Stability and Nash implementation in matching markets with couples

We consider two-sided matching markets with couples. First, we extend a result by Klaus and Klijn (2005, Theorem 3.3) and show that for any weakly responsive couples market there always exists a "double stable" matching, i.e., a matching that is stable for the couples market and for any associated singles market. Second, we show that for weakly responsive couples markets the associated stable correspondence is (Maskin) monotonic and Nash implementable. In contrast, the correspondence that assigns all double stable matchings is neither monotonic nor Nash implementable.