| paper | d4-journals-geb-2025-0053 |
| 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: \((m\tau^4L)^{O(1)}\) bit operations, using an LP with \(O(m\tau^2)\) variables and \(O(\tau^4+m\tau^2)\) constraints. · discrete side: other · adjudication: proved
full proof & adversarial review →
Proposition 3
Given finite male and female type sets with positive rational masses (mu_i) and (nu_j), and a rational matching flow (pi_{ij}\ge 0) with those marginals, specify for each type pair its potential-match prices and income, and for each positive-flow current match its observed bundles and commitment status. Does there exist a type-level split of private consumption, personalized public-good prices, edge weights, and potentials satisfying the type-indexed feasibility constraints of Proposition 3, so that the observed matching flow is rationalizable as a core allocation with transfers? Dummy types may represent singles.
A distribution over complete male and female household profiles, with rational masses and an observed matching flow. Each profile preserves the relevant preferences, consumption, commitment, and budget information; type-pair data give counterfactual prices and income. The decision variables are type-level consumption splits, personalized prices, edge weights, and potentials, and the objective is feasibility of Proposition 3's constraints.
The exact test may depend only on which complete types have positive mass, leaving population proportions and matching-flow shares without a role; the proponent notes profile proliferation but does not address this support-only point.
fatal: False
The mirror covers Theorem 2's with-transfer rationalizability criterion and Proposition 3's LP formulation. It leaves Proposition 2's path-consistency MILP, the identification results, and the simulation analysis aside.
The strongest honest lead is Proposition 3 in Appendix B.2, proved in the paper. It gives an exact linear feasibility formulation for path monotonicity. For each observed couple, the witness splits its private consumption between the partners; for each potential pair, it splits public-good prices into positive personalized prices; and it defines an edge weight \(a_{mw}\) as the value of the partners’ current consumption at the potential match’s prices minus that match’s income. The witness also supplies a scalar \(z_m\) for each man, with \(v_m z_m-v_{\sigma(w)}z_{\sigma(w)}\le a_{mw}\) for every potential pair, where \(v_m\) indicates whether his current couple is committed. Proposition 3 says this system is feasible exactly when the data satisfy path monotonicity. Theorem 2, also proved here, gives that condition its target: a data set is rationalizable as a core allocation with transfers if and only if it satisfies path monotonicity. The paper computes with these conditions; its Appendix B.2 formulation is therefore a suitable computational anchor. ([published article, Appendix B.2](https://repository.essex.ac.uk/39852/1/1-s2.0-S0899825624001805-main.pdf))
My proposed mirror is Path-Monotonicity Rationalizability for Type Distributions. An instance gives finite sets of male and female types, rational masses \(\mu_i\) and \(\nu_j\), and an observed type-to-type matching flow \(\pi_{ij}\) with those marginals. For every type pair it gives the prices and income for a potential match; for observed matches it gives the household consumption bundle and commitment status. Include dummy types for singles where needed. A type describes everything the model uses about its members: their common, unknown utility function, current match and commitment information, observed consumption, and budget data against every opposite-side type.
The question is whether there are type-level private-consumption splits, personalized public-good prices, edge weights, and potentials satisfying the type-indexed version of Proposition 3’s constraints. A feasible witness is a certificate that the observed matching flow is rationalizable as a stable core-with-transfers allocation; infeasibility means it is not. This is a feasibility problem, with no optimization objective, just like the underlying rationalizability test. The given object made continuous is the population measure and its matching flow. Household bundles remain the within-household data the original model tests.
A plausible regime is a very large regional marriage market with many people but a modest number \(\tau_M,\tau_W\) of repeated preference, budget, consumption, and commitment profiles. If potential-match data are given explicitly by type pair, the proposed test is an LP with size polynomial in \(\tau_M,\tau_W\), the numbers of goods, and the binary input length \(L\). I expect this mirror to be Class A. In an atomless population, any finite path involving positive-mass types can recruit a sufficiently small amount of each, so the type-level path constraints do not require selecting particular individuals. This is a proposed type-level extension of the paper’s LP result, not a runtime theorem the authors prove. The paper presents no named hardness result for these tests.
The case is narrow: it mirrors the with-transfers rationalizability result and its LP formulation. It does not cover Proposition 2’s path-consistency MILP, the paper’s identification results, or its simulation analysis. The main weakness is also substantial. The authors emphasize individual-level preference heterogeneity, while this mirror assumes that members of a type share a utility function and the same relevant data. If encoding every preference and counterfactual budget profile makes \(\tau\) approach the population size, the computational gain disappears. The mirror is most plausible in a repeated-profile market; whether the authors would accept that regime as a useful form of their problem is the point an opposing case should press.
The proposed anchor is real: Theorem 2 characterizes rationalizability with transfers, and Proposition 3 gives an exact linear feasibility system for path monotonicity. I would not dismiss it for lack of a computational result. The difficulty is that an exact type must preserve almost the whole household record: its observed consumption and commitment status, its latent preference class, and its prices and incomes against every opposite-side type. The paper allows individual preference heterogeneity and makes counterfactual price variation important to its identification results. In the data regimes it studies, those records are likely to be nearly all distinct; grouping households by broad demographics would discard distinctions used by the stability constraints. The paper itself discusses match-specific prices and incomes as inputs to the test and uses price variation to identify committed couples’ allocations. ([Freer and Surana, “Marital stability with committed couples”](https://repository.essex.ac.uk/39852/1/1-s2.0-S0899825624001805-main.pdf))
A stronger version could posit a large market with a small number of repeated *complete* household profiles and prices determined by type pairs. That is coherent, and a type-level LP could preserve the theorem. It is also a narrow regime: repeated demographics are not enough; the full observed and counterfactual records must repeat. If they do, the theorem’s sharp feasibility test has a further limitation as a population model. Its blocking condition asks whether any finite permissible remarriage path violates an inequality. In an atomless population, every positive-mass type supplies enough distinct people for any finite path, however small its mass. Once the set of positive-mass types is fixed, the test therefore depends on support, not on the mass values or the matching-flow proportions.
That leaves a support-based LP over repeated records, which is a legitimate mirror but gives the population weights no role in the result. Making mass matter would require a different question, such as how much population can participate in a blocking rearrangement or how much data must be adjusted for stability. Those may be worthwhile extensions, but they change the paper’s existential blocking criterion and are not covered by Theorem 2 or Proposition 3.
This is the strongest negative case I can make, and it is not airtight. A stylized market with repeated complete profiles and type-pair prices is plausible enough that the mirror may survive. The objection is that its compelling version is considerably narrower than the proposed “large marriage market” suggests, and its exact stability test mostly compresses the paper’s existing LP rather than asking a mass-sensitive population question.
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.