| paper | d4-journals-jet-2017-0072 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | high |
| authors would recognise it | unclear |
The numbered results give a constructive way to enumerate the core and its extreme allocations, so I count computational character as present despite the lack of a runtime analysis. But the proponent supplied no concrete continuous question for either anchor, which the anchor rule requires. The opponent’s atomless example also shows that preserving individual charge functions can remove the extreme-point object; the type-level alternative is left as a different, unformulated model.
fails bit b-changes — the continuum limit changes the question
No continuous mirror is specified. The closest candidate would cover the core and its extreme allocations after restricting charges to be uniform within each location type; the paper’s solution identities are not formulated over a population distribution.
I can’t make a compliant anchored case for a continuous mirror: the paper gives no named computational-complexity result. No numbered result establishes that a problem is in P, NP-hard, or in another complexity class, and the paper gives no formulation whose computational performance it analyzes.
The closest result is Theorem 1 (proved in this paper): for every cost matrix \(c\), the core is the convex hull of the reduced marginal cost vectors \(y^{r\pi}\) over all permutations \(\pi\). Proposition 3 (also proved here) says those vectors are exactly the extreme core allocations. These are useful constructive characterizations, but enumerating permutations does not establish tractability or hardness. Treating either as a qualifying computational anchor would overstate what the paper proves.
The proponent is right that neither Proposition 3 nor Theorem 1 supplies a complexity result of the kind this programme is looking for. Proposition 3 identifies the extreme core allocations; Theorem 1 says their convex hull is the core. The paper gives a procedure for generating them, but no running-time analysis or complexity classification. Its links to the Bird, folk, and cycle-complete solutions are cost-sharing results, not computational ones. So the closest anchors are constructive characterizations, and treating them as complexity theorems would overstate the paper.
The strongest mirror would use many households at a few service locations, with shared, fixed costs for connecting locations. That is a plausible high-multiplicity setting. Yet the fixed-edge-cost model makes the paper’s central object fragile under continuization. In a one-location example, let the population be atomless, let any positive-measure coalition cost \(K>0\) to connect, and let the grand coalition also cost \(K\). If allocations are measurable per-person charges \(f\), the core consists of nonnegative functions with \(\int_0^1 f=K\). It has no extreme points: any such \(f\) can be perturbed up and down on two positive-measure subsets while preserving nonnegativity and the total. The object characterized by Proposition 3 and Theorem 1—the full set of extreme core allocations—has disappeared.
One can instead require everyone of a type to receive the same per-person charge. That yields a finite-dimensional core over the \(\tau\) occupied locations. The network costs still depend on which locations participate, not on their masses; the equal-treatment convention supplies the mass dependence in the allocation constraints. This is a plausible new continuous cooperative game, but it is an additional modeling choice, and the paper establishes no computational result about it. If the full individual allocation object is retained, the extreme points vanish in the atomless case; if allocation is restricted to type-level charges, the mirror becomes a new finite-support problem.
That is the strongest case against a worthwhile *mirror of this paper’s computational results*. It is not a proof that no useful continuous cost-sharing question exists: the atomless core or the type-level core could warrant study in its own right. The negative case is weakest at that boundary, where a researcher might reasonably regard the new core problem as the contribution rather than as a mirror of the paper’s results.
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.