in weak forms) and strategyproof, using a computer-aided SAT approach plus human-readable MUS proofs.

ABSTRACT

Multiwinner voting rules can be used to select a fixed-size committee from a larger set of candidates. We consider approval-based committee rules, which allow voters to approve or disapprove candidates. In this setting, several voting rules such as Proportional Approval Voting (PAV) and Phragmén's rules have been shown to produce committees that are proportional, in the sense that they proportionally represent voters' preferences; all of these rules are strategically manipulable by voters. On the other hand, a generalisation of Approval Voting gives a non-proportional but strategyproof voting rule. We show that there is a fundamental tradeoff between these two properties: we prove that no multiwinner voting rule can simultaneously satisfy a weak form of proportionality (a weakening of justified representation) and a weak form of strategyproofness. Our impossibility is obtained using a formulation of the problem in propositional logic and applying SAT solvers; a human-readable version of the computer-generated proof is obtained by extracting a minimal unsatisfiable set (MUS). We also discuss several related axiomatic questions in the domain of committee elections.

Motivation & Objective

• Motivate and formalize weak notions of proportionality (JR variants) and strategyproofness in multiwinner approval voting.
• Show that no single rule can satisfy both weak proportionality and weak strategyproofness under a fixed k, n, m.
• Provide a computer-aided impossibility proof and extract human-readable core arguments (MUS).
• Discuss related axioms and extensions in committee elections.

Proposed method

• Define approval-based committee rules and key rules (AV, PAV) for context.
• Propose weak proportionality and several strategyproofness notions (cardinality, Hamming, superset, subset).
• Encode the existence of a proportional and strategyproof rule as a propositional SAT problem.
• Use SAT solvers to establish base-case unsatisfiability for fixed parameters (n, m, k).
• Extract minimal unsatisfiable sets (MUS) to obtain a human-readable core proof.
• Provide induction sketches to extend base-case impossibility to larger parameters.

Experimental results

Research questions

• RQ1 Can an approval-based multiwinner rule be simultaneously proportional (in JR/PJR/EJR-like senses) and strategyproof under dropping manipulations?
• RQ2 Does a base-case impossibility hold for fixed k, n, m, implying broader nonexistence via induction?
• RQ3 What is the minimal axiomatic strength required to obtain impossibility results in open-list committee settings?
• RQ4 How can computer-aided methods (SAT + MUS) illuminate impossibility proofs in social choice?
• RQ5 How do relations among JR-type axioms, apportionment extensions, and efficiency interact in this setting?

Key findings

• There is no approval-based committee rule that is weakly efficient, proportional, and strategyproof for k ≥ 3, n divisible by k, and m ≥ k+1.
• AV-like strategyproofness and proportional representation cannot be achieved simultaneously under these parameters.
• A base-case impossibility is demonstrated for k=3, n=3, m=4 using a SAT-encoded formulation and symmetry-breaking.
• The impossibility extends via induction steps to larger parameter values under the stated weak assumptions.
• The proof combines computer-generated steps with a human-readable MUS extraction to produce a verifiable core argument.