Stable Mechanisms in Controlled School Choice

We consider school choice when the priority of a student for a school is conditional on who the other students are. Students have one of several types and the type distribution in the school matters. As a special class of our priority rules, we introduce the adjusted scoring rules and show that several rules in previous studies, such as those in which the priorities are based on type quotas and/or reserves, are special cases of them. By using them, we can flexibly cope with trade-offs that are particular to controlled school choice problems. We introduce a mechanism which is stable even if the priority order is not completed between different type students. Moreover, we introduce another mechanism that is stable and strategy-proof if all priorities are restricted to be weak orders and is also a student optimal stable if all priorities are further restricted to be strict orders