In this paper, we propose two mechanisms to the two-sided matching games. The first mechanism implements the full set of stable matchings for any preference profile in subgame perfect Nash equilibrium. The second mechanism, which is derived from the first one, implements a matryoshka partition for every preference profile. That is it implements a subset of the full set of stable matchings such that it includes the (wo)men-optimal stable matching and does not include any stable matching that is better than some matching in the set in view of (wo)men. In other words, the second mechanism provides all possible partially biased compact subsets of set of stable matchings in the full domain of preference profiles.