Research

We study many-to-many matching with exact agent demands, exact slot quotas, and spatial composition constraints represented by a partition of agents into proximity clusters. At most one agent from each cluster may be assigned to a given slot. Although this local constraint is matroidal and induces a substitutable fixed-priority choice rule, exact demands and quotas couple completion globally across slots. Consequently, Deferred Acceptance and myopic Serial Dictatorship may terminate without a complete allocation even when one exists. In the quota-two benchmark, Deferred Acceptance has a uniform complete-feasibility guarantee if and only if there are exactly two clusters or the spatial constraint is nonbinding because every cluster is a singleton. We characterize completable partial allocations through cluster-based Hall inequalities and implement the resulting completion test using maximum flow. Building on this characterization, we introduce a completion-preserving priority mechanism that, under heterogeneous demands and capacities, produces a feasible and Pareto-efficient allocation satisfying replica-level no justified-envy under the layered priority order. It also satisfies replica-level pairwise stability when every slot has quota two. Simulations calibrated with administrative data on pharmacy duty schedules in Türkiye show that the mechanism clears the market in every reported scenario and improves model-implied rank outcomes under the generated preference profiles. These conclusions are robust to alternative cluster constructions, geographic specifications, and preference calibrations.

We study the smallest worst rank attainable in fractional allocation under no-envy and ordinal efficiency. The criterion bounds the rank of each object assigned with positive quantity. Pareto efficiency alone does not increase the minimum feasible rank. Under equal claims and arbitrary capacities, we characterize fixed-rank no-envy feasibility by an exact obstruction-and-recovery procedure and derive a direct rank-two system. Under unit claims and unit capacities, whenever no-envy is feasible at rank two, it is jointly feasible with ordinal efficiency at that rank. In this domain with n agents and n objects, the sharp worst-case rank costs of no-envy relative to feasibility and of efficiency conditional on no-envy are max{0, n−2} and max{0, n−3}, respectively, while their instance-wise sum never exceeds max{0, n−2}. For each n ≥ 3, Probabilistic Serial can exceed each of the optimal ordinal-efficiency, no-envy, and joint frontiers by n−2 ranks and by a factor of n/2; one family attains all six bounds. A heterogeneous-claim formulation based on feasible truncation explains why equal claims yield linear no-envy restrictions, whereas unequal claims can generate nonconvexity. All rank-loss bounds concern positive support, not quantity-weighted welfare.