Nash implementation in a many-to-one matching market
In a many-to-one matching market with substitutable preferences, we analyze the game induced by a stable matching rule. We show that any stable matching rule implements the individually rational correspondence in Nash equilibrium when both sides of the market play strategically. We also show that when only workers play strategically in Nash equilibrium and when firms' preferences satisfy the law of aggregate demand, any stable matching rule implements the stable correspondence in Nash equilibrium.