| paper | d4-journals-mss-2011-0028 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | unclear |
Theorem 7 and Theorem 4 give structural characterizations and certificate-length bounds, while Theorem 17 classifies a fixed-size family. None states an algorithm, running-time bound, hardness result, or solver-based formulation, so the paper supplies no qualifying computational result to anchor a continuous-population question.
fails bit a — no named computational result to mirror
The proposed angles concern the paper's weightedness and rough-weightedness characterizations, certificate-length bounds, and small-game results; none yields a computational mirror.
I can’t make a qualifying anchor-based case for a continuous mirror here: the paper has no numbered result that classifies a computational problem as tractable, hard, or parameterized in the sense this task requires.
The closest result is Theorem 7, proved here: an \(n\)-player game is roughly weighted exactly when it has no potent certificate of non-weightedness up to length \((n+1)2^{\frac12 n\log_2 n}\). That gives a finite characterization, but the paper does not establish a complexity bound or analyze an algorithm for deciding rough weightedness. Theorem 4, proved here, similarly bounds the certificate length for weightedness. Theorem 17, proved here, says every six-player constant-sum game is roughly weighted. These are structural results, not computational-complexity anchors. Theorem 1 is a cited characterization of weighted games, not a complexity result.
One could invent a mass-based problem about whether a monotone coalition rule over finitely many agent types admits rough weights. But the paper neither studies that model nor computes such a problem; presenting it as a mirror would create a new research question without a qualifying anchor. Per the task’s criteria, there is no lead mirror problem to state.
The absence of a computational result is the strongest objection. Theorem 7 characterizes rough weightedness by excluding potent trading certificates up to a finite length; Theorem 4 does something similar for weightedness. Neither gives a running time, an input encoding, or a complexity classification. A bound on certificate length is not itself an algorithmic result: the paper does not say how to search the certificates efficiently, and the answer could depend greatly on whether a game is encoded by its full winning-coalition table or by a shorter description. Theorem 17’s six-player conclusion is likewise structural.
The best continuous version of Theorem 7 would define a nonatomic population and a monotone winning rule on its measurable coalitions, then ask whether an integrable weight function and quota represent that rule, perhaps using a mass-based analogue of potent trades. That is a coherent question, but it asks for an axiomatic characterization of continuous voting games. To make it computational, one would also have to supply a finite input representation and pose a decision or optimization problem. The paper supplies neither. Preserving its arbitrary named-player games makes that representation problem especially consequential: collapsing players into types retains the game only when players of a type are interchangeable for every coalition interaction. Imposing that symmetry could yield a sensible new model, but it would be an added restriction, not the paper’s result carried over to a continuous society.
The bounds on \(f(n)\) and \(g(n)\) do not provide another anchor. They concern the maximum shortest certificate among finite games with exactly \(n\) players. A continuous analogue could measure trades by mass, or by some norm or encoding length, but choosing and studying that quantity would define a new invariant. Even a good choice would not turn the paper’s structural bounds into a complexity result. Theorem 17 is still less transferable: a game with six players is not a nonatomic population with six types, and the latter’s variable masses change the object being studied.
This does not show that continuous voting games, or computational questions about them, are uninteresting. It does show why this paper gives no strong reason to pursue them as its continuous mirror. Every plausible repair adds a computational question or a model the paper does not study; the named results themselves do not supply a computational anchor.
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.