Reaching the core of the marriage market through a non-revelation matching mechanism

Each woman indicates a set of acceptable men. Following a line, each man selects his favorite woman among those available to him and to whom he is acceptable. Roth imposes that women select the whole set of men and Alcalde et al. restrict the choices of women to singleton. We relax these restrictions by allowing women to select any set of men and investigate how this change affects the strategic structure of the games induced by the old procedures.