A new dynamic mechanism to the marriage problem with a variant

We know from Gale and Shapley (1962) that every Two-Sided Matching Game has a stable solution. It is also well-known that the number of stable matchings increases with the number of agents on both sides. In this paper, we propose two mechanisms, one of which is a variant of the other, to the marriage problem. Our original mechanism implements the full set of stable matchings for any preference profile. On the other hand, the variant mechanism parititons the domain of preference profiles into two; for one set, it implements the full set of stable matchings like the original mechanism and for the other, it ends up with a proper subset of the set of stable matchings. Besides, for some profiles with multi stability, it gives one of the optimal stable matchings. Namely, the second mechanism coincides either with the original mechanism or it is an improvement for one side; and in some profiles, the algortihm induces Gale and Shapley's algorithm for some profiles. Thus, it is a "middle" mechanism.