Efficiency and stability under substitutable priorities with ties

Many assignment mechanisms appeal to a priority structure to determine how over-subscribed indivisible goods are assigned to unit-demand individuals. We study substitutable priorities with ties which not only nest important classes of priorities and preferences studied in the literature, but also allow us to formalize plausible priority structures not captured in previous literature. Efficiency is typically in conflict with respecting priorities (i.e., stability), and therefore the natural welfare objective is constrained efficiency. A generalization of the deferred acceptance process yields a stable assignment, but this outcome is not necessarily constrained efficient. We identify an easily verifiable sufficient condition for a stable assignment to be constrained efficient, which then leads to an algorithm to compute a constrained efficient assignment. Finally we illustrate practical applications of our framework and algorithm, including a widely studied matching problem with distributional constraints.