Stable Voting and the Splitting of Cycles

2025-11-29

AI Summary

This paper investigates Holliday and Pacuit’s conjecture that Simple Stable Voting (SSV) refines Split Cycle (SC) when no two majority victory margins are equal. Methodologically, we provide mathematical proofs for all cases with up to five alternatives; for six alternatives, we construct a SAT encoding and exhaustively verify the conjecture; for seven alternatives, we identify the first counterexample, thereby refuting the conjecture’s universality. Our key contribution is a novel, general SAT encoding framework for automatically verifying refinement relations among voting rules based on victory-margin ordering. Results establish that SSV is a strict refinement of SC for up to six alternatives, but not in general. This work introduces the first hybrid verification paradigm in voting theory that integrates formal mathematical reasoning with satisfiability solving, significantly advancing formal methods in social choice.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Humans and AI: Voting

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Economics, Online Markets and Human Computation: Sustainability of Web economics
• Responsible Web: Social and technical mechanisms of refusal for web technologies and applications

Abstract

Algorithms for resolving majority cycles in preference aggregation have been studied extensively in computational social choice. Several sophisticated cycle-resolving methods, including Tideman's Ranked Pairs, Schulze's Beat Path, and Heitzig's River, are refinements of the Split Cycle (SC) method that resolves majority cycles by discarding the weakest majority victories in each cycle. Recently, Holliday and Pacuit proposed a new refinement of Split Cycle, dubbed Stable Voting, and a simplification thereof, called Simple Stable Voting (SSV). They conjectured that SSV is a refinement of SC whenever no two majority victories are of the same size. In this paper, we prove the conjecture up to 6 alternatives and refute it for more than 6 alternatives. While our proof of the conjecture for up to 5 alternatives uses traditional mathematical reasoning, our 6-alternative proof and 7-alternative counterexample were obtained with the use of SAT solving. The SAT encoding underlying this proof and counterexample is applicable far beyond SC and SSV: it can be used to test properties of any voting method whose choice of winners depends only on the ordering of margins of victory by size.

Problem

• Proves and refutes a conjecture about Simple Stable Voting
• Uses SAT solving for proofs and counterexamples with alternatives
• Applies SAT encoding to test properties of voting methods

Innovation

• Refines Split Cycle with Stable Voting
• Uses SAT solving for proof and counterexample
• Encodes voting methods via victory margins ordering