Computational complexity of $k$-stable matchings

Abstract: We study deviations by a group of agents in the three main types of matching markets: the house allocation, the marriage, and the roommates models. For a given instance, we call a matching $k$-stable if no other matching exists that is more beneficial to at least $k$ out of the $n$ agents. The concept generalizes the recently studied majority stability. We prove that whereas the verification of $k$-stability for a given matching is polynomial-time solvable in all three models, the complexity of deciding whether a $k$-stable matching exists depends on $\frac{k}{n}$ and is characteristic to each model.