Allocating with Priorities and Quotas: Algorithms, Complexity, and Dynamics

In many applications such as rationing medical care and supplies, university admissions, and the assignment of public housing, the decision of who receives an allocation can be justified by various normative criteria (ethical, financial, legal, etc.). Such settings have motivated the following priority-respecting allocation problem: several categories, each with a quota of interchangeable items, wish to allocate the items among a set of agents. Each category has a list of eligible agents and a priority ordering over these agents; agents may be eligible in multiple categories. The goal is to select a valid allocation: one that respects quotas, eligibility, and priorities, and ensures Pareto efficiency. We provide a complete algorithmic characterization of all valid allocations, exhibiting a bijection between these allocations and maximum-weight matchings under carefully chosen rank-based weights. This recovers and extends known results in this space and enables two wide-reaching extensions: 1. Selecting valid allocations that satisfy additional criteria: Via three examples -- inclusion/exclusion of some chosen agent; agent-side Pareto efficiency vs. welfare maximization; and fairness from the perspective of allocated vs. unallocated agents -- we show that finding priority-respecting allocations subject to some secondary constraint straddles a complexity knife-edge; in each example, one problem variant can be solved efficiently, while a closely related variant is NP-hard. 2. Efficiency-envy tradeoffs in dynamic allocation: In settings where allocations must be made to $T$ agents arriving sequentially via some stochastic process, we show that while insisting on zero priority violations leads to an $\Omega(T)$ loss in efficiency, one can design allocation policies ensuring that the sum of the efficiency loss and priority violations in hindsight is $O(1)$.