| paper | d4-conf-ijcai-2016-0009 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | yes |
Theorem 4.3 gives a probabilistic identity between sampled-game power and power at the generating probabilities, but it does not state an algorithm, computation guarantee, or complexity claim. The proposed quota optimizer is a plausible extension of the paper’s explicit quota-control motivation, yet it is not a continuous analogue of a computational result proved here, so bit (a) fails.
fails bit a — no named computational result to mirror
The proposed optimizer draws on the quota-control and power-equalization motivation; it does not turn the uniform-case fluctuation results or the exponential-case limit theorem into computational results.
The paper has no named computational-complexity result of the kind required for an anchor. Its numbered results are probabilistic or structural statements about Shapley values; the intractability of computing the Shapley value appears only as prior work in the related-work section. So I can’t honestly supply a problem per qualifying anchor.
The closest support for a positive mirror is Theorem 4.3, proved in this paper. It says that for a \(\rho\)-exponential vote distribution, with \(0<\rho<1/2\), sufficiently many voters—\(m\ge C\rho^{-n}(1-2\rho)^{-2}\log n\)—make the sampled weight vector super-increasing with high probability. For quotas in the stated subset-sum gaps and sufficiently far from their boundaries, the Shapley values of the sampled game agree, for every party, with those of the deterministic game whose weights are the probabilities \(p\), with probability at least \(1-O(1/n)\). For all but finitely many normalized quotas, that agreement probability tends to \(1\) as \(m\to\infty\). The paper proves this result; it is not a complexity classification. Its displayed statement also switches between \(T\) and \(\tau\), and appears to mix normalized quotas with unnormalized sampled weights in its margin condition.
A plausible continuous model is: the population consists of a unit mass of voters, each of type \(i\) meaning “votes for party \(i\),” with type mass \(\mu_i=p_i\). The \(n\) parties’ weights are those masses, and a coalition wins at quota \(T\) when its total mass is at least \(T\). The high-multiplicity regime is \(m\gg n\): many voters, but only \(n\) vote types. The parties remain the players in the Shapley calculation; it is the voter population that becomes continuous.
One concrete computational question is Continuous Quota–Power Design. Given rational masses \(p_1,\ldots,p_n\) summing to \(1\) and an allowed quota interval \([a,b]\subset(0,1]\), choose a rational quota \(T\in[a,b]\) minimizing the power disparity \(\max_i\phi_i(p;T)-\min_i\phi_i(p;T)\), and return an optimal quota and its exact Shapley–Shubik vector. This matches the paper’s stated interest in quota choices that equalize power. Theorem 4.3 makes the exponential case a credible limit regime: there, the finite electorate’s power profile eventually agrees with this deterministic mass model for almost every quota. For exact computation, I would expect hardness to transfer: normalizing the weights and quota of a discrete weighted voting game gives a rational mass instance, so the hard combinatorics remain in the number of parties, not voter multiplicity. That expectation is an inference, not a result proved in this paper.
The mirror is narrow. In the uniform case, replacing sampled weights by exact equal masses removes the noise that drives much of the paper’s quota-cycle and power-disparity analysis. Theorem 4.3 supports a meaningful continuous limit for the exponential case, but it does not establish an algorithmic result. That is the weakest point—and why this is a credible extension of the paper’s subject, rather than a continuous mirror of one of its computational-complexity findings.
The strongest negative point is decisive for the programme’s brief: this paper proves no named computational result. Lemma 3.2 is an identity; Theorems 3.6–3.8 are probabilistic guarantees; Theorem 4.3 is a convergence result. The hardness of computing Shapley values appears only as prior work. So there is no complexity classification here to continuize.
The natural population model is sensible: take many anonymous voters, with type \(i\) meaning “votes for party \(i\),” and mass \(\mu_i=p_i\). I would not object that this lacks a high-multiplicity regime. Nor would I treat Theorem 4.3’s link to the deterministic mass game as a collision: it is a genuine discrete-to-continuous result. But it is a probabilistic limit theorem, not a computational question or algorithmic result.
The proposed Continuous Quota–Power Design does not supply the missing anchor. The paper discusses authorities choosing quotas, but its results analyze how power behaves under random weight generation; it neither poses an optimization problem over quotas nor studies its computational complexity. The proposed problem instead takes a deterministic rational mass vector and adds an optimization task. That might be worth studying independently, but it is a new problem motivated by the paper’s topic, not a continuous mirror of one of its results.
The negative case is strongest on that distinction. For fixed masses, the stochastic fluctuations that drive the uniform-case analysis vanish; retaining them would require adding a separate distribution over continuum populations. The exponential-case limit is already identified by Theorem 4.3, but that still supplies no computational result for the programme to chart. I would reject this as a worthwhile *mirror* under the programme’s computational brief, while not claiming that quota-power optimization is inherently unworthy as a separate project.
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.