Group Robust Stability in Matching Markets

We propose a group robust stability notion which requires robustness against a combined manipulation, first misreporting of preferences and then rematching, by any group of students in a school choice type of matching markets. Our first result shows that there is no group robustly stable mechanism even under acyclic priority structures (Ergin (2002)). Then, we define a weak version of group robust stability, called weak group robust stability. Our main theorem shows that there is a weakly group robustly stable mechanism if and only if the priority structure is acyclic, and in that case it coincides with the student-optimal stable mechanism. Hence this result generalizes the main theorem of Kojima (2010). Then as a real-world practice, we add uncertainty regarding an acceptance of an appeal of students to rematch after the announced matching. In that setting, we show that under some conditions along with the acyclicity, the student-optimal stable mechanism is group robustly stable under uncertainty.