| paper | d4-journals-jair-2021-0046 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | yes |
The proposed continuous Shapley–Shubik bound is a recognizable question about a continuum of small voters, but the anchored theorems assert structural bounds on power-index values, not computational results. The recurrence lemmas and finite exhaustive search support calculations, but the paper states no named algorithmic or complexity guarantee for either anchor. Bit (a) therefore fails.
fails bit a — no named computational result to mirror
The proposed extension covers the aggregate Shapley–Shubik ratio bound of Theorem 3.1. It leaves the Banzhaf and Deegan–Packel results, individual power results, false-name results, conjecture, and experimental enumeration without a computational mirror.
The strongest honest positive case is limited: the paper describes a sensible high-multiplicity regime—many interchangeable unit-weight voters alongside a few large voting blocs—and explicitly discusses oceanic games with a continuum of small players. That supports the plausibility of a continuous-population question.
But the paper has no named computational result to anchor that case. Theorem 3.1, proved here, says the aggregate Shapley–Shubik power of big players is at most twice their proportional weight, with the bound tight. Theorem 4.1, also proved here, says the corresponding ratio is unbounded for both absolute and normalized Banzhaf power. These are structural bounds on power indices, not complexity classifications or algorithmic results. Theorem 5.1 and Theorem 6.1 likewise give power bounds. The NP-hardness and PP results in the introduction are attributed to prior work, not presented as numbered results of this paper.
A continuous analogue of the Shapley–Shubik bound might be worth studying for a finite set of large blocs plus a continuum of infinitesimal voters. But that would be a new computational question, not a mirror anchored in a named computational result here. The paper’s recurrences and exhaustive experiment do not supply a runtime or complexity theorem either.
The strongest negative case is that the paper has no result of the kind ChoCo is set up to mirror. Theorem 3.1 bounds a power-index ratio; Theorems 4.1, 5.1, and 6.1 give further bounds on power and false-name manipulation. None classifies a computational problem or gives an algorithm. The cited hardness results concern earlier work, and the exhaustive search supports a conjecture rather than making a complexity claim. So these theorems do not provide computational anchors for the programme.
The best candidate for a continuous question is Theorem 3.1 in a society with a few large blocs and a continuum of small voters. That is a sensible high-multiplicity regime; the paper’s discussion of oceanic games supports its plausibility. But the proposed mirror asks for a continuous Shapley–Shubik value or a bound on it. A continuum version needs a convention for random order, or a specified limit of finite replications. Under either choice, the question remains an analytic power-bound question, not a computational one. The theorem’s bound may make the continuous answer straightforward, but that is not my objection: even a surprising answer would not supply the missing computational question.
The Banzhaf result offers no better anchor. In its unbounded example, the big player’s weight and the quota’s offset from half the small-player population both vanish as proportions, while the decisive pivotality depends on fluctuations at a finer scale. A population distribution alone loses that scale. A Gaussian or other second-order model could retain it, but then the question becomes an asymptotic probability analysis, still not an algorithmic complexity result. For Deegan–Packel, a continuum treatment also needs a rule for measuring minimal winning coalitions; there is no uniform distribution over an atomless set of such coalitions supplied by the population model. The false-name theorem leaves a similar mismatch: keeping the strategic big players finite gives a value bound, while making their aliases continuous changes the strategy space rather than continuizing the society.
This is a strong case against treating the paper as a ChoCo mirror, but it does not prove the universal claim that no worthwhile continuous question could arise from it. One could formulate a new computational problem about computing power indices from rational type masses or optimizing a split. That would be a follow-up motivated by the paper, not a mirror of one of its computational results. The remaining uncertainty is whether the authors would recognize such a new problem as theirs.
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.