Concavity of order statistics in sample size: Implications for auction design

I show that the expected k th -highest order statistic from a monotone hazard rate distribution is concave in sample size. In independent private value auctions with convex marketing costs, this ensures that the seller’s objective is concave, yielding a finite optimum. • Highest order statistics are concave in sample size under monotone hazard rate. • Concavity ensures finite optimizers in auction problems with marketing costs. • Results extend to auctions with reserve prices under mild conditions.