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

François Durand

Nokia Bell Labs France

Massy, France

francois.durand@nokia-bell-labs.com

## 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.

## KEYWORDS

Voting; Coalitional manipulation; Instant-Runoff Voting; Perturbed culture; Phase transition; Super Condorcet Winner

### ACM Reference Format:

François Durand. 2025. Why Instant-Runoff Voting Is So Resilient to Coalitional Manipulation: Phase Transitions in the Perturbed Culture. In Proc. of the 24th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2025), Detroit, Michigan, USA, May 19–23, 2025, IFAAMAS, 9 pages.

# 1 INTRODUCTION

## 1.1 Motivation

The Gibbard-Satterthwaite Theorem [15, 30] shows that all non-trivial voting rules are vulnerable to manipulation (strategic voting), even by a single voter. This vulnerability can only worsen when any number of voters with aligned interests can form a coalition to alter the election outcome, a phenomenon called *coalitional manipulation* (CM). Unlike individual manipulation [29, 31], CM remains significant in large-scale elections and raises several concerns, notably creating moral dilemmas [8, Introduction] and power imbalances between strategic and naive voters, thus undermining the "one person, one vote" principle [9, 12].

However, not all voting rules are equally vulnerable: the CM rate, i.e., the probability that a voting profile is manipulable by a coalition

under a given probabilistic model, can vary significantly between rules. In previous studies, *Instant-Runoff Voting* (IRV) and some of its variants [9] consistently outperform other classical single-winner voting rules in resisting coalitional manipulation, whether the analysis is based on randomly generated profiles [17, 18] [8, Chapters 7–8] or experimental datasets [3, 18] [8, Chapter 9]. This has been confirmed by theoretical calculations in the case of three candidates [24, 26]. This is especially intriguing, as IRV has several theoretical features typically deemed undesirable and seemingly prone to manipulation: most notably, IRV is neither *Condorcet-consistent nor monotonic* [2, Definitions 2.8 and 2.10] [10].

In an effort to shed theoretical light on this phenomenon, we will compare IRV to the two other most widely used voting rules in large-scale single-winner political elections: Plurality and the Two-Round System. We will adopt the *Perturbed Culture* model of random voting profiles, first introduced by Williamson and Sargent [33] and later named by Gehrlein [14, Section 4.3.2]. We will focus on the asymptotic behavior as the number of voters tends to infinity, as it offers more mathematical tractability and is relevant for large-scale elections. We will also examine convergence rates to assess how well this limit approximates scenarios with finite electorates. Our approach is similar to the use of the Ising model in physics, which, despite being unrealistic in its microscopic details, has been remarkably effective in explaining the complex macroscopic phenomenon of phase transitions in ferromagnetism [20].

## 1.2 Contributions

In this paper, we prove that each of the three voting rules undergoes a *phase transition*, with an abrupt change in behavior based on whether the concentration parameter $\theta$ in the Perturbed Culture model exceeds a critical threshold $\theta_c$. Below $\theta_c$, the CM rate tends to 1 for large electorates, while above $\theta_c$, it tends to 0. We compute the critical threshold $\theta_c$ for each voting rule as a function of the number of candidates.

We show through simulations how the CM rate curve as a function of $\theta$, which is continuous for a finite number of voters $n$, converges to a discontinuous curve as $n$ tends to infinity, thereby explaining the phase transition. Additionally, we investigate the critical regime $\theta = \theta_c$, leading to the conjecture that in this case, the CM rate tends to a limit strictly between 0 and 1.

We introduce the concept of a Super Condorcet Winner (SCW), which largely explains IRV's resilience to CM. This leads to one of our most striking results: for IRV, the critical value $\theta_c$ is 0, regardless of the number of candidates. This means that IRV is asymptotically resistant to CM as soon as the Perturbed Culture model shows even the slightest preference concentration. Furthermore, using

<figure><img src="image_1.png" /></figure>

This work is licensed under a Creative Commons Attribution International 4.0 License.

Proc. of the 24th International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2025), Y. Vorobeychik, S. Das, A. Nowé (eds.), May 19–23, 2025, Detroit, Michigan, USA. © 2025 International Foundation for Autonomous Agents and Multiagent Systems (www.ifaamas.org).
experimental datasets, we show that SCWs are frequent in practice and account for most of IRV's resistance to CM.

Finally, we demonstrate that in non-critical regime, i.e., for $\theta \neq \theta_c$, the convergence of the CM rate toward 0 or 1 is exponentially fast. This implies that our results for $n \to \infty$ quickly become relevant even for finite $n$. We also study how this speed varies with $\theta$.

## 1.3 Related Work

In addition to the literature already mentioned, the works most closely related to ours are those that study the CM rate using theoretical tools. Most of them focus on three-candidate elections within models like *Impartial Culture* [25], *Impartial Anonymous Culture* [13, 24], or *Pólya-Eggenberger Urns* [26]. Although they consistently show that IRV is more robust than other rules, they are limited to a specific number of candidates and offer little intuition for IRV's superior performance. Kim and Roush [21] provide key results for large electorates under *Impartial Culture* for Plurality and some other rules (*positional scoring rules* in general, *Maximin*, and *Coombs*) but do not address the Two-Round System or IRV.

Concerning phase transitions, research on this subject is abundant in physics (see Kadanoff [20] for an overview) and in mathematics and computer science[1, 5, 7]. In voting theory, Mossel et al. [27] and Xia [34] also examine phase transitions in coalitional manipulability, focusing on varying numbers of manipulators. In contrast, our study considers the impact of the concentration parameter in the probabilistic distribution of preferences.

## 1.4 Limitations

The limitations of this work stem from its main assumptions. First, while the Perturbed Culture model is useful, it does not capture the full complexity of real-world preferences. Second, our analysis is limited to three voting rules, and extending this to other systems would be valuable. Finally, the concept of coalitional manipulation may face criticism due to coordination challenges or the lack of binding agreements among coalition members (see Durand [8, Introduction] for a response to these critiques).

