Noise Stability of Ranked Choice Voting

TL;DR

This paper conjectures that Borda count is the most stable ranked choice voting method under vote corruption, especially with three candidates, and explores its theoretical properties and implications.

Contribution

It introduces a new conjecture extending the Plurality is Stablest Conjecture to Borda count, analyzing its stability and dimensionality in ranked choice voting.

Findings

Borda count is conjectured to be the most stable method with three candidates.

The conjecture is shown to follow from the Plurality is Stablest Conjecture.

A dimension reduction argument suggests the optimal voting method is low-dimensional.

Abstract

We conjecture that Borda count is the ranked choice voting method that best preserves the outcome of an election with randomly corrupted votes, among all fair voting methods with small influences satisfying the Condorcet Loser Criterion. This conjecture is an adaptation of the Plurality is Stablest Conjecture to the setting of ranked choice voting. Since the plurality function does not satisfy the Condorcet Loser Criterion, our new conjecture is not directly related to the Plurality is Stablest Conjecture. Nevertheless, we show that the Plurality is Stablest Conjecture implies our new Borda count is Stablest conjecture. We therefore deduce that Borda count is stablest for elections with three candidates when the corrupted votes are nearly uncorrelated with the original votes. We also adapt a dimension reduction argument to this setting, showing that the optimal ranked choice voting method is "low-dimensional." The Condorcet Loser Criterion asserts that a candidate must lose an election if each other candidate is preferred in head-to-head comparisons. Lastly, we discuss a variant of our conjecture with the Condorcet Winner Criterion as a constraint instead of the Condorcet Loser Criterion. In this case, we have no guess for the most stable ranked choice voting method.