We consider a single buyer with a combinatorial preference that would like to purchase related products and services from different vendors, where each vendor supplies exactly one product. We study the general case where subsets of products can be substitutes as well as complementary and analyze the game that is induced on the vendors, where a vendor’s strategy is the price that he asks for his product. This model generalizes both Bertrand competition (where vendors are perfect substitutes) and Nash bargaining (where they are perfect complements), and captures a wide variety of scenarios that can appear in complex crowd sourcing or in automatic pricing of related products.
We study the equilibria of such games and show that a pure efficient equilibrium always exists. In the case of submodular buyer preferences we fully characterize the set of pure Nash equilibria, essentially showing uniqueness. For the even more restricted “substitutes” buyer preferences we also prove uniqueness over mixed equilibria. Finally we begin the exploration of natural generalizations of our setting such as when services have costs, when there are multiple buyers or uncertainty about the buyer’s valuation, and when a single vendor supplies multiple products.