## 1.5 Roadmap

The rest of the paper is organized as follows. Section 2 introduces key definitions and notations. Sections 3, 4, and 5 respectively analyze Plurality, Two-Round System, and IRV. Section 6 explores convergence speed. Section 7 concludes with future work.

# 2 DEFINITIONS AND NOTATIONS

## 2.1 Discrete and Continuous Profiles

A discrete profile $P$ consists of three elements: a finite, non-empty set of candidates $C(P)$ with cardinality $m(P)$; a finite, non-empty set of voters $V(P)$ with cardinality $n(P)$; and for each voter $v \in V(P)$, a preference ranking $P_v$ over the candidates in $C(P)$.

For any preference ranking $p$, let $w(p, P)$ denote the weight of $p$ in $P$, i.e., the number of voters in $P$ with ranking $p$. The total weight of a discrete profile is simply the number of voters: $w(P) = \sum_p w(p, P) = n(P)$.

A continuous profile is similarly defined by three components: a finite, non-empty set of candidates $C(P)$ with cardinality $m(P)$;

a total weight $w(P) \in (0, \infty)$; and for each ranking $p$ over the candidates, a weight $w(p, P) \in \mathbb{R}$, such that $\sum_p w(p, P) = w(P)$.

For any profile $P$, whether discrete or continuous, we define the associated normalized profile $\bar{P}$ as the continuous profile where the weight of each ranking $p$ is given by $w(p, \bar{P}) = \frac{w(p, P)}{w(\bar{P})}$. Viewing a profile as a vector of weights, we can naturally define its neighborhood in the usual topological sense.

For any subset $K \subseteq C(P)$, let $P_K$ be the restriction of $P$ to the candidates in $K$. For two distinct candidates $c$ and $d$, let $P^{c>d}$ be the restriction to voters who prefer $c$ over $d$. Similarly, for a candidate $c$ and a position $k \in \{1, \dots, m(P)\}$, let $P^{r(c)=k}$ be the restriction to voters ranking $c$ in the $k$-th position. These notations can be combined to restrict the profile both by candidates and voters.

## 2.2 Voting Rules

A voting rule $f$ maps any profile, discrete or continuous, to a candidate from that profile. In this paper, we focus on homogeneous voting rules, meaning that $f(P) = f(\bar{P})$ for any profile $P$. In other words, the outcome depends only on the relative proportions of preference rankings, not the total weight. Each particular voting rule is formally defined at the beginning of its respective section.

## 2.3 Coalitional Manipulability

When $P$ is a discrete profile, we say that a voting rule $f$ is coalitionally manipulable (also abbreviated as CM) in $P$, or that profile $P$ is CM in rule $f$, if there exists a target profile $Q$ with the same candidates and voters such that $f(Q) \neq f(P)$, and for every voter $v \in V(P)$, if $Q_v \neq P_v$, then $v$ prefers $f(Q)$ to $f(P)$ based on $P_v$. In other words, only voters who benefit from the new outcome may alter their ballots, though some may keep their original votes.

An immediate consequence is as follows. If for a ranking $p$, we have $w(p, Q) < w(p, P)$, then at least one voter with ranking $p$ in $P$ must have changed their ballot in $Q$, i.e., $Q_v \neq P_v$. By the definition, this implies that $f(Q)$ is preferred to $f(P)$ according to $P_v = p$. This observation will now serve as the basis for defining CM in the continuous case.

For a continuous profile $P$, we say that a voting rule $f$ is CM in $P$ (or that $P$ is CM in $f$) if there exists a target profile $Q$ with the same candidates and total weight such that $f(Q) \neq f(P)$ and, for every ranking $p$, if $w(p, Q) < w(p, P)$, then $f(Q)$ is preferred to $f(P)$ according to $p$. In other words, only voters (in a continuous sense) who prefer the new outcome can have changed their ballots.

The relationship between the two notions is clarified by:

**LEMMA 2.1.** If a homogeneous rule $f$ is CM in a discrete profile $P$, then $f$ is also CM in the corresponding normalized profile $\bar{P}$. However, the converse is not true.

The direct implication follows from the definitions, so we will focus on providing a counterexample to show that the converse does not hold. Consider the *positional scoring rule* $f$ with weights $(7, 6, 0, \dots, 0)$, where each candidate's score is given by $s(c) = 7w(P^{r(c)=1}) + 6w(P^{r(c)=2})$, and the candidate with the highest score wins (using a tie-breaking rule if needed). Now, consider a discrete profile $P$ with 8 voters and 3 candidates:

* 3 voters have the ranking $1 > 3 > 2$,

* 5 voters have the ranking $2 > 1 > 3$.
It is straightforward to verify that candidate 1 wins under $f$, and that the rule is CM in the normalized profile $\bar{P}$, but not in the original discrete profile $P$. The issue is that the 5 manipulators supporting candidate 2 must carefully distribute their points between candidates 1 and 3, which is impossible in the discrete case because each manipulator must assign their entire vote to one ranking rather than splitting it fractionally.

## 2.4 Perturbed Culture

Given two positive integers $m$ and $n$, and a concentration parameter $\theta \in (0, 1]$, the *Perturbed Culture* model is defined as follows. A discrete profile $P$ is randomly generated with $C(P) = \{1, \dots, m\}$ and $\mathcal{V}(P) = \{1, \dots, n\}$. Each voter is independently assigned the ranking $(1 > \dots > m)$ with probability $\theta$, and a uniformly random ranking with probability $1 - \theta$.

As $\theta \to 0$, this model converges to the classical *Impartial Culture* model, while for $\theta = 1$, it becomes a deterministic culture where all voters share the ranking $(1 > \dots > m)$.<fnref n="1" />

Since a profile can be represented as a vector giving the weight of each ranking, we can define the *expected normalized profile* (or simply the *expected profile*) under Perturbed Culture. To simplify notation, we denote it by $\hat{P}$, leaving its dependency on $m$ and $\theta$ implicit. In this profile, the ranking $(1 > \dots > m)$ has a weight of $\theta + \frac{1-\theta}{m!}$, while each of the other rankings has a weight of $\frac{1-\theta}{m!}$.

