Finding a stable matching under type-specific minimum quotas

In matching problems with minimum and maximum type-specific quotas, there may not exist a stable (i.e., fair and non-wasteful) assignment (Ehlers et al., 2014). This paper investigates the structure of schools' priority rankings which guarantees stability. First, we show that there always exists a fair and non-wasteful assignment if for each type of students, schools have common priority rankings over a certain number of bottom students. Next, we show that the pairwise version of this condition characterizes the maximal domain of two schools' priority rankings over same type students to guarantee the existence of stable assignments. To prove the existence theorem, we propose a new mechanism Deferred Acceptance with Precedence Lists (DAPL), which is feasible, non-wasteful, strictly PL-fair and group strategy-proof for any priority rankings. Strict PL-fairness is weaker than fairness, but DAPL satisfies fairness under our sufficient condition. We also show that there is no strategy-proof mechanism that Pareto dominates DAPL whenever the outcome of DAPL is Pareto dominated by a stable assignment.