| paper | d4-journals-geb-2017-0078 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | yes |
P→P discrete result → its continuous mirror
Answer: Deterministic polynomial-time solvable in \(\tau,L\), hence in \(m,\tau,L,1/\varepsilon\). · discrete side: polynomial · adjudication: proved_with_minor_gaps
full proof & adversarial review →
Theorem 26
statement extracted from the paper’s text layer
Given binary-rational type masses μ_t summing to one, a target aggregate power vector p, accuracy ε, and fixed η in (0,1/2), find rational type weights w_t ≥ 0 and a quota fraction q such that, writing W = ∑_t μ_t w_t, ηW ≤ q ≤ (1-η)W, the type-level limits Φ_t exist and ∥Φ-p∥_2 ≤ ε. Here Φ_t is the limit, as N tends to infinity through sizes with integral Nμ_t, of the total ordinary Shapley–Shubik index of the Nμ_t voters of type t in the replicated game with quota Nq. Treat this as a promise inverse-design problem: a type-homogeneous reasonable scheme achieves error at most ε/2.
A finite set of interchangeable voter types with rational population masses μ_t; nonnegative type weights w_t; and a quota fraction q. The objective is to make the limiting aggregate Shapley–Shubik power vector Φ approximate p in ℓ_2 distance. The ratio p_t/μ_t is the target power density for type t.
The mirror covers Theorem 26's approximate inverse-design guarantee. It does not cover Theorem 27's bounded-weight refinement or the paper's conjecture about exact inverse design.
The strongest positive mirror is Continuum Inverse Shapley Design. A type is a class of voters who are identical for this problem: they all vote yes or no on the same proposal, have the same eligibility and other voting attributes, and have the same desired Shapley power per member. The instance gives \(\tau\) such types, their population fractions \(\mu_t>0\) with \(\sum_t\mu_t=1\), and a target vector \(p\) of desired *aggregate* power shares, with \(\sum_t p_t=1\). A mass \(\mu_t\) represents the fraction of the population in type \(t\); its per-member target is \(p_t/\mu_t\).
The decision is to assign each type a nonnegative voting weight \(w_t\) and choose a quota. To define continuous Shapley power precisely, replicate the society to \(N\) voters, with exactly \(N\mu_t\) voters of each type, and use the type’s weight for each of its voters. For quota \(Nq\), let \(\phi_{i,N}\) be voter \(i\)’s ordinary Shapley–Shubik index in that finite weighted voting game, and let \(\Phi_{t,N}=\sum_{i\text{ of type }t}\phi_{i,N}\). Define \(\Phi_t\) as the limit of \(\Phi_{t,N}\) along population sizes for which every \(N\mu_t\) is integral, when that limit exists. A solution is a type-homogeneous scheme for which \(\|\Phi-p\|_2\le\varepsilon\). Require a reasonable quota: if \(W=\sum_t\mu_t w_t\), then \(q\in[\eta W,(1-\eta)W]\) for fixed \(\eta>0\). This uses the conventional Shapley normalization; the paper’s generalized indices differ by a constant factor.
The regime is a large membership organization with many voters but few standardized membership or desired-power classes: millions of members, perhaps tens of types. Assigning each type a weight and comparing its resulting power with a desired allocation preserves the paper’s central design question—construct a weighted voting rule to realize specified Shapley values. It aggregates the target by type because a continuum has no named voter whose individual index could be the output. The society is what becomes continuous; the vote and the weighted threshold rule remain the same.
My lead anchor is Theorem 26, proved in this paper. It gives algorithm IS the target vector \(a(1),\ldots,a(n)\), accuracy \(\varepsilon\), and confidence \(\delta\). If some monotone increasing, \(\eta\)-reasonable LTF \(f\) has Shapley distance from \(a\) at most \(1/\mathrm{poly}(n,2^{\mathrm{poly}(1/\varepsilon)})\), then with probability at least \(1-\delta\), IS outputs an LTF \(h\) with \(d_{\text{Shapley}}(f,h)\le\varepsilon\). Its running time is \(\mathrm{poly}(n,2^{\mathrm{poly}(1/\varepsilon)},\log(1/\delta))\). The theorem also states a probability-at-most-\(\delta\) bound on a bad output for any input target vector. The continuous mirror is the corresponding promise problem with \(n\) replaced by \(\tau\) and the individual Shapley vector replaced by the type-level limit vector \(\Phi\): given \((\mu_t)_t\), \(p\), \(\varepsilon\), and \(\delta\), output type weights and a reasonable quota that approximate \(p\), provided a type-homogeneous reasonable scheme can approximate it closely enough. The expected running time is polynomial in \(\tau\), the input bit length \(L\), \(2^{\mathrm{poly}(1/\varepsilon)}\), and \(\log(1/\delta)\).
I expect this mirror to be tractable, but the key continuous fact still needs to be established. In the regular regime of fixed type weights and an interior quota, the natural conjectured limit is \(\Phi_t=\mu_t w_t/W\). If so, a target \(p\) is realized by choosing \(w_t\) proportional to \(p_t/\mu_t\) and setting the quota at half the total weight. Proving the limit and controlling its convergence—especially with zero weights, ties, or boundary quotas—are substantive questions. They also give a finite-to-continuous bridge: a convergence rate would say when a continuum solution gives a useful approximation for a large finite electorate.
The case’s weakest point is that Shapley–Shubik power is defined through permutations of discrete voters. The replica limit is a reasoned co-modification of the model, but it may discard the finite-population effects that make the paper’s inverse problem difficult; the limiting allocation could reduce to weight shares. That makes this a credible mirror for designing influence in a large society, while leaving open whether its continuum complexity is as rich as the finite problem. It covers the paper’s inverse-design guarantee, not every finite-voter detail or the exact inverse problem raised in the conclusion.
Theorem 26 is a real computational result, so the decisive objection that there is nothing to mirror does not apply. The best case against this anchor is that its object is an \(n\)-coordinate reconstruction problem: it takes desired powers for named voters and returns an LTF on those voters. The proposed mirror replaces that vector with \(\tau\) aggregate targets and defines power through a limit of replicated finite games. That is a substantial change of model, and the limit may be the central mathematical result rather than a routine translation.
But that objection does not defeat the stronger version of the mirror. In a large membership organization with a few classes of interchangeable voters, type-homogeneous weights and aggregate type power are natural. Replicating rational type masses gives a precise finite-to-continuous bridge; symmetry recovers the per-member target from each aggregate. This is a population continuization, and the changes to the target and power measure are recognizable co-modifications. Neither the paper’s individual labels nor its \(\ell_2\) metric is essential to its inverse-design question.
One could challenge the replica limit as an added modeling choice, but the proponent has specified the sequence, fixed type weights, and an interior quota. Asking whether the limit exists and how quickly finite games approach it is a well-posed problem. If the limit makes inverse design simple, that is an answer to the mirror, not a reason to reject it.
So the negative case is weak. I can identify a genuine burden—prove the limiting power measure and a useful convergence rate—but not a fundamental obstruction. Theorem 26 survives as a plausible, worthwhile continuous mirror, and I cannot honestly support the universal claim that no scenario works.
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.