## 2.5 CM Rate

We denote by $\rho(f, m, n, \theta)$ the CM rate, i.e., the probability that a voting rule $f$ is CM in a profile drawn from the Perturbed Culture model with $m$ candidates, $n$ voters and concentration $\theta$.

# 3 PLURALITY

We will begin our study with the *Plurality* voting rule, which assigns each candidate $c$ in a profile $P$ a score equal to the total weight of voters ranking $c$ first: $s_{Plu}(c, P) = w(P^{(c)=1})$. The winner is the candidate with the highest score (using a tie-breaking rule if needed): Plu($P$) = arg max $s_{Plu}(c, P)$. The specific tie-breaking method will not affect our findings.

## 3.1 Theoretical Results for Plurality

The intuition behind our theoretical results is as follows. First, we analyze Plurality’s behavior in the expected normalized profile $\hat{P}$ as a function of $\theta$. For small $\theta$, Plurality is CM in this profile, but for large enough $\theta$, it is not. Using the weak law of large numbers, we then show that as $n \to \infty$, the normalized random profile $\bar{P}$ will, with high probability (i.e., with a probability that tends to 1 when $n \to \infty$), be close enough to $\hat{P}$, ensuring that Plurality behaves similarly. Throughout this subsection, we assume $m \ge 2$.

We begin by analyzing the expected profile $\hat{P}$. In this profile, the plurality score for candidate 1 is $s_{Plu}(1, \hat{P}) = \theta + \frac{1-\theta}{m}$, while the number of voters inclined to manipulate for any candidate $c \ne 1$ is $w(\hat{P}^{c>1}) = \frac{1-\theta}{2}$. If all manipulators vote optimally for $c$, they succeed if $w(\hat{P}^{c>1}) > s_{Plu}(1, \hat{P})$, which simplifies to $\theta < \frac{m-2}{3m-2}$.

Defining the critical value $\theta_c(\text{Plu}, m) = \frac{m-2}{3m-2}$, we conclude that Plurality is CM for $\theta < \theta_c(\text{Plu}, m)$ and not CM for $\theta > \theta_c(\text{Plu}, m)$ (the equality case is not needed for our forthcoming analysis).

We now apply the weak law of large numbers to show that as $n \to \infty$, these results hold with high probability. We start by examining the supercritical regime $\theta > \theta_c(\text{Plu}, m)$, relying on the following lemma.

**LEMMA 3.1.** *Assume there exists a neighborhood of the expected normalized profile $\hat{P}$ where the homogeneous rule $f$ is not CM. Then $\lim_{n \to \infty} \rho(f, m, n, \theta) = 0$.*

**PROOF.** Applying the weak law of large numbers, the following statements hold with high probability: denoting $P$ the random profile, its normalized version $\bar{P}$ lies in the desired neighborhood of $\hat{P}$, hence (by assumption) $f$ is not CM in $\bar{P}$, hence (by Lemma 2.1) $f$ is also not CM in the random discrete profile $P$. $\square$

This lemma applies easily to Plurality. For $\theta > \theta_c(\text{Plu}, m)$, we have shown that for every candidate $c \ne 1$, $w(\hat{P}^{c>1}) < s_{Plu}(1, \hat{P})$. As this is a strict inequality, it holds in a neighborhood of the profile, allowing us to apply Lemma 3.1. Hence, $\lim_{n \to \infty} \rho(\text{Plu}, m, n, \theta) = 0$.

We now turn to the *subcritical regime* $\theta < \theta_c(\text{Plu}, m)$. Unfortunately, we cannot directly apply the same reasoning: even if the normalized profile $\bar{P}$ is CM near $\hat{P}$, it does not necessarily follow that the discrete profile $P$ is also CM, as Lemma 2.1 does not hold in the reverse direction.

However, for Plurality, manipulators can always employ a common strategy. Formally, a voting rule $f$ is *unison-manipulable* (UM) in profile $P$ (or $P$, in $f$) if manipulation can succeed even when all interested voters cast the same ballot [9, 32].<fnref n="2" /> Clearly, UM implies CM. Unlike CM, UM holds equivalently for both a discrete profile $P$ and its normalized profile $\bar{P}$, which leads to the following lemma.

**LEMMA 3.2.** *Assume there exists a neighborhood of the expected normalized profile $\hat{P}$ where the homogeneous rule $f$ is UM. Then $\lim_{n \to \infty} \rho(f, m, n, \theta) = 1$.*

The proof is similar to Lemma 3.1: by the weak law of large numbers, with high probability, the normalized random profile $\bar{P}$ is in the desired neighborhood, making it UM, hence the random discrete profile $P$ is also UM, and thus CM. Applied to Plurality, Lemma 3.2 directly leads to $\lim_{n \to \infty} \rho(f, m, n, \theta) = 1$ for $\theta < \theta_c(\text{Plu}, m)$.

The following theorem summarizes our results so far.

**THEOREM 3.3.** Let $\theta_c(\text{Plu}, m) = \frac{m-2}{3m-2}$ with $m \ge 2$.

* If $\theta < \theta_c(\text{Plu}, m)$, then $\lim_{n \to \infty} \rho(\text{Plu}, m, n, \theta) = 1$.

* If $\theta > \theta_c(\text{Plu}, m)$, then $\lim_{n \to \infty} \rho(\text{Plu}, m, n, \theta) = 0$.

For $m=2$, the theorem indicates $\theta_c(\text{Plu}, 2) = 0$, which is expected, as Plurality cannot be manipulated with only two candidates. Similarly, for $m=1$, we would reach the same conclusion by conventionally setting $\theta_c(\text{Plu}, 1) = 0$. The theorem becomes more interesting for $m \ge 3$, where it describes a *phase transition* around $\theta_c(\text{Plu}, m)$, meaning a sudden change in behavior as the parameter crosses this threshold. This raises key questions: What causes this discontinuity, and how do we approach it as $n$ increases? What happens when $\theta$ is equal to or near the critical value?

