Group robust stability in matching markets

We introduce the notion of group robust stability which requires robustness against a combined manipulation, first misreporting preferences and then rematching, by any group of students in the school choice type of matching markets. Our first result shows that there is no group robustly stable mechanism even under acyclic priority structures. Next, we define a weak version of group robust stability, called weak group robust stability. Our main theorem, then, proves that there is a weakly group robustly stable mechanism if and only if the priority structure of schools is acyclic, and in that case, it coincides with the student-optimal stable mechanism.