Dynamic reserves in matching markets

Theorem 4 supplies a plausible continuous-population extension, but it is a structural comparative-static theorem rather than a named computational result. The proposed mirror also requires new choices for priority ties, mass splitting, and representations of transfer functions. Because condition (a) fails, the paper cannot provide a qualifying ChoCo anchor.

The objection that survived

The mass-valued mechanism requires new priority-boundary and transfer-function representations, so its computational formulation is not inherited from the paper.

What the mirror covers

The proposed mirror covers Theorem 4's comparative statics under dynamic reserves; it does not provide computational counterparts for the stability, strategy-proofness, improvement, or representation theorems.

Open questions for a prover

The case FOR (proponent)

The strongest honest conclusion is that this paper does not provide a qualifying ChoCo anchor. Its continuous-population mirror is quite plausible, but the paper contains no named computational result of the required kind.

The numbered results proved in the paper are all structural or mechanism-design theorems. Theorem 1 proves that the cumulative offer mechanism (COM) is stable under dynamic-reserves choice functions. Theorem 2 proves weak group strategy-proofness. Theorem 3 proves respect for unambiguous priority improvements. Theorem 4 proves that making one school’s monotone capacity-transfer scheme more flexible makes the COM outcome weakly Pareto dominate the original outcome. Theorem 5 proves that every slot-specific-priorities choice rule can be generated by a dynamic-reserves choice rule. Propositions 1–4 establish completion, irrelevance of rejected contracts, substitutability, and the law of aggregate demand. None asserts membership in \(\mathrm{P}\), NP-hardness, parameterized tractability, approximation, or another computational classification. The cited results from Hatfield and Kominers and others do not become anchors merely because this paper invokes them.

The best substantive mirror would be:

\[ \textsc{Continuous-Dynamic-Reserve-COM}. \]

An instance has finite schools \(S\), privilege categories \(T\), and a finite set of complete student types \(\Theta\). A type \(\theta\) specifies a preference order \(P^\theta\) over school–privilege contracts and the outside option, the privileges \(E^\theta\subseteq T\) that it may claim, and its priority-relevant score or rank \(\rho^\theta_s\) at every school. The society is a rational distribution \(\mu\) over \(\Theta\), where \(\mu_\theta\) is the fraction of applicants of type \(\theta\). Each school has capacity per unit population, a precedence sequence, target reserve shares, and monotone capacity-transfer functions defined on vacancy masses rather than integer vacancy counts.

The decision variable is an aggregate assignment \(x_{\theta,s,t}\), recording the mass of type \(\theta\) assigned to school \(s\) through privilege \(t\). Each infinitesimal student still receives one integral contract; only the aggregate mass is divisible. The continuous COM is the type-level cumulative-offer process: mass proposes contracts in preference order, schools retain the highest-priority offered mass subject to dynamic reserves, and rejected mass continues proposing. Given two transfer schemes \(q\) and \(\widetilde q\), with \(\widetilde q\) everywhere at least as flexible as \(q\), the problem is to compute both COM assignments and determine whether the \(\widetilde q\)-assignment Pareto dominates the \(q\)-assignment, interpreted typewise through an ordinally improving coupling of the two mass assignments.

This is a credible mirror of Theorem 4. The regime could be national or statewide admissions: millions of applicants, a modest number of schools and privilege categories, and many repeated combinations of preference template, eligibility, and priority band. Capacities and reserves naturally scale as fractions of the applicant population. The type count can therefore be much smaller than the number of students. It is not merely outcome-space fractionalization: schools and contracts remain discrete, while only the population is continuized.

I would expect this problem to be Class A under a natural representation in which transfer functions are rational piecewise-linear rules. A type-level COM can plausibly process each type through a bounded list of contracts, while school choices are weighted priority selections over finitely many types. The interesting follow-up questions would be exact bit-complexity bounds, convergence from finite replicated markets, sensitivity to reserve-transfer functions, and whether stability, group strategy-proofness, and Pareto monotonicity survive arbitrary mass splitting at priority boundaries.

