Strategy-proof matching with regional minimum quotas

This paper considers the matching problem with regional quotas, in particular, regional minimum quotas. Although such quotas are relevant in many real-world settings, there is a lack of strategy-proof mechanisms that consider regional minimum quotas. We first show that without any restrictions on the region structure, finding a feasible matching that satisfies all quotas is NP-complete. Then, assuming that regions have a hierarchical structure (in this case, a tree), and maximum quotas are imposed only on individual schools, we show that checking the existence of a feasible matching can be done in a linear time in the number of regions. Furthermore, we develop strategy-proof matching mechanisms based on the Deferred Acceptance mechanism (DA), which we call Multi-Stage DA with Regional minimum Quotas (MSDA-RQ) and Round-robin Selection DA with Regional minimum Quotas (RSDA-RQ). When minimum quotas are imposed, fairness and nonwastefulness are incompatible. We prove that RSDA-RQ is fair but wasteful, while MSDA-RQ is nonwasteful but not fair. Moreover, we compare our mechanisms with artificial cap mechanisms whose individual maximum quotas are adjusted beforehand so that all regional quotas can be automatically satisfied. Our simulation reveals that our mechanisms substantially outperform artificial cap mechanisms in terms of student welfare. Furthermore, it illustrates the trade-off between our mechanisms.