| paper | d4-journals-geb-2019-0092 |
| authors | — |
| venue | ? |
| filed under | unclassified |
| judged by | gpt-6-luna / xhigh (triple__luna__xhigh__c6r1) |
| judge confidence | medium |
| authors would recognise it | yes |
other→P discrete result → its continuous mirror
Answer: polynomial-time solvable · discrete side: other · adjudication: proved
full proof & adversarial review →
Proposition 10
statement extracted from the paper’s text layer
Given a finite set of lawyer types (Theta), rational masses (mu_theta), common strict preferences over court-period contracts and the outside option, court-specific priority categories, and rational per-period capacity shares (q_{c,t}), compute a feasible mass allocation (x_{theta,c,t}) with no positive-mass blocking set under the extended time-specific choice rule.
Each type (theta) has mass (mu_theta), a common preference ranking, and a priority category at each court. Decision variables (x_{theta,c,t}) allocate type mass to court-period contracts, with unmatched mass included; capacities are (q_{c,t}). Courts process periods in the paper's precedence order, accept higher priority categories first up to capacity, and split mass proportionally within ties. A block is positive mass that weakly prefers proposed contracts and would be selected by the courts' extended choice rule when those offers are considered with the current allocation.
The paper ranks individual lawyers, so grouping them by priority category requires the category ties and mass-splitting rule to represent a real applicant regime; the proponent specifies that rule but does not establish how often the paper's strict individual priorities permit such grouping.
fatal: False
The mirror covers the stable-allocation output in Proposition 10. It does not establish continuous analogues of group strategy-proofness or lawyer-optimality, or cover the paper's Berlin-mechanism comparisons and flexible-capacity results.
There is no anchor-qualified computational result in this paper. Its numbered results concern mechanism properties—stability, strategy-proofness, efficiency, envy, and incentives—not whether a problem is in P, NP-hard, or in another complexity class. So I cannot honestly supply a continuous problem “per anchor” under your rules.
The closest result is Proposition 10, proved in the paper using cited results from Kominers and Sönmez (2016) and the paper’s Lemma 6. It says that the time-specific lawyer-offering stable mechanism is stable and group strategy-proof; if lawyers are weakly impatient, it also selects a lawyer-optimal stable allocation. That is a substantial algorithmic mechanism result, but it makes no complexity claim. Proposition 11 says the mechanism respects improvements, Proposition 12 says it eliminates justifiable envy, and Corollary 1 says it is weakly Pareto efficient. None supplies a complexity anchor either.
If I were allowed to offer an unanchored follow-on question, the strongest positive case would be a continuous version of the stable-allocation component of Proposition 10. Consider a large intake of applicants, grouped into types \(\theta\) that share a complete preference ranking over court–start-date pairs and the same priority category at each court. Let \(\mu_\theta\) be each type’s population share, and let \(q_{c,t}\) be court \(c\)’s capacity in period \(t\), expressed as a share of the cohort. The decision variable is mass \(x_{\theta,c,t}\) assigned from each type to each contract, with \(x_{\theta,\varnothing}\) left unassigned. Require \(\sum_{c,t}x_{\theta,c,t}+x_{\theta,\varnothing}=\mu_\theta\) and \(\sum_\theta x_{\theta,c,t}\le q_{c,t}\). Extend the paper’s time-specific choice rule to mass offers, allowing it to split mass at priority ties. The problem is: given rational masses, preferences, priorities, and capacities, find a feasible mass allocation with no positive mass block under that extended choice rule, or certify that none exists. A solution is the exact rational allocation \(x\).
A plausible regime is a large yearly intake in a court district where many applicants share the same court and timing preferences, exam-grade band, waiting-time band, and social-priority category, so the number of types \(\tau\) is much smaller than the number of applicants. This keeps the paper’s court–time contracts and capacity constraints while making the population the continuous object. I would tentatively expect this narrow problem to be tractable by aggregating a cumulative-offer process or using a linear formulation, but the paper establishes neither that algorithm nor its complexity. It covers the stable-allocation part of Proposition 10; it does not yet mirror the result’s group-strategy-proofness claim.
The authors could plausibly recognize that as a population version of their matching-with-waiting-times problem. The weakest point is that their model gives courts rankings over individual lawyers. If exact individual rank is part of a lawyer’s type, repeated types disappear. A mass mirror therefore needs finite priority categories, with ties resolved proportionally or by lottery. That is a real co-modification, and the continuous strategy-proofness question would need its own definition. The paper supports the *subject* of this candidate mirror; it does not provide a named computational result to anchor it.
The proponent has not named an eligible anchor. Proposition 10 is the closest result, but it establishes stability and group strategy-proofness of a mechanism, with lawyer-optimality under an extra preference assumption. It does not classify the complexity of a computational problem. Under the programme’s stated test, there is no numbered computational result here to mirror.
The proposed mass-allocation problem is a plausible follow-on, but it does not repair that gap: it asks a new question about computing stable allocations. Nor does its type definition yet preserve the paper’s model. The paper gives each lawyer a position in each court’s priority ranking. If that exact position is part of a lawyer’s type, lawyers with distinct ranks cannot be grouped into positive-mass types. Grouping by priority category instead requires a new court choice rule for ties and a precise definition of blocking when a category’s mass exceeds remaining capacity. The paper’s slot-specific choice functions are sensitive to how choices are made; its results do not specify that mass rule. Such a rule may be well worth studying, but it would need independent justification as the authors’ problem.
The candidate also sets aside group strategy-proofness, one of Proposition 10’s central claims. In an anonymous model given only by type masses, a single lawyer has measure zero, so a unilateral report cannot be evaluated from the aggregate input alone. Positive-mass deviations by a whole type are different from the individual and group deviations in the paper. Preserving the incentive result would therefore require further structure beyond the proposed stable-allocation question.
That is the strongest negative case this paper supports: it fails the anchor requirement, and the proposed candidate is an unanchored extension whose priority and incentive models still need work. The broader claim that *no scenario* admits a worthwhile continuous mirror is much weaker. Large applicant cohorts grouped by shared preferences and priority criteria are a sensible high-multiplicity regime, and the paper itself points to university admissions and teacher training as related applications. With capacities scaled to cohort size and a well-defined rule for priority ties, a continuous matching question could be recognized as theirs. I would reject the claimed mirror as an anchored result of this paper; I would not claim the paper rules out worthwhile mirrors in every scenario.
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.