This paper provides sufficient conditions under which the negotiated wage in unionized oligopolistic industries with centralized negotiations is independent of a number of product market features (such as the number of firms, the degree of product substitutability, or the type of market competition). This wage independence property is shown to hold in a broad class of industry specifications widely-used in the literature, both when negotiations are conducted over wages alone (Right-to-Manage), and over wages and employment (Efficient Bargains). In particular, it holds for the Dixit-Stiglitz preference-for-diversity model, the symmetric linear demands-linear one factor (labor) technology model, and the constant elasticity demand and cost functions model. In these models the negotiated wage is independent of the bargaining institution, too. Unions are then better-off as the market becomes more competitive since aggregate employment increases.