Controlled choice over public schools attempts giving parents selection options while maintaining diversity of di erent student types. In practice, diversity constraints are often enforced by setting hard upper bounds and hard lower bounds for each student type. We demonstrate that, with hard bounds, there might not exist assignments that satisfy standard fairness and non-wastefulness properties; and only constrained non-wasteful assignments which are fair for same type students can be guaranteed to exist.We introduce the student exchange algorithm with hard bounds (SEAHB) that nds aPareto optimal assignment among such assignments. To achieve fair (across all types)and non-wasteful assignments, we propose the control constraints to be interpreted assoft bounds{exible limits that regulate school priorities dynamically. In this setting,the deferred acceptance algorithm with soft bounds (DAASB) nds an assignment thatis Pareto optimal among fair assignments while eliciting true preferences. Thus, wedemonstrate DAASB has clear bene ts over SEAHB.
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