| paper | d4-journals-mss-2022-0008 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | yes |
The typed continuous least-core question is concrete, and its scalar version has the exact value \(\varepsilon^*=1-q/A\), where \(A=\sum_t\mu_ta_t\); that simple answer does not itself undermine the mirror. But the only anchor raised is Problem P2, not an eligible numbered computational result under the stated criterion. Bit (a) therefore fails, without accepting the opponent’s broader claim that no worthwhile mirror exists.
fails bit a — no named computational result to mirror
The proposed mirror covers least-core payoff selection for scalar weighted threshold games with finitely many weight types; it leaves the paper’s vector-weighted extension and finite-player network formulation aside.
The strict anchor check is the weak point: the supplied paper has no numbered Theorem, Lemma, Corollary, or Proposition asserting a computational result. Its main contribution is Problem P2, a numbered LP formulation, with its size and pseudo-polynomial-time claims stated in prose. The introduction also cites Elkind et al. [6] for coNP-hardness of checking \(\varepsilon\)-core nonemptiness, but that is neither a numbered result of this paper nor the same problem as computing a least-core payoff. So P2 is the strongest available lead, but it qualifies only if a numbered formulation counts as an anchor.
P2 (Section 3; formulated here and used in the paper’s experiments) computes a payoff vector in the least core of a weighted voting game \([q;w_1,\ldots,w_n]\). The authors build a directed acyclic graph with \(O(nW^+)\) vertices and arcs, where \(W^+=\sum_i w_i\); its source-to-winning-terminal paths correspond to winning coalitions. They combine the dual shortest-path constraints with the payoff variables \(x_i\) and \(\varepsilon\), obtaining an LP of size \(O(nW^+)\). Solving it gives a least-core payoff in pseudo-polynomial time in the integer weights. The paper develops this formulation and then solves it computationally in Section 4.
The strongest mirror I can make from P2 is Typed Continuous Weighted Voting Least Core. Its input is a finite set \(T\) of voting-weight types, rational masses \(\mu_t>0\) summing to \(1\), a nonnegative weight \(a_t\) for each type, and a rational quota \(q\) with \(0<q\leq\sum_t\mu_ta_t\). A coalition is any measurable subpopulation described by masses \(z_t\in[0,\mu_t]\), and wins when \(\sum_t a_tz_t\geq q\). Find \(\varepsilon\) and type-level per-capita payoffs \(p_t\geq0\) minimizing \(\varepsilon\), subject to
\[
\sum_t\mu_tp_t=1
\quad\text{and}\quad
\sum_tz_tp_t\geq1-\varepsilon
\quad\text{for every winning }z.
\]
A solution is an optimal least-core value and a payoff density attaining it. Restricting payoffs to be equal within each type is natural here: players of the same type have identical effects on every coalition’s status, and averaging payoffs within a type preserves feasibility.
A plausible regime is a large shareholder or member body with many participants but few voting-weight classes: the unit-vote members, several standardized weight bands, and perhaps a few institutional classes. The type is the member’s voting weight; mass is that class’s population share. The intended scale is \(n\gg\tau\), with quota and total voting weight growing proportionally as the body grows. The underlying question remains the paper’s: how should a normalized unit of transferable payoff be allocated so that the worst winning coalition is treated as well as possible?
This mirror looks like Class A. For proposed \((\varepsilon,p)\), finding the winning coalition with minimum payoff is a fractional-knapsack problem over \(\tau\) types: choose mass from types in increasing payoff-per-vote order, up to each type’s available mass, until the quota is met. That gives a polynomial-time separation procedure, and hence a polynomial-time algorithm in \(\tau\) and the binary input length via LP optimization. The continuous question is recognizable to the authors because it preserves the weighted voting game, winning-coalition definition, least-core objective, and payoff constraints; it replaces each finite type class by divisible mass.
The soft spot is the move from indivisible players to divisible subpopulations. It enlarges the set of possible coalitions, so the continuous least-core value need not equal the value in any particular finite game. Establishing a precise replication limit or approximation bound is a further question, as is whether the paper’s vector-weighted extension admits an efficient pricing method when the number of weight dimensions grows. Thus P2 supports a concrete positive lead, but under the literal numbered-theorem anchor rule this is an unanchored candidate mirror, not a fully compliant anchored case.
P2 is not a numbered complexity theorem. It is a linear program; the paper states its \(O(nW^+)\) size and pseudo-polynomial consequence in prose. The cited coNP-hardness result is earlier work, and it concerns checking \(\varepsilon\)-core nonemptiness. So the proponent has not supplied the kind of named computational result the anchor requirement calls for.
There is also a stronger objection to the proposed scalar mirror. Let \(A=\sum_t\mu_t a_t\) be total voting weight, \(Q\) the quota, and \(\theta=Q/A\). For any normalized payoff density \(p\), the coalition taking the same fraction \(\theta\) of every type has weight \(Q\) and receives payoff \(\theta\). Thus no allocation can make every winning coalition receive more than \(\theta\). But setting \(p_t=a_t/A\) gives every winning coalition at least \(\theta\). The continuous least-core value is therefore exactly \(\varepsilon^*=1-\theta\), regardless of the type distribution. The fractional coalition available in the limit removes the subset-selection structure that motivates the paper’s shortest-path formulation.
That is a genuine degeneracy of this mirror, though it does not make the question ill-posed: it gives the question a simple answer. And that limits the negative case. The paper also sketches vector weighted games. In a union of thresholds, a coalition can win through different routes, so the scalar argument need not settle the least-core allocation. That could yield a worthwhile continuous question, but it is not the anchor the proponent raised.
So the strongest honest case against is narrow: the raised anchor does not meet the strict theorem criterion, and its natural scalar continuous version collapses to a type-independent formula. I cannot support the universal claim that no worthwhile mirror exists; the paper’s vector extension leaves a plausible route around that objection.
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.