<footnote n="1">¹We exclude the case $\theta = 0$ from our theoretical analysis to simplify the proofs: as $n \to \infty$, assuming $\theta > 0$ guarantees that candidate 1 wins under sincere voting for all three voting rules considered. Nevertheless, our results hold even in the case $\theta = 0$, and we will also include it in our figures.</footnote>

<footnote n="2">²The term *unison* was introduced by Walsh [32] but we follow the slightly different definition proposed by Durand [9].</footnote>
<figure><img src="image_2.png" /><figcaption>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 3.3.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: CM rate of Plurality as a function of $n$ for different values of $m$ with $\theta = \theta_c(\text{Plu}, m)$. Monte Carlo simulations with 1,000,000 profiles per point.</figcaption></figure>

## 3.2 Simulations for Plurality

To understand the origin of the discontinuity, Figure 1 shows the CM rate of Plurality as a function of $\theta$ for various $n$ with $m = 4$. Curves for finite $n$ are based on Monte Carlo simulations,<fnref n="3" /> with 1,000,000 profiles per point, leading to error margins of $\frac{1}{\sqrt{1000000}} = 0.1\%$. The limiting curve for $n \to \infty$ is derived from Theorem 3.3. For finite $n$, the curve is continuous. As $n$ increases, it becomes sigmoid-shaped and steepens, ultimately converging to a step function as $n \to \infty$.

The observed behavior mirrors what occurs in physics: since a finite combination of continuous functions remains continuous, non-analyticity can only arise in an infinite system [20, Section 11.6]. As in physics, a phase transition occurs beyond a certain level of disorder: while a ferromagnetic metal loses its magnetization above the Curie temperature [4], Plurality loses its resistance to coalitional manipulation below the critical value of the concentration parameter $\theta$.

Theorem 3.3 describes the behavior in the subcritical and supercritical regimes, but what happens in the critical regime, i.e., when $\theta = \theta_c(\text{Plu}, m)$? Figure 2 shows the CM rate in that case as a function of the number of voters $n$, for different values of $m$. This leads to several conjectures:

* The critical CM rate $\rho(\text{Plu}, m, n, \theta_c(\text{Plu}, m))$ converges to a limit as $n \to \infty$.

* This limit is strictly less than 1.

* This limit increases with $m$.

It is beyond the scope of this paper to theoretically prove these results. In the study of phase transitions, analyzing the critical behavior is often challenging [6, 22, 23, 28]. With that, we conclude our study of Plurality and proceed to the Two-Round System.

# 4 TWO ROUND SYSTEM

The Two-Round System (TR) is as follows. In the first round, each candidate $c$ receives a score $s_{\text{TR}}^1(c, P) = s_{\text{Plu}}(c, P)$, and the set $K$ of the two candidates with the highest scores advances to the second round. These two candidates then receive scores $s_{\text{TR}}^2(c, P) = s_{\text{Plu}}(c, P_K)$, and the candidate with the highest score wins. A tie-breaking rule is applied if necessary.<fnref n="4" />

## 4.1 Theoretical Results for the Two-Round System

For $n = 2$, the Two-Round System is equivalent to Plurality, so we focus on the case $m \ge 3$. As with Plurality, we first analyze the expected profile and then extend the results using the weak law of large numbers. The key difference is of technical nature: unison manipulation is generally insufficient in the Two-Round System.

We begin by examining the expected normalized profile $\hat{P}$. Candidate 1 clearly wins the election, with the second-round opponent determined by the tie-breaking rule. For a manipulation to succeed in favor of a candidate $c \ne 1$, candidate $c$ must reach the second round. However, if candidate 1 also advances, the manipulation will fail. Therefore, the second round must involve a candidate $d \notin \{1, c\}$, which is possible since we assumed $m \ge 3$. Now, consider the portion of the first-round scores for candidates 1, $c$, and $d$ coming from sincere voters:

$$ \left\{ \begin{array}{l} s_{\text{TR}}^{1}(1, \hat{P}^{1>c}) = \theta + \frac{1-\theta}{m}, \\ s_{\text{TR}}^{1}(c, \hat{P}^{1>c}) = 0, \\ s_{\text{TR}}^{1}(d, \hat{P}^{1>c}) = \frac{1-\theta}{2m}, \end{array} \right. $$

where, for example, $s_{\text{TR}}^1(d, \hat{P}^{1>c})$ denotes the first-round score, in the two-round system, of candidate $d$ in the restriction of the expected normalized profile $\hat{P}$ to the voters who prefer candidate 1 to candidate $c$ (i.e., “sincere” voters).

For both candidates $c$ and $d$ to surpass candidate 1's score, at least $\theta + \frac{1-\theta}{m}$ manipulators must vote for $c$, while $\theta + \frac{1-\theta}{m} - \frac{1-\theta}{2m}$ must vote for $d$. Therefore, the total number of manipulators, given by $w(\hat{P}^{c>1}) = \frac{1-\theta}{2}$, must be at least the sum of these two quantities.

<footnote n="3">³The code is available at https://github.com/francois-durand/irv-cm-aamas-2025.</footnote>

<footnote n="4">⁴For simplicity, we consider an “instant” version of TR, where voters cast their ballots once. In most actual implementations, voters participate in two rounds. While this is equivalent for sincere voting, the instant version restricts some manipulation strategies [8, Table 1.1]. However, our results apply to both variants.</footnote>
Simplifying, the necessary condition becomes $\theta \le \frac{m-3}{5m-3}$. In other words, coalitional manipulation is impossible for $\theta > \frac{m-3}{5m-3}$.

Let us now show that for $\theta < \frac{m-3}{5m-3}$, manipulation is possible. Consider the specific case where $c = 2$ and $d = 3$. Suppose that:

* A fraction $\frac{2(1-2\theta)}{3(1-\theta)}$ of manipulators vote $(2 \succ \dots \succ m \succ 1)$,

* And a fraction $\frac{1+\theta}{3(1-\theta)}$ of them vote $(3 \succ \dots \succ m \succ 1 \succ 2)$.

