| paper | d4-conf-atal-2008-0179 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | unclear |
Theorem 1 concerns how an approximation changes when a player’s weight changes; it does not state that a solution can be computed, give an approximation guarantee, or establish complexity. The inverse method’s \(O(n^2)\) per-iteration bound is unnumbered, and the exact \(\#\mathrm{P}\)-completeness claim is background. The proposed cohort extension would also need a normalization such as \(n\phi_i\), changing the finite-player quantity at issue.
fails bit a — no named computational result to mirror
The paper’s inverse-design procedure has an unnumbered \(O(n^2)\) per-iteration bound and experimental convergence analysis; its exact Shapley \(\#\mathrm{P}\)-completeness claim is attributed to prior work. Neither supplies a named computational result for a continuous mirror.
The strongest honest finding is that this paper has no anchor that meets your rule. The sole numbered result is Theorem 1, proved here. It says that for two weighted voting games with the same quota and mean weight and approximately the same variance, Algorithm 1’s *approximate* Shapley value for a player does not decrease when that player’s weight increases, does not increase when it decreases, and stays the same when it is unchanged. That is a monotonicity property of the approximation, not a named complexity result about a problem.
The paper’s \(O(n^2)\)-per-iteration claim for its inverse method appears in the algorithm analysis, without a numbered theorem. The \(\#\mathrm P\)-completeness claim for exact Shapley-value computation is background attributed to reference [2], not a result established here. Neither can serve as an anchor under the stated rules.
So I can’t responsibly supply a continuous problem as a mirror of a qualifying named result. A type-level inverse Shapley design problem may be a plausible research direction, but presenting it as this paper’s anchored mirror would overstate the evidence.
Theorem 1 does not provide a computational result to continuize. It proves monotonicity for Algorithm 1’s Gaussian approximation to a player’s Shapley value, under assumptions on the games’ means and variances. Its conclusion is about that approximation, not the complexity of computing Shapley values or solving the inverse problem. The exact \(\#\mathrm P\)-completeness claim is background attributed to prior work. The paper’s own \(O(n^2)\) bound is per iteration; it gives no bound on the iterations needed to reach a specified error, and convergence is assessed experimentally. So the proponent is right that there is no qualifying named complexity result to anchor.
The best alternative mirror would group shareholders or voters into types with identical target powers and other relevant parameters, then ask for voting weights that achieve a target distribution of power. That is a coherent high-multiplicity scenario. But in a continuum, each individual has zero mass, so their unscaled Shapley value vanishes. Recovering a nonzero quantity requires a normalized limit, such as \(n\phi_i\). That asks about a limiting power density rather than the paper’s finite vector of individual Shapley values; the finite-player pivotality that drives its computation is precisely what the limit may erase. This is a substantial reason not to count that model as a mirror of the paper’s result.
It is not a proof that the normalized inverse problem would be uninteresting. A cohort model could still make it worthwhile, and the authors might recognize it as their problem. So the negative case is strong against an *anchored* mirror of this paper, but weak against the universal claim that no scenario could yield a worthwhile continuous mirror. The proponent raises no surviving named-result anchor; the broader inverse-design question remains plausible.
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.