Probabilistic Rationing with Categorized Priorities: Processing Reserves Fairly and Efficiently

The strongest mirror is a high-multiplicity version of the paper’s rationing problem, not merely its already-fractional outcome. The paper allocates shares to named people; the continuous version allocates population mass among administratively meaningful applicant/patient types.

My lead question is Typed Maximum Rationing Eating\(_\infty\), mirroring Theorem 10 (proved in this paper). Strictly speaking, the paper has no numbered theorem stated in complexity-class language; Theorem 10 is the named result to anchor on because it defines MRE and establishes its maximum-size and fairness/efficiency properties, while the surrounding construction invokes polynomial-time Hopcroft–Karp and polynomial-time VER for linear convex constraints.

An instance has a finite set \(T\) of applicant types, rational masses \(\mu_t\) summing to one, categories \(C\) with rational capacities \(Q_c\), and, for each category, an eligibility relation and a priority order over types. A type is complete: it records the policy-relevant attributes used by every category—say occupation, qualification, region, medical-risk score band, veteran status, and category-specific eligibility. Thus all members of a type are interchangeable for this allocation problem. The unknown is a nonnegative mass allocation \(x_{t,c}\), where \(x_{t,c}\) is the fraction of the whole population of type \(t\) receiving a unit from category \(c\). It must satisfy
\[ \sum_c x_{t,c}\leq \mu_t,\qquad \sum_t x_{t,c}\leq Q_c, \]
and eligibility.

Let \(S^\star\) be the maximum total allocated mass among feasible allocations. The task is to compute the allocation returned by the type-level MRE process: categories simultaneously eat their highest-priority still-available eligible types, but a proposed increment is permitted only if it can still be extended to an allocation of total size \(S^\star\). Equivalently, it is Algorithm 2 with people replaced by type masses and the maximum-size face of the transportation polytope used as VER’s feasibility constraint. A solution is the rational matrix \(x\).

The desirable guarantees are exactly the typed forms of Theorem 10: eligibility compliance, priority respect (a category cannot allocate to a lower-priority type while a higher-priority type remains unsatiated), non-wastefulness, maximum allocated mass, and category sd-efficiency. This is not a softened version of the question. It preserves the central conflict between simultaneous priority processing, maximum utilization, and fairness among categories; it simply removes irrelevant personal labels.

The natural regime is a large national immigration, school-reserve, or healthcare programme. For example, an annual skilled-immigration intake may contain millions of applicants but only thousands or tens of thousands of relevant type vectors. Its categories are visa streams or reserves, and their quotas scale with the applicant cohort. Priority is often already assigned by score bands and statutory classifications; any within-band lottery is operationally real rather than an artificial mathematical concession. In that regime “allocate 0.7% of applicants of this profile to this reserve” is closer to the policy decision than “allocate to applicant 446,873.” The number of people can grow dramatically while the attribute space, categories, and policy tiers remain stable or grow much more slowly.

I expect Typed MRE\(_\infty\) to be Class A when the support \(T\), eligibility relation, and priority orders are explicit. Maximum size is a capacitated bipartite-flow computation on \(T\times C\), and the MRE/VER feasibility region remains a rational linear polytope. The interesting computational boundary is not the explicit model, but compactly represented types: for example, types described by many attributes and eligibility/priority rules specified by Boolean or scoring predicates. Then the allocation LP may have exponentially many latent types, and the real question becomes separation or pricing over feasible attribute profiles—the same structural shift the continuization programme is meant to expose.

A second, somewhat simpler anchor is Typed Rationing Eating\(_\infty\), mirroring Theorem 8 (also proved here). Its input is the same, but there is no maximum-size constraint. Every category eats its most-preferred eligible unsatiated type at equal rate until it reaches capacity or no eligible mass remains. The requested output is the resulting \(x\). The theorem’s conclusions transfer naturally: typed eligibility, priority respect, non-wastefulness, anonymity with respect to mass within a type, neutrality of category names, category sd-envy-freeness, and category sd-efficiency. For explicit types this should be plainly tractable—essentially probabilistic serial with divisible type supplies—and is likely the most direct continuous computational analogue of the paper’s linear-time RE claim.

The MRE question is stronger than RE as a programme anchor because it retains the paper’s substantive efficiency constraint. It also retains a useful negative boundary. The paper’s Proposition 11 (proved here) survives at positive mass: take two types \(A,B\), each of mass \(1/2\), two categories each of capacity \(1/2\), with the first ranking \(A\succ B\) and the second eligible only for \(A\). Maximum size forces the second category to take \(A\) and the first to take \(B\), violating category sd-envy-freeness. Thus the MRE/RE tradeoff is not a finite-agent artifact; it is present in the continuous population model.

The main weakness is real: the paper assumes strict priority rankings over named agents, whereas a useful high-multiplicity model needs priority tiers over types, with equal treatment or a lottery within a tier. If every person has a legally decisive idiosyncratic priority score, then each is effectively a different type and there is no compression. I do not think that defeats the mirror, because large rationing systems often do use coarse statutory classes, score bands, and lotteries; but it confines the claim to that regime. A further research task would be to formalize convergence from increasingly fine strict tie-breakings to the type-mass process, and to identify which tie-breaking assumptions preserve the typed RE/MRE outcome.