Interdependent Values without Single-Crossing

Abstract

We consider a setting where an auctioneer sells a single item to n potential agents with interdependent values. That is, each agent has her own private signal, and the valuation of each agent is a known function of all n private signals. This captures settings such as valuations for oil drilling rights, broadcast rights, pieces of art, and many more.In the interdependent value setting, all previous work has assumed a so-called single-crossing condition. Single-crossing means that the impact of a private signal, si , on the valuation of agent i , is greater than the impact of si on the valuation of any other agent. It is known that without the single-crossing condition, an efficient outcome cannot be obtained. We ask what approximation to the optimal social welfare can be obtained when valuations do not exhibit single-crossing.We show that, in general, without the single-crossing condition, one cannot hope to approximate the optimal social welfare any better than assigning the item to a random bidder. Consequently, we consider a relaxed version of single-crossing, c-single-crossing, with some parameter c≥1 , which means that the impact of si on the valuation of agent i is at least 1/ c times the impact of si on the valuation of any other agent ( c =1 is single-crossing). Using this relaxed notion, we obtain a host of positive results. These include a prior-free universally truthful 2√ nc3/2 -approximation to welfare, and a prior-free deterministic ( n -1) c -approximation to welfare. Under appropriate concavity conditions, we improve this to a prior-free universally truthful 2 c -approximation to welfare as well as a universally truthful O(c2)-approximation to the optimal revenue.