The first fraction is positive since $\theta < \frac{m-3}{5m-3} < \frac{1}{2}$, and the two fractions clearly sum to 1. We chose these values to ensure candidate 2's lead over candidate 1 in the first round is double that of candidate 3's, though the reasoning holds for slightly different values. Standard calculations then show that candidates 2 and 3 lead the first round, with candidate 2 winning the second.

In summary, the behavior of the Two-Round System in the expected normalized profile shifts at $\theta_c(\text{TR}, m) = \frac{m-3}{5m-3}$. The next step is to extend this result to discrete profiles as $n \to \infty$. For the supercritical regime $\theta > \theta_c(\text{TR}, m)$, the reasoning mirrors that for Plurality, relying on Lemma 3.1.

In the subcritical regime $\theta < \theta_c$, the situation is more subtle. Unison manipulation is generally insufficient for the Two-Round System [8, Table 1.1], so we cannot directly apply Lemma 3.2.

Let us take a step back for the intuition of the problem. Suppose that a rule $f$ is CM in a neighborhood of the expected profile $\hat{P}$, and consider a discrete profile $P$ whose normalized form $\bar{P}$ lies in this neighborhood, similarly to Lemma 3.2 for unison manipulation. This means that $f$ is CM from $\bar{P}$ toward a target continuous profile $Q$. The challenge arises from discretization: manipulators in the discrete profile $P$ may only be able to target a profile whose normalized form approximates $Q$.

However, as $n$ grows large, these finite-size effects diminish, allowing manipulators to approach $Q$ closely. For CM to succeed, it suffices that $f$ produces the same outcome when a profile is sufficiently close to $Q$. Thus, we will rely on two neighborhoods: one for manipulability near $\hat{P}$, and another for stability of the outcome around the target profile. Technically, to obtain the desired result, we will require the second neighborhood to be uniform with respect to the first, meaning that for different choices of $\bar{P}$ near $\hat{P}$, the neighborhood of $Q$ that guarantees outcome stability does not shrink arbitrarily. Let us now formalize all this.

**Definition 4.1.** Let $\delta > 0$. We say that a rule $f$ is $\delta$-stable-CM in a continuous profile $P$ (or that $P$ is $\delta$-stable-CM in $f$) if there exists a continuous profile $Q$ such that:

* $f$ is CM from $P$ to $Q$,

* For any profile $Q'$ with $d(Q, Q') < \delta$, we have $f(Q') = f(Q)$. Here, $d(Q, Q')$ denotes the $t^\infty$-distance between profiles, viewed as vectors of weights. However, our results hold for any norm-derived distance, as all norms are equivalent in finite-dimensional spaces.

**LEMMA 4.2.** Assume there exists $\delta > 0$ and a neighborhood of the expected normalized profile $\hat{P}$ where the homogeneous rule $f$ is $\delta$-stable-CM. Then $\lim_{n \to \infty} \rho(f, m, n, \theta) = 1$.

The uniformity property mentioned earlier lies in that the same value of $\delta$ applies consistently on the entire neighborhood of $\hat{P}$.

**PROOF.** Consider $n$ large enough such that $\frac{1}{n} < \delta$. The weak law of large numbers ensures that, with high probability:

(1) The normalized random profile $\bar{P}$ lies in the desired neighborhood and is therefore $\delta$-stable-CM to a target profile $Q$.

(2) The manipulators in the discrete profile $P$ can approach $Q$ with a margin of one manipulator, causing deviations of at most $\frac{1}{n} < \delta$ in each normalized weight.

(3) By assumption, this implies that the voting outcome for the approximated target profile matches that of $Q$.

(4) Therefore, $f$ is CM in $P$. $\square$

We can now address the Two-Round System for $\theta < \frac{m-3}{5m-3}$.

Let $\zeta > 0$ (its value will be specified later). Consider a continuous profile $P$ such that $d(P, \hat{P}) < \frac{\zeta}{m!}$. Then, all the scores and score gaps calculated for $\hat{P}$ still hold for $P$, with an additive error of at most $\zeta$. It follows that if $\zeta < \theta$, candidate 1 wins in $P$, just as in $\hat{P}$.

As we did in the case of the expected profile, consider the profile $Q$ obtained from $P$ by modifying the ballots of voters who prefer candidate 2 over candidate 1 as follows:

* A fraction $\frac{2(1-2\theta)}{3(1-\theta)}$ vote $(2 \succ \dots \succ m \succ 1)$,

* And a fraction $\frac{1+\theta}{3(1-\theta)}$ vote $(3 \succ \dots \succ m \succ 1 \succ 2)$.

To apply Lemma 4.2, it suffices to show that, sufficiently close to $Q$ (including the profile $Q$ itself), candidate 2 wins.

Let $\zeta > 0$ (specified later) and set $\delta = \frac{\zeta}{m!}$. For a profile $Q'$ such that $d(Q', Q) < \delta$, the first-round scores are bounded as follows:

$$ \left\{ \begin{array}{l} s_{\text{TR}}^{1}(1, Q') < \left( \frac{1-\theta}{m} + \theta + \zeta \right) + 0 + \zeta, \\ s_{\text{TR}}^{1}(2, Q') > 0 + \left( \frac{1-\theta}{2} - \zeta \right) \frac{2(1-2\theta)}{3(1-\theta)} - \zeta, \\ s_{\text{TR}}^{1}(3, Q') > \left( \frac{1-\theta}{2m} - \zeta \right) + \left( \frac{1-\theta}{2} - \zeta \right) \frac{1+\theta}{3(1-\theta)} - \zeta, \\ s_{\text{TR}}^{1}(c, Q') < \left( \frac{1-\theta}{2m} + \zeta \right) + 0 + \zeta, \quad \text{for any } c > 3. \end{array} \right. $$

Each score above is presented as the sum of three terms: the first comes from sincere voters in $P^{1>2}$ and stays within $\zeta$ of the corresponding value for $\hat{P}^{1>2}$ (except for candidate 2, by definition of $P^{1>2}$); the second comes from manipulators in $Q$; and the third is an error between $Q$ and $Q'$, bounded by $\zeta$. If $\zeta = \zeta < \frac{m-3-(5m-3)\theta}{30m}$, standard calculations show that candidates 2 and 3 secure the highest scores and thus advance to the second round.

