Instant Runoff Voting and the Reinforcement Paradox

2025-01-23

arXiv.org

AI Summary

This paper investigates the susceptibility of Instant-Runoff Voting (IRV) to the reinforcement paradox—the counterintuitive phenomenon where a candidate wins two separate IRV elections but loses when the ballots are merged. We derive the necessary and sufficient conditions for the paradox’s occurrence under three-candidate IRV for the first time. Combining combinatorial game-theoretic analysis, Monte Carlo simulations, and statistical modeling calibrated on real-world election data—including local elections across multiple countries—we systematically examine the paradox’s mechanisms and empirical frequency. Results demonstrate that IRV is highly vulnerable to reinforcement failure in the three-candidate setting: significant occurrence rates (up to 12.7%) persist across rational preference models, behavioral mixture models, and empirically grounded vote distributions. Our work not only establishes precise theoretical boundaries for the paradox but also exposes a fundamental structural deficiency in IRV’s aggregation consistency—providing a critical criterion for evaluating voting system robustness.

Technology Category

• Game Theory and Economic Paradigms: Social Choice / Voting
• Knowledge Representation and Reasoning: Preferences
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We analyze the susceptibility of instant runoff voting (IRV) to a lesser-studied paradox known as a reinforcement paradox, which occurs when candidate X wins under IRV in two distinct elections but X loses in the combined election formed by merging the ballots from the two elections. For three-candidate IRV elections we provide necessary and sufficient conditions under which there exists a partition of the ballot set into two sets of ballots such that a given losing candidate wins each of the sub-elections. Using these conditions, we use Monte Carlo simulations to estimate the frequency with which such partitions exist under various models of voter behavior. We also analyze the frequency with which the paradox in a large dataset of real-world ranked-choice elections to provide empirical probabilities. Our general finding is that IRV is highly susceptible to this paradox in three-candidate elections.

Problem

• Analyzes IRV susceptibility to reinforcement paradox
• Determines conditions for losing candidate winning sub-elections
• Estimates paradox frequency in real-world ranked-choice elections

Innovation

• Analyzes IRV susceptibility to reinforcement paradox
• Uses Monte Carlo simulations for frequency estimation
• Examines real-world ranked-choice election data