| paper | d4-journals-mss-2022-0027 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | yes |
The proponent identifies the key gap: Proposition 11 concerns voting behavior, not computation. A weighted population version of \(L_p\) winner determination is a sensible question, but the paper does not establish a named computational result whose analogue it would be.
fails bit a — no named computational result to mirror
The proposed model concerns winner determination for \(L_p\) deepest voting; it does not mirror the paper’s axiomatic or paradox results as computational results.
This paper has no named computational complexity result of the required kind: no numbered theorem, lemma, corollary, or proposition establishes that an underlying problem is in P, NP-hard, fixed parameter tractable, or similar. Its numbered propositions concern voting axioms and paradoxes. For example, Proposition 11 (proved here) says that \(L_p\) deepest voting has reinforcement and no-show paradoxes for \(p\in[1,\infty]\setminus\{2\}\), and neither paradox for \(p=2\). That is a result about voting behavior, not computational complexity, so it cannot serve as an anchor under your criteria.
The closest plausible mirror would take a voter type to be a complete grade vector \(z\in[0,1]^d\), with mass \(\mu_z\) representing the fraction of a large electorate sharing that vector. For finite support, \(L_p\) deepest voting chooses \(x\) to minimize \(\sum_z\mu_z\lVert z-x\rVert_p^p\), then elects the candidate with the largest coordinate of \(x\). This is a sensible population version of the paper’s model; for \(p=1\) and \(p=2\), it gives weighted medians and weighted means, respectively. But the paper does not state a named complexity result about computing that winner, nor does it analyze a continuous population. Presenting this as an anchored continuous mirror would therefore overstate the paper’s results.
The proposed anchor does not survive the programme’s computational test. Proposition 11 concerns reinforcement and no-show behavior; it gives no complexity bound, algorithm, or hardness result. The other nearby results are also axiomatic or behavioral: Propositions 1–5 study voting properties, Proposition 8 identifies familiar rules within the \(L_p\) family, and Propositions 10–11 exhibit paradoxes. Replacing voter counts with masses could restate some of those claims, but it would not turn any of them into a computational result.
The best version of the proposed \(L_p\) mirror takes each grade vector as a type and minimizes \(\sum_z \mu_z\lVert z-x\rVert_p^p\). That is a sensible model, but for the paper’s familiar cases it mostly yields weighted medians or means; its \(L_\infty\) case yields coordinate-wise midranges. Asking for their continuous-population behavior would not uncover a complexity landscape the paper established. For Proposition 11 specifically, removing one voter from a nonatomic population changes no mass, so the individual no-show question becomes vacuous. Removing a positive-mass cohort repairs that, but asks a different behavioral question, not a computational one. Reinforcement can be phrased for mass distributions too, with the same limitation.
One could instead study the complexity of finding deepest points for halfspace, projection, or Oja depth from rational weighted support. That may be worthwhile as a computational project, but it is a new complexity analysis of winner determination, not a continuous mirror of a result in this paper. The paper’s general depth-function notation already accommodates distributions, while its voting results are stated for finite grading matrices; a finite-support mass input is also a weighted version of those grade vectors. So the negative case is strong under ChoCo’s anchoring criterion, though not a proof that adjacent computational work lacks value.
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.