A modified deferred acceptance algorithm for many-to-one matching markets with externalities among firms

We consider a many-to-one matching market with externalities among firms where each firm's preferences satisfy substitutability, increasing choice and no external effect by unchosen workers, which are defined by  Bando (2012). We first illustrate that a sequential version of the deferred acceptance (DA) algorithm with worker-proposing may not find a worker-optimal quasi stable matching. Then, we provide a modified DA algorithm in which (i) each worker simultaneously proposes to his most preferred firm that has not rejected him and (ii) each firm chooses its acceptable workers from the cumulative set of workers who have ever proposed to it, assuming that the other workers proposing to its rival firms are hired. We show that this algorithm finds a worker-optimal quasi stable matching. We also show that this algorithm can be generalized into a fixed point algorithm.