  Who gains and who loses from a manipulable school-choice mechanism? Studying
the outcomes of sincere and sophisticated students under the manipulable Boston
Mechanism as compared with the strategy-proof Deferred Acceptance, we provide
robust "anything-goes" theorems for large random markets with coarse priority
structures. I.e., there are many sincere and sophisticated students who prefer
the Boston Mechanism to Deferred Acceptance, and vice versa. Some populations
may even benefit from being sincere (if also perceived as such). Our findings
reconcile qualitative differences between previous theory and known empirical
results. We conclude by studying market forces that can influence the choice
between these mechanisms.
