School choice with controlled choice constraints: Hard bounds versus soft bounds

Controlled choice over public schools attempts giving parents selection options while maintaining diversity of different 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 that are fair for same type students can be guaranteed to exist. We introduce the student exchange algorithm that finds a constrained efficient assignment among such assignments. To achieve fair (across all types) and non-wasteful assignments, we propose control constraints to be interpreted as soft bounds-flexible limits that regulate school priorities dynamically. In this setting, (i) the student-proposing deferred acceptance algorithm produces an assignment that Pareto dominates all other fair assignments while eliciting true preferences and (ii) the school-proposing deferred acceptance algorithm finds an assignment that minimizes violations of controlled choice constraints among fair assignments.