Equilibria in second-price auctions with private participation costs

We study equilibria in second-price auctions where bidders are independently and privately informed about both their values and participation costs, and where the joint distributions of these values and costs across bidders are not necessarily identical. We show that there always exists an equilibrium in this general setting with two-dimensional types of ex ante heterogeneous bidders. When bidders are ex ante homogeneous, there is a unique symmetric equilibrium, but asymmetric equilibria may also exist. We provide conditions under which the equilibrium is unique (not only among symmetric ones). We find that the marginal density of participation costs and the concentration of values matter for the uniqueness. The presence of private information on participation costs tends to reduce multiplicity of participation equilibria, although multiplicity still persists.