Agreement toward stability in matching markets

For firms with responsive preferences, we prove that the set of stable matchings unanimously preferred by workers to any firm quasi-stable matching has a lattice structure. This follows from a generalization of the Decomposition Lemma. The result does not hold when firms have q-substitutable preferences. Nevertheless, we show that the set of stable matchings unanimously preferred by workers to a firm quasi-stable matching contains an element which is unanimously least preferred by workers, and most preferred by firms. When a firm quasi-stable matching is fed into our extension of the Deferred Acceptance algorithm (where firms propose), the existence of this matching guarantees the success of the algorithm.