| paper | d4-journals-geb-2018-0071 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | yes |
The named results establish analytical limits and a fairness rule, but none has the required computational character. Section 6 formulates the inverse Shapley-weight optimization and reports heuristic computation, yet that formulation is not a named computational result of the paper. The proposed mirror is recognizable, but it cannot supply the missing anchor.
fails bit a — no named computational result to mirror
The proposed question covers the inverse-weight design problem in Section 6, equation (18), using constituency masses \(\alpha_i\); it does not turn Proposition 1 or Corollary 1 into computational results.
The strongest honest case is a narrow one: this paper makes a very plausible high-multiplicity fair-representation problem, but it contains no named computational result of the kind your anchor rule requires. It proves asymptotic probability claims and discusses an inverse optimization problem; it does not prove that a problem is in P, NP-hard, FPT, or similar. So I can’t honestly supply a compliant anchor-by-anchor case.
The closest analytical foothold is Proposition 1, proved here: as the constituency-level shocks grow relative to the idiosyncratic components, the ratio of two delegates’ probabilities of being pivotal approaches the ratio of their Shapley values in the weighted voting game. Corollary 1, also proved here, applies this to large constituencies: if individual ideal points have the stated idiosyncratic and constituency components, then weights whose Shapley values are proportional to constituency populations give approximately equal individual influence, for any relative quota \(q\in[0.5,1)\). These are the right results to motivate a computational question, but neither is a complexity result.
The best follow-up mirror I can formulate is the paper’s inverse-weight problem from Section 6. Given \(m\) constituencies with rational population shares \(\alpha_i\ge 0\), \(\sum_i\alpha_i=1\), a rational quota \(q\in[0.5,1)\), and an accuracy \(\varepsilon>0\), find normalized rational weights \(w_i\ge0\), \(\sum_iw_i=1\), minimizing
\[
\sum_{i=1}^{m}\left|\phi_i([q;w_1,\ldots,w_m])-\alpha_i\right|.
\]
Equivalently, for a supplied threshold \(r\), decide whether weights exist with this distance at most \(r\), and return such weights. Here each constituency is a mass block of share \(\alpha_i\); voters within a block share its relevant preference-distribution parameters, while the paper’s affiliated-constituency regime supplies the asymptotic link between a delegate’s Shapley value and voter influence. The action is to choose assembly weights, and the objective is to match delegate pivotality to population mass. The paper itself writes down this inverse problem in equation (18) and computes heuristic examples, so the authors would likely recognize it as their design problem in a high-multiplicity regime.
My provisional expectation is that any hardness would be continuum-specific: the remaining combinatorics concern finding a weighted voting game with a desired Shapley vector as \(m\) grows, not the number of citizens within each constituency. The paper gives no complexity classification for this optimization, so that is a research expectation, not an anchored result.
The weakest point is also the reason this is not a full positive case: Section 6’s inverse problem is a finite-assembly design problem fed by population shares. The paper does not analyze its complexity, and its named results do not establish that continuizing the population changes that complexity. It is a credible question the authors might accept, but it cannot satisfy the requested named-anchor standard.
The strongest case against starts with the anchor requirement: this paper states no complexity result. Proposition 1 and Corollary 1 are probability limits and fairness claims, not algorithmic results. Their best population-continuous versions remain analytical. With an atomless preference distribution inside each constituency, a single voter has zero mass, so the paper’s literal event that a voter determines the outcome disappears. Preserve its per-capita meaning by taking the normalized finite-population limit, and the result is essentially the one already proved: with identical continuous constituency shocks, delegate orderings are equally likely, so pivotal probabilities are Shapley values; fair representation then asks for \(\phi(v)\propto\alpha\), where \(\alpha\) is the vector of constituency shares. That reformulation supplies no computational result to mirror.
The inverse-weight problem is the proponent’s strongest candidate, and it deserves a fair hearing. But it is the optimization in equation (18) of the paper itself: choose weights whose Shapley vector is close to the normalized population vector. Replacing \(n_i/n\) with rational shares \(\alpha_i\) leaves that objective unchanged. The paper already discusses heuristic and exact approaches and cites work on the inverse power-index problem. This may be a worthwhile computational project, but it is an algorithmic agenda the paper explicitly raises, not a computational result obtained by continuizing its population.
That defeats the proposed anchors under the requested standard; it does not establish the universal claim that no worthwhile continuous mirror exists. A federation with large constituencies is a credible high-multiplicity setting, and the authors would likely recognize the inverse-weight question. The honest negative case is therefore narrow: this paper offers no named computational result to anchor a compliant mirror, while its best computational candidate is already formulated in the paper. I would not claim more than that.
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.