Optimal auctions with positive network externalities

We consider the problem of designing auctions in social networks for goods that exhibit single-parameter submodular network externalities in which a bidder's value for an outcome is a fixed private type times a known submodular function of the allocation of his friends. Externalities pose many issues that are hard to address with traditional techniques; our work shows how to resolve these issues in a specific setting of particular interest. We operate in a Bayesian environment and so assume private values are drawn according to known distributions. We prove that the optimal auction is APX-hard. Thus we instead design auctions whose revenue approximates that of the optimal auction. Our main result considers step-function externalities in which a bidder's value for an outcome is either zero, or equal to his private type if at least one friend has the good. For these settings, we provide a e/e+1-approximation. We also give a $0.25$-approximation auction for general single-parameter submodular network externalities, and discuss optimizing over a class of simple pricing strategies.