We study a set of bilateral Nash bargaining problems between an upstream input supplier and several differentiated but competing retailers. If one bilateral bargain fails, the supplier can sell to the other retailers. We show that, in a disagreement, the other retailers' behavior has a dramatic impact on the supplier's outside options and, therefore, on input prices and welfare. We revisit the countervailing buyer power hypothesis and obtain results in stark contrast with previous findings, depending on the type of outside option. Our results apply, more generally, to the literature that incorporates negotiated input prices using bilateral Nash bargaining. (JEL C72, C78, D43, L13, L14, L81)