We consider "pure" matching problems, where being unmatched ("being single") is not a feasible choice or it is always the last choice for every agent. We show that there exists a proper subsolution of the stable solution that is implementable in Nash equilibria. Moreover, if the number of men M and the number of women W are less than or equal to 2, then any subsolution of the stable solution is implementable. However, if M=W≥3, there exists no implementable single-valued subsolution of the stable solution. All these results should be contrasted with the results in the recent literature on the matching problems with a single status.
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