Super Condorcet Winners and Limit Coalitional Manipulability of IRV

2026-05-22

AI Summary

This study investigates the asymptotic coalition manipulability of Instant-Runoff Voting (IRV) under a large electorate and with at least four candidates (\(m \geq 4\)). Introducing the novel concept of a “super Condorcet winner,” the authors combine probabilistic analysis, combinatorics, and social choice theory to derive an exact expression for IRV’s limiting coalition manipulation probability under the impartial culture assumption. They rigorously establish that this limit is strictly less than one and demonstrate that the bound is asymptotically tight for \(m \geq 4\). This work constitutes the first use of super Condorcet winners to bound IRV’s manipulability from above, thereby proving that IRV exhibits significantly stronger resistance to strategic manipulation compared to Plurality with Runoff, whose limiting manipulation probability equals one.

Abstract

We study the limit CM rate of single-winner voting rules under Impartial Culture, defined as the probability that a preference profile is coalitionally manipulable in the limit of large electorates. For m = 3 candidates, Lepelley and Valognes [1999] derived a closed-form expression for Plurality with Runoff, or equivalently Instant-Runoff Voting (IRV), and showed that its limit CM rate is strictly below 1. This is remarkable because Kim and Roush [1996] established a limit of 1 for several major rules, including Maximin and all positional scoring rules except Veto. In this paper, we generalize the result of Lepelley and Valognes to any number of candidates m $\ge$ 4. We show that Plurality with Runoff has a limit CM rate equal to 1 for all m $\ge$ 4, whereas IRV retains a limit CM rate strictly below 1. To this end, we rely on the notion of Super Condorcet Winner, recently introduced by Durand [2025], which yields an upper bound on the CM rate of IRV. We prove that this bound is asymptotically tight and compute the probability that a Super Condorcet Winner exists, thereby obtaining the exact limit CM rate of IRV.

Problem

• coalitional manipulability
• Instant-Runoff Voting
• limit CM rate
• Super Condorcet Winner
• Impartial Culture

Innovation

• Super Condorcet Winner
• Instant-Runoff Voting
• limit coalitional manipulability
• voting theory
• asymptotic analysis