In the second round, we have:

$$ s_{\text{TR}}^{2}(2, Q') - s_{\text{TR}}^{2}(3, Q') > \left( \theta - \frac{1-\theta}{6} - \zeta \right) + \left( \frac{1-\theta}{2} - \zeta \right) \frac{1-5\theta}{3(1-\theta)} - \zeta. $$

If $\zeta = \zeta < \frac{\theta}{12}$, standard calculations show that this quantity is positive, and therefore candidate 2 wins, as desired.

Overall, it thus suffices to take $\zeta = \zeta < \min\left(\frac{m-3-(5m-3)\theta}{30m}, \frac{\theta}{12}\right)$. We summarize our findings in the following theorem.

**THEOREM 4.3.** Let $\theta_c(\text{TR}, m) = \frac{m-3}{5m-3}$ with $m \ge 3$.

* If $\theta < \theta_c(\text{TR}, m)$, then $\lim_{n \to \infty} \rho(\text{TR}, m, n, \theta) = 1$.

* If $\theta > \theta_c(\text{TR}, m)$, then $\lim_{n \to \infty} \rho(\text{TR}, m, n, \theta) = 0$.

Recall that for $m = 2$, the same conclusions hold by setting $\theta_c(\text{TR}, 2) = 0$, since Two-Round is equivalent to Plurality in this case. For $m = 3$, the theorem also gives $\theta_c(\text{TR}, 3) = 0$, which is remarkable: according to the Gibbard-Satterthwaite theorem, manipulability becomes an issue from $m = 3$, yet Two-Round avoids this with high probability in Perturbed Culture as soon as $\theta > 0$.
<figure><img src="image_4.png" /><figcaption>Figure 3: CM rate of the Two-Round System as a function of n for different values of m with $\theta = \theta_c(\text{TR}, m)$. Monte Carlo simulations with 1,000,000 profiles per point.</figcaption></figure>

Since Theorems 3.3 and 4.3 share similar structures, a natural question arises: does every voting rule $f$ have a critical parameter $\theta_c(f, m)$ with similar properties? The answer is no. Consider a rule $f$ that uses Plurality when $n$ is even and Two-Round when $n$ is odd.<fnref n="5" /> From Theorems 3.3 and 4.3, it follows that for $\theta < \frac{m-3}{5m-3}$, the CM rate converges to 1, while for $\theta > \frac{m-2}{3m-2}$, it converges to 0. However, for $\theta \in (\frac{m-3}{5m-3}, \frac{m-2}{3m-2})$, the CM rate tends to 1 for even $n$ and to 0 for odd $n$: overall, it does not converge.

It is still possible to define a lower critical value $\theta_l(f, m)$ and an upper critical value $\theta_u(f, m)$ as, respectively, the largest and smallest values in $[0, 1]$ such that:

* If $\theta < \theta_l(f, m)$, then $\lim_{n \to \infty} \rho(f, m, n, \theta) = 1$,

* If $\theta > \theta_u(f, m)$, then $\lim_{n \to \infty} \rho(f, m, n, \theta) = 0$.

We can then define $\theta_c(f, m)$ as their common value when it exists. With this convention, Theorems 3.3 and 4.3 are summarized as:

$$ \theta_c(\text{Plu}, m) = \frac{m-2}{3m-2}, \quad \theta_c(\text{TR}, m) = \frac{m-3}{5m-3}. $$

## 4.2 Simulations for the Two-Round System

The Two-Round equivalent of Figure 1 is similar, so we proceed directly to the counterpart of Figure 2: Figure 3, showing the critical CM rate as a function of $n$ for different $m$. We use SVVAMP 0.12.0 [11], a Python package for studying the manipulability of voting rules. As for Plurality, the critical CM rate appears to converge to a limit in $(0, 1)$ that increases with $m$.

# 5 IRV (INSTANT-RUNOFF VOTING)

Let us now proceed to *Instant-Runoff Voting* (IRV), where the winner is determined through multiple rounds. In each round, the candidate with the lowest Plurality score is eliminated, until only one remains. Formally, let $K(r, P)$ be the set of remaining candidates at the start of round $r$ and $\ell(r, P)$ be the candidate losing at round $r$. We have:

$$ \left\{ \begin{array}{l} K(1, P) = C(P), \\ \ell(r, P) = \arg \min s_{\text{Plu}}(c, P_{K(r,P)}), \\ K(r+1, P) = K(r, P) \setminus \{\ell(r, P)\}, \end{array} \right. $$

using a tie-breaking rule for elimination when necessary. The winner IRV($P$) is the last remaining candidate in $K(m(P), P)$.

## 5.1 Theoretical Results for IRV

As usual, we start by examining the expected normalized profile $\hat{P}$. Since $\theta > 0$, candidate 1 clearly wins. Now suppose that IRV is CM in $\hat{P}$ to a target profile $Q$, where candidate $c \neq 1$ wins. Candidate 1 must be eliminated in some round $r$. For conciseness, denote $K = K(r, Q)$ and $k = |K|$. Obviously $c$ must belong to $K$. The sincere voters' contribution to candidate 1's score at this round is:

$$ s_{\text{Plu}}(1, \hat{P}_K^{1>c}) = s_{\text{Plu}}(1, \hat{P}_K) = \theta + \frac{1-\theta}{k}. $$

Thus, $s_{\text{Plu}}(1, Q_K) \ge \theta + \frac{1-\theta}{k}$, and since $k \ge 2$, this is strictly greater than $\frac{1}{k}$. Therefore, the score of candidate 1 exceeds the average score at this round, hence it cannot be minimal. This contradiction proves that IRV is not CM in $\hat{P}$.

