Ballot Length in Instant Runoff Voting

The paper’s numbered theorems are existential and structural, while its NP-hardness statements are citations rather than results proved in the paper. The proposed continuous questions are precise and sociologically plausible, but they add new electorate-design objectives rather than continuizing a computational result of this paper. Therefore the paper fails the computational entry gate.

The objection that survived

Theorem 9 does not assert complexity or an algorithm, so minimum-support continuous truncation design cannot serve as its computational mirror; it changes the paper’s fixed-profile analysis into electorate synthesis.

What the mirror covers

The proposed mirrors cover the paper’s worst-case truncation sensitivity and single-peaked constructions, but not its simulations, cited hardness results, or a named robustness algorithm.

Open questions for a prover

The case FOR (proponent)

There is a credible positive case, strongest around the paper’s explicit separation between voters and ballot types. One caveat comes first: the paper contains no numbered theorem classifying a problem as NP-hard, polynomial-time, W[1]-hard, or FPT. Its NP-hardness statements about possible winners and IRV margin are cited results, not named results proved in this paper. So, under a strictly literal anchor rule, there is no qualifying complexity-theoretic anchor. The best available positive case therefore rests on the paper’s named constructive and structural theorems.

The mirror I would use is this. A type is a complete ballot behaviour \(t=(\pi,\ell)\), where \(\pi\) is a full ranking and \(\ell\) is an intrinsic stopping point; at ballot length \(h\), the voter submits the first \(\min(h,\ell)\) candidates. This encodes the paper’s partial ballots without treating stopping behaviour as an individual idiosyncrasy. A continuous society is a distribution \(\mu\) over these types. For each \(h\), run exactly the paper’s IRV procedure on the truncated mass distribution. The outcome remains a discrete IRV winner; only the electorate has become continuous.

This is plausible in elections with millions of voters but a modest number of repeated preference blocs: party supporters, demographic or occupational cohorts, or ideological bands whose members submit the same ranking and ballot length. The regime is \(n\gg\tau\), with many agents per type. The paper’s own simulations even use an infinite voter population in its 1-Euclidean model, although there the continuum is used for simulation rather than computational analysis.

My lead anchor is Theorem 9, proved by the authors: “Given \(k>3\) candidates, there is a tie-free profile producing \(k-1\) truncation winners with \(\Theta(k^3)\) voters of \(\Theta(k)\) types.” This is unusually well suited to continuization because it explicitly says that the extreme phenomenon does not require one distinct voter per preference. It occurs with many repeated types.

The corresponding problem is *Minimum-Support Continuous Truncation Design*. Given \(k\), a finite permitted type set \(T\), and a rational tie margin \(\delta>0\), find a distribution \(\mu\in\Delta(T)\) such that every IRV run at \(h=1,\ldots,k-1\) is tie-free with margin at least \(\delta\), and \(k-1\) distinct candidates win across those ballot lengths, while minimizing \(|\operatorname{supp}(\mu)|\). Equivalently, its decision version asks whether such a society exists using at most \(q\) types.

A solution is the mass vector \(\mu\), together with the resulting elimination orders and winner sequence. This is recognisably the paper’s question: the rankings and truncation rule are unchanged, and the objective asks how economically a population can realize the paper’s extreme ballot-length sensitivity. Total voter count disappears, as it should in a high-multiplicity model. Theorem 9 gives a feasible construction with \(\Theta(k)\) types; \(k-1\) types are also necessary in the extreme case because each distinct winner must receive first-place mass somewhere. Thus the continuous version inherits the theorem’s tight \(\Theta(k)\) type scale.

I expect the fixed-elimination-order version to be Class A. Once the order at each ballot length is fixed, every IRV tally is linear in \(\mu\), so feasibility and maximum tie margin are linear programs. The global minimum-support version is more likely Class C: its difficulty comes from selecting support types and elimination histories, not from the number of named voters. That would be a meaningful continuum-specific boundary rather than a defect in the mirror.

A second, independent anchor is Theorem 5, also proved here: when \(k=\kappa(\kappa+1)/2\), there is a single-peaked consequential-tie-free profile with \(3\kappa(\kappa+1)/2\) partial ballots and \(\kappa\) distinct truncation winners. Its mirror is *Single-Peaked Continuous Truncation Capacity*. The instance consists of a candidate axis, the finite set \(T_{\mathrm{SP}}\) of admissible single-peaked ballot types, and a tie-breaking rule. The decision variable is \(\mu\in\Delta(T_{\mathrm{SP}})\); the objective is to maximize
\[ \left|\{W_h(\mu):h=1,\ldots,k-1\}\right|. \]
A solution is the mass distribution and the induced winner sequence. The theorem predicts capacity at least \(\kappa=\Theta(\sqrt{k})\) in the continuous single-peaked regime.

This version is especially plausible as a population mirror: a large electorate can consist of mass distributed across ideological intervals or recurring left-right preference cohorts. I would expect the fixed-axis version to be tractable, using the one-dimensional structure together with LP or interval dynamic programming; the exact upper bound is itself an open question in the paper and becomes a natural continuous-computational problem. If hardness appears, it would be worth determining whether it comes from the continuous mass design or from the candidate/elimination combinatorics.

