Why Instant-Runoff Voting Is So Resilient to Coalitional Manipulation: Phase Transitions in the Perturbed Culture

TL;DR

This work analyzes how three common single-winner voting rules—Plurality, the Two-Round System, and Instant-Runoff Voting (IRV)—behave under coalitional manipulation (CM) within the Perturbed Culture model. It reveals phase transitions in the CM rate as a function of the concentration parameter $\theta$, identifying a critical threshold $\theta_c$ that separates CM-possible and CM-resistant regimes, with IRV exhibiting the strongest resilience by proving $\theta_c(\mathrm{IRV},m)=0$ via the Super Condorcet Winner (SCW) concept. The paper provides theoretical proofs, finite-$n$ and asymptotic results, and extensive simulations, showing exponential convergence of CM rate away from the critical point and validating the SCW-based explanation using real-world datasets where SCWs are prevalent. The results illuminate why IRV remains resistant to CM even under modest preference concentration and offer a structured framework for evaluating CM susceptibility across voting rules. These insights have implications for electoral design and the study of strategic voting in large-scale elections.

Abstract

Previous studies have shown that Instant-Runoff Voting (IRV) is highly resistant to coalitional manipulation (CM), though the theoretical reasons for this remain unclear. To address this gap, we analyze the susceptibility to CM of three major voting rules-Plurality, Two-Round System, and IRV-within the Perturbed Culture model. Our findings reveal that each rule undergoes a phase transition at a critical value theta\_c of the concentration of preferences: the probability of CM for large electorates converges exponentially fast to 1 below theta\_c and to 0 above theta\_c. We introduce the Super Condorcet Winner (SCW), showing that its presence is a key factor of IRV's resistance to coalitional manipulation, both theoretically and empirically. Notably, we use this notion to prove that for IRV, theta\_c = 0, making it resistant to CM with even minimal preference concentration.

Figures (7)

• Figure 1: CM rate of Plurality as a function of $\theta$ for different values of $n$ with $m = 4$. Curves for finite $n$ are based on Monte Carlo simulations with 1,000,000 profiles per point. The limiting curve as $n \to \infty$ follows from Theorem \ref{['thm_plurality_theta_c']}.
• Figure 2: CM rate of Plurality as a function of $n$ for different values of $m$ with $\theta = \theta_c(\textnormal{Plu}{}, m)$. Monte Carlo simulations with 1,000,000 profiles per point.
• Figure 3: CM rate of the Two-Round System as a function of $n$ for different values of $m$ with $\theta = \theta_c(\textnormal{TR}{}, m)$. Monte Carlo simulations with 1,000,000 profiles per point.
• Figure 4: CM rate of IRV as a function of $\theta$ for different values of $n$ with $m = 4$. Curves for finite $n$ are based on Monte Carlo simulations with 1,000,000 profiles per point. The limiting curve as $n \to \infty$ follows from Theorem \ref{['thm_irv_theta_c']}.
• Figure 5: CM rate of IRV as a function of $n$ for different values of $m$ with $\theta = \theta_c(\textnormal{IRV}{}, m) = 0$ (Impartial Culture). Monte Carlo simulations with 1,000,000 profiles per point.

Theorems & Definitions (9)

• lemma 1
• lemma 2
• lemma 3
• theorem 1
• definition 1
• lemma 4
• definition 2
• lemma 5
• theorem 3