In this reasoning, IRV's resistance to coalitional manipulation stems from the fact that in any subset of candidates $K$ containing candidate 1, this candidate has a Plurality score that exceeds the average score. This motivates the following definition:

**Definition 5.1.** A candidate $c$ is a Super Condorcet Winner (SCW) in a profile $P$ if, for every subset of candidates $K$ containing $c$, the following holds:

$$ s_{\text{Plu}}(c, P_K) > \frac{w(P)}{|K|}. $$

This concept strengthens the classical notion of a Condorcet Winner, which only requires the condition to hold for subsets $K$ of size 2. We summarize its relevance to IRV as follows:

**LEMMA 5.2.** If $c$ is an SCW in profile $P$, then $\text{IRV}(P) = c$ and $\text{IRV}$ is not CM in $P$.

The same result easily extends to several IRV variants, such as Exhaustive Ballot [9], Condorcet-IRV [10, 18], Benham rule, Tideman rule, Smith-IRV, and Woodall rule [16]. However, the converse is not true: there exists profiles without an SCW where IRV is still not CM (see Durand [8, Table 1.1] for an example).

Now, consider the neighborhood of the expected profile $\hat{P}$. Since the SCW condition involves a finite number of strict inequalities that depend continuously on the profile's coefficients, candidate 1 is an SCW not only in $\hat{P}$ but also in its neighborhood. From here, we can follow two proof strategies that only differ in the order of their steps.

One approach is to first apply Lemma 5.2 to deduce that IRV is not CM in this neighborhood. Using Lemma 3.1 (based on the weak law of large numbers), we then deduce $\lim_{n \to \infty} \rho(\text{IRV}, m, n, \theta) = 0$. Alternatively, we could first use the weak law of large numbers to show that candidate 1 is an SCW with high probability, then apply Lemma 5.2 to show $\lim_{n \to \infty} \rho(\text{IRV}, m, n, \theta) = 0$.

Since this holds for any $\theta > 0$, we obtain a remarkable result:

**THEOREM 5.3.** For IRV, the critical value of the concentration parameter in Perturbed Culture is

$$ \theta_c(\text{IRV}, m) = 0. $$

<footnote n="5">⁵For simplicity, this counter-example involves a non-homogeneous rule.</footnote>
<figure><img src="image_5.png" /><figcaption>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 5.3.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: CM rate of IRV as a function of $n$ for different values of $m$ with $\theta = \theta_c(\text{IRV}, m) = 0$ (Impartial Culture). Monte Carlo simulations with 1,000,000 profiles per point.</figcaption></figure>

In summary, within the Perturbed Culture model, IRV has the smallest possible critical value. Even a slight concentration of preferences favoring candidate 1 is enough for IRV to become resistant to coalitional manipulation with high probability. And for the same reasons, this also holds for the IRV variants mentioned earlier.

## 5.2 Simulations for IRV

For IRV, as for Two-Round, our simulations for finite $n$ are carried out using SVVAMP. Similar to Figure 1 for Plurality, Figure 4 shows the CM rate as a function of $\theta$ for different $n$. For large $n$, the curve takes on a sigmoidal shape that converges to the theoretical curve from Theorem 5.3. The behavior near $\theta = 0$ suggests that in the Impartial Culture model, the CM rate converges to a limit within (0, 1), as proven for $m = 3$ by Lepelley and Valognes [25], conjectured in the general case by Durand [8, Conjecture 7.8], and further supported by Figure 5. This second figure also indicates that, as with Plurality and Two-Round, the limit CM rate in the critical regime appears to increase with the number of candidates $m$.

<table><caption>Table 1: Empirical study of Super Condorcet Winners (SCW) and IRV in two datasets, with the presence of a Condorcet Winner (CW) included as a reference.</caption><thead><tr><th></th><th>Netflix dataset</th><th>FairVote dataset</th></tr></thead><tbody><tr><td>Profiles</td><td>11,215</td><td>10,044</td></tr><tr><td>– with a CW</td><td>99.30%</td><td>99.98%</td></tr><tr><td>– where IRV is not CM (a)</td><td>95.87%</td><td>96.30%</td></tr><tr><td>– with an SCW (b)</td><td>94.05%</td><td>96.20%</td></tr><tr><td>Ratio (b) / (a)</td><td>98%</td><td>&gt; 99%*</td></tr></tbody></table>

## 5.3 Empirical Results for IRV

In the Perturbed Culture model, the presence of an SCW explains IRV's resistance to coalitional manipulation. However, as noted earlier, this is not a necessary condition: IRV can be non-manipulable in profiles without an SCW. This raises the question of whether the presence of an SCW often accounts for IRV's non-manipulability in realistic scenarios.

To investigate this, Table 1 analyzes the Netflix and FairVote datasets [9], which respectively contain 11,215 profiles derived from slight perturbations of 2,243 empirical profiles and 10,044 profiles based on 162 empirical profiles. It provides two key insights. First, an SCW is very common in real-world datasets, here appearing in 94% or 96% of profiles. Second, in most cases where IRV resists CM, this can be explained by the presence of an SCW—98% in the Netflix dataset and over 99% in the FairVote dataset. This confirms that SCWs are a crucial factor in IRV's resilience to manipulation.

The frequent appearance of SCWs may seem surprising, but it becomes intuitive when revisiting the definition. For a candidate $c$ and a subset $K$ of candidates that includes $c$, if preferences were perfectly balanced, we would expect $s_{Plu}(c, P_K) = \frac{w(P)}{|K|}$. The condition for being an SCW is simply to exceed this average. Therefore, even a slight bias in favor of $c$ makes it likely for $c$ to be an SCW.

# 6 CONVERGENCE SPEED

We will now study the convergence speed, to assess how fast the results found for $n \to \infty$ become relevant for finite values of $n$.

## 6.1 Theoretical Bound

We will first show that in the non-critical regime, the CM rate converges exponentially fast as $n \to \infty$. Next, we will bound this speed of convergence depending on the parameter $\theta$.

