  In a many-to-one matching market, we analyze the matching game induced by a
stable rule when firms' choice function satisfy substitutability. We show that
any stable rule implements the individually rational correspondence in Nash
equilibrium when both sides of the market play strategically. Moreover, when
only workers play strategically and firms' choice functions satisfy the law of
aggregate demand, we show that the firm-optimal stable rule implements the
stable correspondence in Nash equilibrium.
