Second-Price Auctions with Different Participation Costs

This paper studies equilibria of second price auctions in independent private value envi- ronments with different participation costs. Two types of equilibria are identified: monotonic equilibria in which a bidder with a lower participation cost results in a lower cutoff for sub- mitting a bid, and non-monotonic equilibria in which a lower participation cost results in a higher cutoff. We show that there always exists a monotonic equilibrium, and further, that the monotonic equilibrium is unique for either concave distribution functions or strictly convex distribution functions with non-increasing reverse hazard rates. There exist non- monotonic equilibria when the distribution functions are strictly convex and the difference of the participation costs is sufficiently small. We also provide comparative static analysis and study the limiting properties of equilibria when the difference in bidders’ participation costs approaches zero.