A bargaining set for roommate problems

Since stable matchings may not exist, we adopt a weaker notion of stability for solving the roommate problem: the bargaining set. Klijn and MassÛ (2003) show that the bargaining set coincides with the set of weakly stable and weakly e¢ cient matchings in the marriage problem. First, we show that a weakly stable matching always exists in the roommate problem. However, weak stability is not su¢ cient for a matching to be in the bargaining set. Second, we prove that the bargaining set is always non-empty. Finally, as Klijn and MassÛ (2003) get for the marriage problem, we show that the bargaining set coincides with the set of weakly stable and weakly e¢ cient matchings in the roommate problem.