Stability against robust deviations in the roommate problem

We propose a new solution concept in the roommate problem, based on “robustness” of deviations (i.e., blocking coalitions). call deviation from matching robust up to depth k, if none deviators gets worse off than at original after any sequence most k subsequent deviations. say that is stable against (for short, SaRD) no it k. As smaller imposes stronger requirement for be SaRD, we investigate existence SaRD with minimal constructively demonstrate always exists k=3 and establish sufficient conditions k=1 2.