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arxiv: 2605.23742 · v1 · pith:JOZF5MB2new · submitted 2026-05-22 · 💻 cs.GT

Super Condorcet Winners and Limit Coalitional Manipulability of IRV

Fran\c{c}ois Durand , \'Elie de Panafieu , Guillem Perarnau (UPC) This is my paper

Pith reviewed 2026-05-25 02:27 UTC · model grok-4.3

classification 💻 cs.GT
keywords coalitional manipulabilityIRVinstant runoff votingsuper condorcet winnerimpartial cultureplurality with runofflimit probabilityvoting rules
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The pith

IRV has a limit coalitional manipulability rate strictly below 1 for any number of candidates, while Plurality with Runoff reaches rate 1 for m at least 4.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes earlier findings on three candidates to show that Instant Runoff Voting cannot be manipulated by coalitions with probability approaching 1, even for large numbers of candidates. The protection comes from the existence of a Super Condorcet Winner, which the authors prove supplies both an upper bound and the exact asymptotic value of the manipulability rate. Plurality with Runoff lacks this protection and becomes fully manipulable in the limit once there are four or more candidates. The results are derived under the impartial culture model for large electorates.

Core claim

The limit coalitional manipulability rate of IRV equals the probability that a Super Condorcet Winner exists, a quantity strictly less than one for every m. Plurality with Runoff has limit rate exactly one when m is at least four. The Super Condorcet Winner condition yields an upper bound on IRV manipulability that is shown to be asymptotically tight, allowing the exact rate to be computed from the existence probability.

What carries the argument

Super Condorcet Winner, a candidate whose pairwise margins are strong enough to block all coalitional manipulations of IRV.

If this is right

  • Plurality with Runoff is fully coalitionally manipulable in the limit for every m at least 4.
  • The exact limit manipulability rate of IRV equals the probability a Super Condorcet Winner exists under impartial culture.
  • For m equals 3 the new result recovers the closed-form value obtained by earlier authors.
  • The Super Condorcet Winner supplies a matching lower bound on the manipulability rate in addition to the known upper bound.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Runoff procedures that eliminate candidates sequentially may systematically differ from other rules in their asymptotic resistance to coalitions.
  • The probability of a Super Condorcet Winner could be used as a quick proxy for manipulability resistance in related single-winner or multi-winner settings.
  • Explicit formulas or efficient computation methods for the Super Condorcet Winner probability at large m would make the exact IRV rate immediately usable.

Load-bearing premise

The upper bound on the coalitional manipulability rate of IRV from the existence of a Super Condorcet Winner is asymptotically tight for large electorates.

What would settle it

Numerical computation or simulation for increasing electorate sizes showing that the coalitional manipulability frequency of IRV converges to a number different from the probability of a Super Condorcet Winner.

Figures

Figures reproduced from arXiv: 2605.23742 by \'Elie de Panafieu, Fran\c{c}ois Durand, Guillem Perarnau (UPC).

Figure 1
Figure 1. Figure 1: Probability that IRV is susceptible to coalitional manip [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper generalizes Lepelley and Valognes (1999) on the limit coalitional manipulability (CM) rate under Impartial Culture. For m=3, IRV (equivalent to Plurality with Runoff) has limit CM rate strictly below 1. The authors prove that for m≥4, Plurality with Runoff has limit CM rate exactly 1, while IRV retains a limit CM rate strictly below 1. They rely on the Super Condorcet Winner (from Durand 2025) to upper-bound the CM rate of IRV, prove this bound is asymptotically tight as electorate size tends to infinity, and compute the exact limit probability of a Super Condorcet Winner existing.

Significance. If the central claims hold, the work provides a precise characterization of limit CM rates for two runoff-based rules across all m, showing IRV's superior resistance to coalitional manipulation relative to Plurality with Runoff for m≥4. The asymptotic tightness result converts the Super Condorcet Winner existence probability into an exact limit CM rate, extending the m=3 case with a falsifiable quantitative prediction under Impartial Culture.

major comments (1)
  1. [§4] §4 (proof of asymptotic tightness): the argument that the Super Condorcet Winner upper bound is tight relies on constructing a sequence of profiles where CM occurs exactly when a Super Condorcet Winner exists in the limit; this step must be verified to ensure the probability computation in Theorem 5.2 is not merely an upper bound.
minor comments (2)
  1. [Introduction] Notation for the limit CM rate (e.g., the expression involving the probability of Super Condorcet Winner) should be introduced earlier, before its use in the main theorems, to improve readability.
  2. [Theorem 3.1] The statement of Theorem 3.1 on Plurality with Runoff achieving limit rate 1 for m≥4 would benefit from an explicit reference to the section containing the construction showing CM with probability approaching 1.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending minor revision. We address the major comment below.

read point-by-point responses

  1. Referee: [§4] §4 (proof of asymptotic tightness): the argument that the Super Condorcet Winner upper bound is tight relies on constructing a sequence of profiles where CM occurs exactly when a Super Condorcet Winner exists in the limit; this step must be verified to ensure the probability computation in Theorem 5.2 is not merely an upper bound.

    Authors: We appreciate the referee drawing attention to this aspect of the proof. In Section 4, the argument for asymptotic tightness proceeds by constructing an explicit sequence of preference profiles (P_n) indexed by electorate size n. For this sequence, we establish that, in the limit as n tends to infinity, a coalition can successfully manipulate the IRV outcome if and only if a Super Condorcet Winner exists. The construction specifies the limiting vote shares and the precise conditions under which a manipulating coalition can force a different winner in the runoff stages when the Super Condorcet Winner is present, while showing that no such manipulation is feasible in the limit when it is absent. This equivalence is verified by analyzing the first-round plurality scores and the pairwise comparisons that determine the second-round outcome. Consequently, the probability of coalitional manipulability converges exactly to the probability that a Super Condorcet Winner exists, which is computed in Theorem 5.2. The details appear in the proof of Theorem 4.3. revision: no

Circularity Check

0 steps flagged

Minor self-citation for upper bound; central derivation and tightness proof are independent

full rationale

The paper invokes the Super Condorcet Winner concept and resulting upper bound from Durand [2025] (co-author overlap) to bound the IRV CM rate, then proves asymptotic tightness and computes the SCW existence probability under Impartial Culture to obtain the exact limit rate. This computation and tightness argument constitute new, self-contained content that does not reduce by construction to the cited bound or any fitted input. No self-definitional, fitted-prediction, or renaming patterns appear. The self-citation is not load-bearing for the final result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the impartial culture model for generating random preferences and on the properties of the Super Condorcet Winner concept introduced in prior work.

axioms (1)
  • domain assumption Impartial Culture: each voter independently draws a uniform random ranking of the m candidates
    Defines the probability space in which the limit CM rate is computed as electorate size tends to infinity.
invented entities (1)
  • Super Condorcet Winner no independent evidence
    purpose: Provides an upper bound on the coalitional manipulability rate of IRV
    Introduced in Durand 2025; the paper proves the bound is tight for the limit rate.

pith-pipeline@v0.9.0 · 5781 in / 1284 out tokens · 23461 ms · 2026-05-25T02:27:49.169082+00:00 · methodology

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Reference graph

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