Tyranny of the Minority in Social Choice: a Call to Arms

The paper contains no numbered computational theorem, lemma, corollary, proposition, algorithmic result, or complexity classification, so bit (a) fails. The proposed abstention-robustness problem is a plausible high-multiplicity question, but it cannot serve as a mirror of a result in this paper.

The objection that survived

The proposed robustness problem is a new formalization rather than a computational result established by the paper; under the named-result gate, this is fatal.

What the mirror covers

The proposed mirror covers worst-case abstention and outcome distortion, but leaves strategic abstention, engagement, sampling, delegation, PPV, and artificial-agent directions untreated.

Open questions for a prover

The case FOR (proponent)

The strict answer is that this paper contains no eligible computational anchor. It has no numbered Theorem, Lemma, Corollary, or Proposition, and it proves no \(P\), NP-hardness, FPT, approximation, or parameterized-complexity result. Statements such as “we showed” refer to cited or unpublished work, especially [30] and [31], rather than results formally established in this paper. The Condorcet Jury Theorem is mentioned as background, not proved or computationally classified here.

The strongest honest positive case is therefore a proposed mirror of the paper’s central open direction, not a mirror of a named theorem.

Call it Continuous Worst-Case Abstention Robustness. An instance contains candidates \(C\), a positional scoring rule \(s\), a finite set of complete voter types \(T\), a rational society distribution \(\mu\in\mathbb{Q}^{T}\), an abstention budget \(\alpha\in[0,1]\cap\mathbb{Q}\), and a target candidate \(c^\star\). A type includes its ranking and any participation parameter used by the model. The adversary chooses abstention masses \(x_t\) satisfying \(0\le x_t\le\mu_t\) and \(\sum_t x_t\le\alpha\). The active society is \(\mu-x\). The task is to compute the minimum score margin of \(c^\star\) over all admissible abstention patterns, namely \( \rho=\min_{x}\min_{d\ne c^\star}\sum_t(\mu_t-x_t)(s(t,c^\star)-s(t,d)) \), and to output a witnessing abstention vector if \(\rho<0\). Thus \(c^\star\) is guaranteed to win exactly when \(\rho\ge0\), subject to the paper’s chosen tie convention.

This is a natural high-multiplicity regime for national referenda, participatory budgeting, shareholder votes, and large online polls: millions of voters fall into a relatively small number of repeated preference and participation types, with \(N\gg |T|\). The mass \(x_t\) is not an illicit individual-level price or identity deletion; it is precisely the fraction of a homogeneous voter type that remains silent. For rational masses, clearing denominators recovers a finite election with repeated voter clones, so the bridge to high multiplicity is exact.

The authors should recognise this as their own question. It formalizes the running \(60\%\)-versus-\(8\%\) example and the “Worst-case and Best-case abstention” discussion, while preserving the paper’s core concern: the active electorate chooses the winner, but the outcome affects the entire population. For explicitly listed types and scoring rules, the problem is likely Class A: each rival comparison is a linear optimization problem, and the worst abstention pattern removes the most damaging available mass first. If rankings are represented implicitly, the resulting pricing problem may become substantially harder.

The mirror covers only the paper’s distortion and worst-case-robustness agenda. It does not cover strategic abstention, engagement, delegation, PPV, or artificial agents. Its weakest point is consequently serious: it converts a broad normative call into a new robust-optimization problem, and it does not preserve individual pivotality or equilibrium incentives in an atomless society. Further questions would include continuous best-case sampling, optimal incentive allocation to increase turnout, rounding guarantees for finite elections, and type-level delegation models.

So the positive case is that the paper points unusually clearly toward a sensible continuous population model. But under the programme’s named-result rule, it has no qualifying anchor, and this candidate cannot honestly be presented as mirroring a theorem of the paper.

The case AGAINST (opponent, writing after the proponent)

The decisive objection is that this paper has no eligible computational anchor. It is a blue-sky position paper: it contains no numbered theorem, lemma, corollary, proposition, algorithmic result, or complexity classification. The claims attributed to “we showed” belong to cited or unpublished work, especially [30] and [31], not to this paper. It does not define a computational input-output problem whose complexity could be mirrored.

The proposed Continuous Worst-Case Abstention Robustness problem is nevertheless a sensible new ChoCo problem. With finitely many explicit types, its objective

\[ \rho=\min_{x}\min_{d\ne c^\star} \sum_t(\mu_t-x_t)\bigl(s(t,c^\star)-s(t,d)\bigr) \]

is a rational linear optimization problem, and clearing denominators gives repeated voter clones. Thus the proponent is right that this is a legitimate high-multiplicity regime for referenda, shareholder votes, participatory budgeting, and online polls. That also means the usual objections about mass being meaningless or type-level prices being artificial do not work here.

But those virtues establish only that the paper inspires a worthwhile new problem; they cannot turn it into a mirror of a result in the paper. The proposed problem selects one paragraph—worst-case abstention—and supplies the missing computational formulation itself. Under the programme’s named-result gate, that is a re-modeling prompted by the paper, not a continuous analogue of one of its results. Calling it “the paper’s central open direction” does not repair the absence of a source theorem.

The stronger strategic version fares worse in the continuum. A unilateral voter has mass zero, so changing that voter’s ballot or turnout changes every aggregate score by zero. The pivotality term central to the paper’s strategic-abstention discussion therefore vanishes. With positive voting costs, abstention becomes the degenerate prediction; preserving nontrivial incentives requires finite-\(N\) noise, costs scaled with \(N\), or positive-mass coalitional deviations. Each rescue is meaningful, but each adds a second asymptotic parameter or changes individual deviations into coalition deviations. It is then a new mean-field or finite-size model rather than a direct continuous population mirror.

Delegation and PPV cannot provide a better rescue without the same problem in another form. Delegation depends on individual links, proxy loads, paths, accountability, and sometimes histories. A type containing all such relational information is generally unique to an individual, destroying high multiplicity. Aggregating to type-to-type flows or a graphon can be interesting, but it changes the relational object and is not determined by the population marginal \(\mu\). Artificial-agent and VAA versions likewise require an information-access model: if the population distribution is given explicitly, the learning problem has been removed; if it is hidden, the central issue becomes statistical elicitation or sample access, not population continuization.

So the negative verdict is strong under the programme’s strict screening rule: this paper supplies no qualifying computational anchor, and the best proposed mirror is an unanchored new optimization problem. The universal claim that no worthwhile continuous question can be inspired by the paper is weaker and should not be overstated. The abstention-robustness formulation is genuinely plausible; it simply cannot honestly be presented as a mirror of a computational result contained in this paper.

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.