| paper | d4-conf-aaai-2026-2265 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | unclear |
The named results characterize equilibrium existence, limits, stability, and winning probabilities; none states an algorithm, hardness result, or computational formulation. Computing thresholds from finite encodings of the support and pivotality functions would be a new computational problem, not a mirror of a computational result in this paper.
fails bit a — no named computational result to mirror
No computational result is covered; the paper’s equilibrium, stability, and Jury theorem results remain analytical.
I can’t give an anchor-compliant positive case: the paper contains no numbered result classifying a computational problem as tractable, hard, FPT, or similar. Its numbered results are analytical. For example, Proposition 3 identifies the Calculus of Voting pivotality with a PPM, while Theorem 4 says that under weakly vanishing, \(q\)-tie-sensitive pivotality, each pivot point below \(q\) is the limit of left and right equilibria, and—when there are finitely many pivot points—every equilibrium limit is either \(0\) or a pivot point. Neither is a complexity result.
A plausible *follow-up* would represent voters by a mass distribution over preference and cost types, then ask for stable equilibrium thresholds satisfying \(c=p\!\left(N(s_A(c)+s_B(c)),\,|s_A(c)-s_B(c)|/(s_A(c)+s_B(c))\right)\), and classify their limits and winner probabilities as \(N\) grows. But that would be a new computational question based on the paper’s model, not a mirror of one of its named computational results.
The weak point for that follow-up is substantial: the paper’s winner probabilities come from finite-population sampling fluctuations. A deterministic continuum population removes those fluctuations; retaining them requires specifying a stochastic finite-\(N\) model alongside the mass description. So there are no qualifying anchors, and I would not claim an expected complexity class for this proposed problem from the paper.
There are no named computational results in this paper to anchor a continuous mirror. Its propositions and theorems concern equilibrium existence, limiting thresholds, stability, and winner probabilities. None makes a claim about complexity or algorithms. The proponent identifies this correctly, so there is no paper result for the reader to green as a computational mirror.
The proposed follow-up—compute stable equilibrium thresholds and classify their limits and winner probabilities—does not supply an anchor. The paper already represents voter costs and preferences through a distribution \(\mathcal D\) and its support functions \(s_A,s_B\); its equilibrium condition is the scalar fixed-point equation
\[
c=p\!\left(N(s_A(c)+s_B(c)),\frac{|s_A(c)-s_B(c)|}{s_A(c)+s_B(c)}\right).
\]
Turning that analysis into an algorithmic problem would require choosing finite encodings for the support functions and pivotality model, along with a precision and output specification. Those choices define a new computational problem. For ordinary finite descriptions, the unknown remains a single threshold; any hardness would need to come from the chosen representation or an expanded model, rather than from a computational result in this paper.
The stronger obstacle is that the paper’s jury-theorem classification depends on finite-population sampling fluctuations. Its slow, moderate, and fast cases turn on the scale \(\sqrt{N}|c_N-c^*|\). An exact continuum of mass removes the random vote counts whose fluctuations determine those winner probabilities. Retaining them requires keeping a finite sample of \(N\) voters from \(\mathcal D\), which preserves the paper’s model rather than making the realized society a continuous mass distribution.
One could formulate a new problem using repeated cost cohorts with fractional masses, perhaps allowing partial participation by a marginal cohort. That might be worth studying. The paper assumes an atomless positive-cost distribution, though, and does not analyze that cohort model or its algorithmic complexity. It would be a related project, not a continuous mirror of one of the paper’s results. The negative case is therefore strong for rejecting a mirror of this paper; the broader claim that no useful computational project could be built from its ideas is less certain.
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.