Stable Voting and the Splitting of Cycles

TL;DR

This paper investigates the properties of Stable Voting and Simple Stable Voting, refinements of the Split Cycle method, proving conjectures for small numbers of alternatives and providing counterexamples for larger sets using SAT solving.

Contribution

It proves the conjecture that SSV refines SC for up to 5 alternatives and refutes it for more than 6, employing SAT solving for complex cases.

Findings

SSV refines SC for up to 5 alternatives

Counterexample shows SSV does not refine SC for more than 6 alternatives

SAT encoding can test properties of voting methods based on margin orderings

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.