| paper | d4-conf-wine-2025-0017 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | no |
Each named candidate theorem establishes axiomatic properties of a tournament rule; none states an algorithm, complexity bound, or computational hardness result. The unnumbered claim that the rules can be implemented in \(O(n^2)\) time does not satisfy the named-result gate, leaving no computational anchor to continuize.
fails bit a — no named computational result to mirror
No results are covered by a valid computational mirror: Theorems 5.7, 4.4, and A.5 give axiomatic guarantees, while the \(O(n^2)\) implementation claim is unnumbered.
I can’t give an anchor-backed positive case under your criteria: this paper has no numbered result asserting a computational complexity classification.
The closest candidates are Theorem 5.7, which proves that the paper’s \(\lambda\)-rule is Condorcet consistent, monotone, and \(2\)-NM\(_{11}\) for every tournament size; Theorem 4.4, which proves those first two properties and \(3.5\)-multiplicative pairwise non-manipulability for its other rule; and Theorem A.5, which proves top-cycle consistency, monotonicity, and \(5\)-multiplicative pairwise non-manipulability. These are constructive axiomatic guarantees, not results that a problem is in P, NP-hard, or otherwise classified. The paper’s quadratic implementation claim appears in unnumbered prose, so it does not qualify as an anchor either.
There is a possible limit-object intuition: in a large league, team types could represent recurring team archetypes, mass could be their share of the league, and a tournament could be represented by a type-level match-outcome kernel. A rule would output a probability density over types. But the resulting question—whether such a rule satisfies continuum versions of fairness and manipulation axioms—is an axiomatic rule-design problem, not the computational question the programme is seeking. In an atomless model, changing one team’s match changes the kernel on a measure-zero set, making the paper’s pairwise manipulation condition vacuous. Giving a whole type positive-mass power to manipulate would change the problem.
So the strongest honest positive observation is that the rules’ score formulas suggest a possible continuum extension. It does not yield a valid continuous computational problem anchored in this paper’s named results.
The decisive problem is that none of the three named results is a computational result. Theorem 5.7 gives an axiomatic guarantee for the \(\lambda\)-rule; Theorem 4.4 gives one for the geometric-score rule; Theorem A.5 gives one for the top-cycle rule. Each says what a rule guarantees on every finite tournament, not how hard it is to compute an answer to a problem. The paper even says its proposed rules run in quadratic time. Its main open question—whether a rule can satisfy \(2\)-NM\(_1\)—is a question about existence of a rule, so a continuous version would be axiomatic rule design, outside ChoCo’s computational scope.
The strongest population-based alternative is to model many voters by a distribution over preference types and derive a majority tournament from that distribution. But the paper takes the tournament itself as input: its rules discard the voter profile, and the manipulations concern teams changing the outcome of their match. A continuous voter electorate therefore supplies context for the tournament without continuizing the object these results study.
One could instead make teams a continuum and represent match outcomes by a kernel, then ask for continuous versions of all three theorems. That fails as a computational mirror across Theorems 5.7, 4.4, and A.5 for the same reason: changing one pairwise match affects a null set in an atomless model, so the pairwise manipulation condition becomes unobservable if inputs identify kernels up to null sets. Giving positive mass to a type and letting it change matches on behalf of its members restores a nonzero effect, but changes a two-team manipulation into a bloc manipulation. And if the finite-step kernel is encoded by \(\tau\) team types, the rules’ score calculations reduce to sums over that encoding; the paper offers no hard computational task for the continuous formulation to expose.
The negative case is strongest as a claim about fit with this programme, not as proof that no interesting continuous tournament theory can be developed. A graphon-based axiomatic study might be worthwhile on its own. The material here gives little reason to count it as a continuous computational mirror of these results.
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.