Controlled choice over public schools attempts giving options to parents while maintaining diversity, often enforced by setting feasibility constraints with hard upper and lower bounds for each student type. We demonstrate that there might not exist assignments that satisfy standard fairness and non-wastefulness properties; whereas constrained non-wasteful assignments which are fair for same type students always exist.We introduce a "controlled" version of the deferred acceptance algorithm with an improvement stage (CDAAI) that finds a Pareto optimal assignment among such assignments. To achieve fair (across all types) and non-wasteful assignments, we propose the control constraints to be interpreted as soft bounds-flexible limits that regulate school priorities. In this setting, a modified version of the deferred acceptance algorithm (DAASB) finds an assignment that is Pareto optimal among fair assignments while eliciting true preferences. CDAAI and DAASB provide two alternative practical solutions depending on the interpretation of the control constraints.JEL C78, D61, D78, I20. * An earlier version (Ehlers, 2010) of this paper emerged from a joint project of the first author with Atila Abdulkadiroglu. We are grateful for his extensive comments and contribution to that paper. Ehlers acknowledges financial support from the SSHRC (Canada).

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Roth and Rothblum [Roth, A. E., U. G. Rothblum. 1999. Truncation strategies in matching markets—In search of advice for participants. Econometrica 67 21–43] showed that for matching markets using the deferred acceptance algorithm a physician with symmetric (incomplete) information possibly gains only by truncating her true ranking. We show that in symmetric information environments this result is identical for all priority mechanisms and all linear programming mechanisms introduced in British entry-level medical markets and in public school choice in some American cities.