Revenue optimal auction for single-minded buyers

We study the problem of characterizing revenue optimal auctions for single-minded buyers. We identify revenue optimal auctions with a simple structure, if the conditional distribution of any buyer´s valuation is nondecreasing, in the hazard rates ordering of probability distributions, as a function of the bundle the buyer is interested in. The revenue optimal auction is given by the solution of a maximum weight independent set problem. We provide a novel graphical construction of the weights and highlight important properties of the resulting auction.