In a matching problem between students and schools, a mechanism is said to be
robustly stable if it is stable, strategy-proof, and immune to a
combined manipulation, where a student first misreports her preferences and then blocks the matching that is produced by the
mechanism. We find that even when school
priorities are publicly known and only students can behave
strategically, there is a priority structure for which no robustly stable mechanism exists. Our main
result shows that there exists a
robustly stable mechanism if and only if the priority structure of
schools is acyclic (Ergin, 2002), and in that case, the
student-optimal stable mechanism is the unique robustly stable
mechanism.