Reserve Matching with Thresholds

TL;DR

This paper develops a general threshold reserve framework for allocating scarce resources to unit-demand agents across independently prioritized categories, enabling both hard and soft reserves and overlapping categories. It introduces IMMAM, a three-stage mechanism that (i) finds a max-in-max allocation, (ii) enforces eligibility-driven max-in-max matching, and (iii) uses a DA-like refinement to respect category priorities, all within a universal domain. A key theoretical contribution is an impossibility result showing the tension between maximizing beneficiary assignments and total allocations, along with a demand-law condition under which all units can be allocated to beneficiaries. To address practicality, the authors propose Max-in-Max algorithms with improved efficiency (IMMAM-M and IMMAM-MB) that leverage path independence for computational gains and stability across multiple institutions. The framework and mechanisms hold broad policy relevance for vaccine allocation and similar resource-distribution problems where flexible categorization, overlapping eligibility, and priority-ordering are important.

Abstract

We develop a general framework for reserve systems that allocate scarce resources such as vaccines to unit-demand agents under prioritization and eligibility constraints, along with a computationally efficient mechanism. Reserve systems allocate scarce resources --such as vaccines, medical units, school seats, or government positions-- to essential groups by creating categories with prioritized beneficiaries. Prior work typically assumed a common baseline priority ordering and featured either hard or soft reserves. The threshold reserve model we introduce supports independent priority orderings, mixtures of hard and soft reserves, and overlapping categories, thereby capturing both beneficiary designations and eligibility constraints while offering policymakers greater flexibility. Our Iterative Max-in-Max Assignment Mechanism (IMMAM) satisfies all desirable properties in this domain: it respects priorities within categories, maximizes resource utilization, and then lexicographically maximizes beneficiary assignments. IMMAM is path independent and therefore well-behaved in settings with multiple institutions making simultaneous allocation decisions. We leverage path independence to obtain comparative statics and to significantly improve the mechanism's computational efficiency. We outline applications of our framework in the context of vaccine allocation.

Figures (4)

• Figure 1: Illustration of \ref{['impossibility example']}. Null matching, maximum in beneficiary assignment matching, and maximum in resource allocation matching are given respectively. Being beneficiary is indicated with a red edge, and eligibility alone is indicated with a black edge.
• Figure 2: A patient-initiated even alternating chain.
• Figure 3: The even alternating chain from \ref{['fig: an even alternating chain']} after the process of augmentation.
• Figure 4: A patient-initiated even alternating chain with positive potential. Red edges represent beneficiary assignment.

Theorems & Definitions (53)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Example 4.1
• Theorem 4.2
• Definition 4.3
• Theorem 4.4: Demand law
• proof