The serious weakness is that this is not a computational theorem from the paper. Extending the paper’s integer-valued capacity functions and strict individual priority orders to mass-valued capacities and repeated priority types requires new modelling choices and proofs. Moreover, the resulting algorithm may simply be a continuous restatement of deferred acceptance rather than an instance where discrete computational hardness dissolves. Thus I would present this as a natural research problem inspired by Theorem 4, not as a compliant theorem-anchored continuization of the paper.

The case AGAINST (opponent, writing after the proponent)

The decisive objection is formal: this paper has no qualifying computational anchor. Theorem 1 proves stability, Theorem 2 strategy-proofness, Theorem 3 respect for improvements, Theorem 4 a Pareto comparative-static, and Theorem 5 a representation result. None states a complexity bound, an algorithmic classification, an approximation guarantee, or hardness result. The cumulative-offer process is an algorithm in the informal sense, but the paper proves no computational theorem about it. Thus any continuous-computation result would be a new project inspired by the paper, not a continuization of one of its named computational results.

Theorem 4 is the strongest possible rescue, but its proposed mirror does not repair that defect. Give it the best formulation: let a type specify preferences, eligibility, and every school’s priority-relevant information; let \(\mu\) be a rational type distribution; scale capacities and reserves per capita; run a mass-valued cumulative-offer process; and compare the outcomes under two transfer schemes through a type-preserving Pareto coupling. This is coherent only after choosing what happens to the paper’s strict priority orders.

There is a basic dichotomy. If strict priority is preserved faithfully, priority position is part of the agent’s complete description. In a generic instance, almost every student has a distinct priority vector, so \(\tau\) is essentially \(n\). Then \(\mu\) is merely a normalized encoding of the original finite market, not a high-multiplicity society. If priority positions are compressed into bands so that many students share a type, the mechanism has changed: the paper has strict individual priorities, whereas the mirror needs ties, a mass tie-breaking rule, or fractional splitting at priority boundaries. The resulting allocation kernel and Pareto relation are new modelling choices, not consequences of Theorem 4.

The same problem affects the other structural theorems. In an atomless population, one student’s misreport or priority improvement has measure zero and cannot change the aggregate outcome, so individual strategy-proofness and respect for improvements become vacuous. Allowing a positive-mass deviation produces bloc strategy-proofness, not the paper’s individual or finite-coalition concept. Likewise, stability must be reformulated in terms of measurable blocking masses. Those may be worthwhile new notions, but they are not preserved automatically by replacing counts with masses.

There is also no well-defined computational input for the proposed problem as stated. The paper permits arbitrary monotone capacity-transfer functions \(q_s^k:\mathbb Z_+^{k-1}\to\mathbb Z_+\), without specifying their encoding or evaluation cost. A continuous version needs functions on \(\mathbb R_+^{k-1}\), together with a representation such as rational piecewise-linear formulas. Choosing that representation creates a new algorithmic model; leaving it implicit makes complexity undefined. Once it is fixed, computing the type-level COM is essentially a weighted version of the already finite deferred-acceptance process, which makes at most \(|X|\) proposals. Theorem 4’s dominance conclusion is already proved for every finite instance; normalizing multiplicities does not expose a new computational obstruction.

Theorem 5 offers no escape: its constructive proof creates one artificial type for every contract, precisely eliminating rather than exploiting high multiplicity.

So the strongest negative verdict is that this is not a qualifying ChoCo anchor. A coherent continuous admissions problem could certainly be built from it, especially with finite priority bands and rational piecewise-linear transfer rules, but that would be a new high-multiplicity matching programme. The honest weakness in the negative case is that such a new programme may still be worthwhile; what fails is the claim that this paper itself supplies a computational result worth continuizing.

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.