Cost-Free Neutrality for the River Method

TL;DR

This paper introduces a polynomial-time algorithm for the River voting method with Parallel-Universe Tiebreaking, ensuring neutrality and efficient computation, unlike the NP-complete complexity in similar methods.

Contribution

It demonstrates that River with PUT can be computed efficiently, revealing structural advantages over Ranked Pairs, and provides a new algorithm called FUN for this purpose.

Findings

River with PUT is computable in polynomial time.

The FUN algorithm efficiently simulates all tiebreakings in one pass.

River's structure offers advantages over Ranked Pairs in neutrality and complexity.

Abstract

Recently, the River Method was introduced as novel refinement of the Split Cycle voting rule. The decision-making process of River is closely related to the well established Ranked Pairs Method. Both methods consider a margin graph computed from the voters' preferences and eliminate majority cycles in that graph to choose a winner. As ties can occur in the margin graph, a tiebreaker is required along with the preferences. While such a tiebreaker makes the computation efficient, it compromises the fundamental property of neutrality: the voting rule should not favor alternatives in advance. One way to reintroduce neutrality is to use Parallel-Universe Tiebreaking (PUT), where each alternative is a winner if it wins according to any possible tiebreaker. Unfortunately, computing the winners selected by Ranked Pairs with PUT is NP-complete. Given the similarity of River to Ranked Pairs, one might expect River to suffer from the same complexity. Surprisingly, we show the opposite: We present a polynomial-time algorithm for computing River winners with PUT, highlighting significant structural advantages of River over Ranked Pairs. Our Fused-Universe (FUN) algorithm simulates River for every possible tiebreaking in one pass. From the resulting FUN diagram one can then directly read off both the set of winners and, for each winner, a certificate that explains how this alternative dominates the others.