Position auctions with endogenous supply

We consider a multi-object private values setting with quantity externalities. In this setting, a value to a bidder from an object may depend on the total number of objects sold. For example, the likelihood a customer will respond to an advertisement is higher the fewer other advertisements are shown; a spectrum license is more valuable the fewer licenses are being allocated. We raise and solve the problem of finding revenue maximizing and efficiently allocating auctions in such a setting. We show that both optimal and efficient auctions have the property that the quantity of objects sold depends non-trivially on the whole profile of players’ valuations. That is, the quantity to sell is determined endogenously, within the auction. We demonstrate that auctions currently used for allocating advertising positions are suboptimal and offer simple designs that can implement (or approximate) optimal and efficient auctions under quantity externalities.