These mirrors cover the paper’s worst-case truncation sensitivity and its single-peaked restriction. They do not claim that the paper’s simulations, resampling experiments, or cited NP-hard margin-of-victory result are already continuous computational social choice. Nor do they treat fractional outcomes or random winners as the relevant continuity. The continuous object is specifically the population distribution.

The weakest point is that the paper’s central results are existential rather than algorithmic. Theorem 9 already establishes the unrestricted \(k-1\)-winner phenomenon, so the continuous problem must add a genuine computational objective—such as minimum support, maximum margin, or minimum mass movement—to avoid merely renaming the theorem. A second weakness is regime dependence: the mirror is convincing for elections with repeated blocs or ideological cohorts, but much less so for a small committee whose members have genuinely idiosyncratic rankings.

The strongest honest conclusion is therefore: this paper does not supply a named complexity-theoretic anchor, but it does supply an unusually good structural anchor for a continuous population mirror. Theorem 9, in particular, already contains the high-multiplicity pattern the programme needs: \(\Theta(k^3)\) agents compressed into \(\Theta(k)\) meaningful types while preserving extreme IRV behaviour.

The case AGAINST (opponent, writing after the proponent)

The negative case is strongest at the programme’s entry gate: this paper supplies no qualifying computational anchor. Theorem 1–9 are existential and combinatorial statements about which winner sequences can be realized. They do not define an input problem, an optimization task, or a complexity classification. The NP-hardness statements in the paper are citations to prior work, not results proved here. Thus the proponent is trying to manufacture a computational problem around structural theorems rather than continuize one of the paper’s computational results.

Theorem 9 does establish that the high-multiplicity regime is sensible. A profile with \(\Theta(k^3)\) voters and \(\Theta(k)\) ballot types is exactly the kind of repeated-bloc electorate the programme permits. But that is evidence for the modelling regime, not evidence for a worthwhile continuous mirror. Passing from its integer counts to normalized masses preserves the already-known winner sequence and erases the theorem’s main quantitative content: multiplying every count changes the voter total but not the election. The \(\Theta(k^3)\) lower-bound phenomenon simply disappears.

The proposed minimum-support problem adds a new objective that the paper never studies. Support is not a resource, cost, or constraint in the paper; it is merely a descriptive statistic. Its lower bound is also nearly tautological: \(k-1\) distinct winners require at least \(k-1\) first-place ballot types, while Theorem 9 already supplies an \(O(k)\)-type construction. The fixed-elimination-order LP is a feasibility certificate for a prescribed winner history, not an algorithmic counterpart of Theorem 9. A global support-minimization problem might be an interesting profile-synthesis problem, but it would be a new problem attached to the theorem, not a computational mirror of it.

The same difficulty defeats better variants of that anchor. One could optimize the normalized margin, minimize the mass that must be moved to alter the winner sequence, or ask for the smallest denominator of a rational society. The first two are legitimate IRV robustness or manipulation questions; the last is an encoding question. None is supplied by Theorem 9, and each changes the paper’s fixed-profile analysis into electorate design or fractional control. That may deserve a separate project, but it cannot make this paper a computational-continuization anchor.

Theorem 5 is more defensible sociologically. A large electorate distributed across ideological cohorts can quite naturally be single-peaked, and replacing repeated voters by masses loses no relevant information. So “there is no possible high-multiplicity story” would be a bad objection here. But its proposed continuous capacity problem again reverses the quantifiers: the paper constructs a profile; the mirror chooses a distribution \(\mu\) in order to manufacture one. Given \(\mu\), the direct question—compute the winners for \(h=1,\ldots,k-1\)—is just \(k-1\) finite IRV simulations over a weighted ballot histogram.

Choosing a more sophisticated type representation does not repair the problem. With a fixed candidate axis, single-peaked rankings form a finite type space; with ideal-point locations, the finitely many ranking regions collapse the continuum to finitely many ballot types. Thus the “continuous” capacity question is either a weighted version of the same finite profile-synthesis problem or requires an additional distributional model not present in the paper. Theorem 5’s \(\Omega(\sqrt{k})\) construction transfers verbatim as an existence witness, but it does not create a computational task.

The closest escape hatch is the paper’s closing suggestion of a ballot-length-sensitive margin of victory. A fractional mass-perturbation version could indeed be a worthwhile IRV robustness problem. But it is explicitly an open direction beyond the paper, not a named result of it; its natural anchor would be the cited IRV margin literature or a new paper defining that notion. The same is true of optimizing margins or support under single-peakedness. These are plausible follow-ups, but their plausibility cannot retroactively supply the missing computational anchor.

So the proponent has shown that both the unrestricted and single-peaked settings admit sensible high-multiplicity stories. They have not shown that this paper contains a worthwhile continuous computational mirror. The honest qualification is that a strong researcher could build a valuable new paper on fractional IRV robustness or electorate synthesis. The case against is that doing so would be inventing the computational problem, not continuizing one that this paper actually establishes.

The adversarial triple: the proponent anchors on up to three named results; the opponent sees that case and must defeat every anchor; the judge decides which case convinced it. These are the pipeline’s own outputs, generated by tools/triple_run.py — no human edited them. The paper’s own text is not reproduced here beyond the quoted statement above.