Which Voting Rules Are More Resilient to Coalitional Manipulation

Jul 16, 2026

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The study investigates which ordinal voting rules are more resistant to coalitional manipulation by developing a deliberately minimal probabilistic model. Under the Perturbed Culture framework, parameterized by the extra weight of one ranking, each rule shows a sharp phase transition: manipulation succeeds below a critical concentration and fails above it. The resulting thresholds partition rules into natural families linked to strengthened Condorcet-winner notions. Empirical tests on Netflix and FairVote confirm that the model predicts relative vulnerability, its evolution with candidate count, and cluster persistence.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages46

Reading Time1 h 56 min

Abstract

Introduction

Motivation

Contributions

Related Work

Limitations

Roadmap

Framework

Voting Theory and Model

Voting Rules

Known Results: Phase Transitions in the Perturbed Culture

Theoretical Results

Pair-Safe Condorcet Winner (PSCW)

Set-Safe Condorcet Winner (SSCW)

Resistant Condorcet Winner (RCW)

Critical Concentration Parameters

Numerical Results

Overall CM Rates

Variation with the Number of Candidates

Comparison of Voting Rules for Fixed Candidates

Future work

References

arXiv :2607 .00758v 1 [ cs .GT] 1 Jul 2026 Which Voting Rules Are More Resilient to Coalitional Manipulation? FRANÇOIS DURAND (NOKIA BELL LABS FRANCE) Which voting rules are more resilient to coalitional manipulation? We find that a deliberately minimal model, capturing only the degree of advantage of one preference ranking over the others, can predict their relative vulnerability remarkably well. Extending prior work on three rules, we systematically analyze all standard ordinal voting rules under the Perturbed Culture model, a variant of Impartial Culture parameterized by the extra weight assigned to one ranking. Each rule exhibits a sharp phase transition: manipulation succeeds with high probability below a critical concentration threshold, and fails above it. This structure reveals natural families of rules: seemingly distinct methods such as Maximin, Ranked Pairs, Schulze, and Young share identical thresholds, while Baldwin, Nanson, Kemeny, and Dodgson form another. These groupings are driven by new, strengthened notions of Condorcet winners. In addition, we identify a third family based on a previously introduced Condorcet notion: Black, Slater, and Copeland. Empirically, the model displays strong predictive power. Tested on real-world datasets (Netflix and FairVote), it accurately ranks rules by vulnerability, predicts how this ranking evolves with the number of candidates, and explains why empirically similar clusters persist despite large absolute differences in manipulation rates, with a more nuanced picture for Bucklin and veto-based rules. Thus, an extremely parsimonious model with no tuning captures the comparative vulnerability of voting rules: which rules to prefer depends largely on the number of candidates alone. Presentation video: [https://www.youtube.com/watch?v=hY4233TGUGw](https://www.youtube.com/watch?v=hY4233TGUGw). Contents Abstract 0 1 Introduction 1 1.1 Motivation 1 1.2 Contributions 1 1.3 Related Work 2 1.4 Limitations 2 1.5 Roadmap 2 2 Framework 3 2.1 Voting Theory and Model 3 2.2 Voting Rules 4 2.3 Known Results: Phase Transitions in the Perturbed Culture 5 3 Theoretical Results 7 3.1 Pair-Safe Condorcet Winner (PSCW) 7 3.2 Set-Safe Condorcet Winner (SSCW) 9 3.3 Resistant Condorcet Winner (RCW) 11 3.4 Critical Concentration Parameters 12 4 Numerical Results 14 4.1 Overall CM Rates 14 4.2 Variation with the Number of Candidates 16 4.3 Comparison of Voting Rules for Fixed 􀀼 16 5 Future work 18 References 19 François Durand (Nokia Bell Labs France) 1 1 Introduction Which voting rules are more resilient to coalitional manipulation? This question has long motivated research in social choice theory, yet remains difficult to answer in general terms. The Gibbard–Satterthwaite theorem establishes that no non-trivial voting rule can be entirely immune to manipulation, but it leaves open the possibility of meaningful comparisons between rules. This paper develops a principled framework for such comparisons, based on a deliberately simple probabilistic model that captures the essential structure of voting rule vulnerability. 1.1 Motivation A voting rule is coalitionally manipulable (CM) in a given profile if a subset of voters could obtain a preferred outcome by misreporting their preferences. This property can be interpreted ex ante asa vulnerability to strategic voting, or expost as a source of regret for sincere voters, potentially undermining trust in electoral outcomes [Durand, 2015, Eggers and Nowacki, 2024] . While the Gibbard–Satterthwaite theorem implies that any non-trivial voting rule is susceptible to this phenomenon [Gibbard, 1973, Satterthwaite, 1975], it remains possible to compare the vulnerability of voting rules, in particular through their CM rate, defined as the theoretical or empirical proportion of profiles in which the rule is coalitionally manipulable. To explain the low empirical vulnerability of Instant-Runoff Vo

FAQs

What question does the paper address about voting rules?

It asks which standard ordinal voting rules are more resilient to coalitional manipulation and how their vulnerability can be compared in a principled way.

How does the Perturbed Culture model relate to manipulation vulnerability?

The model varies preference concentration by assigning extra weight to one ranking. Each voting rule exhibits a phase transition: coalitional manipulation succeeds with high probability below a critical threshold and fails above it.

What theoretical structure explains why different voting rules share the same thresholds?

The thresholds group seemingly distinct rules into families driven by strengthened notions of Condorcet winners, including an additional family based on a previously introduced Condorcet notion.

How well does the minimal model perform on real data?

Experiments on Netflix and FairVote show strong predictive power: it ranks rules by vulnerability, tracks how that ranking changes with the number of candidates, and clarifies why empirically similar clusters persist despite large differences in manipulation rates.

Jul 16, 2026

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