As an example, consider Lemma 3.1, where we assume the existence of a neighborhood where the rule $f$ is not CM. By definition, there exists $\epsilon > 0$ such that this neighborhood contains an open ball of diameter $\epsilon$ for the infinity norm. Our approach is to apply Hoeffding's inequality [19] to bound the probability that the normalized random profile $\bar{P}$ falls outside this ball. Since Hoeffding's inequality applies to scalar random variables, we use the union bound to extend it to the weight vector representing a voting profile. Formally,

<footnote>* We omit the next digit of the raw result (99.9%), not significant given the sample size.</footnote>
<figure><img src="image_7.png" /><figcaption>Figure 6: CM rate of Plurality as a function of n for different supercritical values of θ with m = 4. Monte Carlo simulations with 1,000,000 profiles per point.</figcaption></figure>

denoting by P the probability:

$$
\begin{align}
\rho(f, m, n, \theta) &\le \mathbb{P}(d(\bar{P}, \hat{P}) \ge \epsilon), \nonumber \\
&\le \sum_{\rho} \mathbb{P}(|w(p, \bar{P}) - w(p, \hat{P})| \ge \epsilon) \quad \text{(union bound)}, \nonumber \\
&\le 2m! e^{-2\epsilon^2 n} \quad \text{(Hoeffding's inequality).} \tag{1}
\end{align}
$$

Thus, the convergence is exponentially fast in $n$, and we can quantify the rate if the size of the neighborhood is known. By the same reasoning, similar results hold for Lemmas 3.2 and 4.2, and consequently for Theorems 3.3, 4.3, and 5.3. In the literature on phase transitions, this is known as *sharp* transitions, meaning that the limiting curve quickly approximates the behavior even for finite $n$.

Let us now bound the speed of convergence more precisely. For example, consider Plurality in the supercritical regime. If a profile $P$ lies within an open ball of radius $\epsilon$ centered at $\hat{P}$, the score of candidate 1 is bounded from below: $s_{Plu}(1, P) > \theta + \frac{1-\theta}{m} - m!\epsilon$, and the number of manipulators is bounded from above: $w(P^{2>1}) < \frac{1-\theta}{2} + m!\epsilon$. To ensure $s_{Plu}(1, P) > w(P^{2>1})$, we set $\epsilon = \frac{(3m-2)\theta-(m-2)}{2m!}$, which can be rewritten as $\epsilon = (\theta - \theta_c(\text{Plu}, m)) \frac{3m-2}{2m!}$. Using our bound (1), there exists a coefficient $A^+(\text{Plu}, m)$—which we could explicitly compute—such that:

$$ \rho(\text{Plu}, m, n, \theta) = O\left(e^{-A^{+}(\text{Plu}, m)} (\theta - \theta_c(\text{Plu}, m))^2\right). $$

This reasoning for Plurality in the supercritical regime generalizes to the subcritical regime, with a coefficient $A^{-}(\text{Plu}, m)$, and to the other voting rules in this paper: since all relevant quantities (scores, numbers of manipulators) are linear in the profile weights, we can take a value of $\epsilon$ that depends linearly on $\theta - \theta_c$, leading to a term in $(\theta - \theta_c)^2$ via Hoeffding's inequality. Thus, we obtain:

* Supercritical regime: $\rho = O(e^{-A^{+}(f,m)(\theta-\theta_c)^2n})$,

* Subcritical regime: $\rho = 1 - O(e^{-A^{-}(f,m)(\theta_c-\theta)^2n})$,

where $\rho = \rho(f, m, n, \theta)$ and $\theta_c = \theta_c(f, m)$.

## 6.2 Simulation Study of the Convergence Speed

Figure 6 shows the CM rate of Plurality as a function of $n$ for various values of $\theta$ (whereas Figure 1 does the reverse). Each curve

<figure><img src="image_8.png" /><figcaption>Figure 7: Asymptotic negative slope $C(4, \theta)$ from Figure 6, plotted as a function of $\theta - \theta_c(\text{Plu}, 4)$. The vertical blue lines represent error margins.<fnref n="6" /></figcaption></figure>

has an oblique asymptote on a semi-log scale, indicating not only that it is bounded by a decreasing exponential (as predicted by theory), but that it follows the form $\rho \sim n \to \infty B(m, \theta)e^{-C(m, \theta)n}$, with $m = 4$ here. This figure also allows us to measure the slopes of the asymptotes, providing the values of $C(4, \theta)$ for each $\theta$.

To analyze how convergence speed varies with $\theta$, Figure 7 plots the measured asymptotic slopes against $\theta - \theta_c$ in log-log scale (the values of $\theta$ were specifically chosen to be evenly spaced in that figure). When $\theta$ is far from $\theta_c$, the dependency is in $(\theta - \theta_c)^2$, in line with the upper bound found previously. However, close to $\theta_c$, the dependency seems to involve a smaller exponent (estimated at 1.265). In the terminology of phase transition, this is called the *critical exponent* of the convergence speed.

We repeated this for $m \in \{5, 6, 7\}$, the subcritical regime, and the other voting rules, with similar results but various critical exponents. This suggests a long-range dependency in $|\theta - \theta_c|^2$ but smaller critical exponents near the critical regime. This intriguing behavior will deserve further theoretical investigation.

## 7 FUTURE WORK

A natural direction for future work is to compute the critical parameter $\theta_c$ for other voting systems. Another key area of research would be a deeper analysis of the critical regime, including the calculation of the limiting CM rate at $\theta = \theta_c$ and the asymptotic behavior of the slope of the sigmoid $\rho(\theta)$ at $\theta = \theta_c$, which is linked to a finer analysis of the convergence speed in the non-critical regime. Expanding the study to other models, such as Mallows, is also promising. Preliminary analysis shows that the qualitative results observed in this paper, particularly the key finding that IRV's limit CM rate drops to zero with even slight concentration of preferences, also hold true under the Mallows model.

## ACKNOWLEDGMENTS

This work has been carried out at LINCS (www.lincs.fr).

<footnote n="6">⁶The computation of the error margins is detailed in the code repository https://github.com/francois-durand/irv-cm-aamas-2025.</footnote>
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