Posted prices vs. negotiations: an asymptotic analysis

The design of optimal auctions focuses on ways to negotiate with the bidders for eliciting relevant information that they hold. Sometimes, however, decisions should be made very quickly, and the auctioneer cannot allow a costly iterative procedure of negotiation or waiting for bidders to determine their exact valuation. One solution that has been used in practice is to post prices for the bidders, without collecting any information from the bidders, and ask for their immediate take-it-or-leave-it response. Our paper compares the expected revenue in full-revelation auctions to that in posted-price auctions. We focus on single-item auctions in a Bayesian model where the values that the bidders are willing to pay for the item are independently identically distributed according to a known distribution. This paper provides an exact asymptotic characterization of the optimal expected revenue achieved by each one of the above auctions. For posted price auctions, we also present the exact prices that achieve the optimal results.Our results are given up to terms with lower asymptotic order, that is, up to factor of 1--o(1). We provide two sets of results; one for distributions on a support that is bounded from above, and a second set for distributions on unbounded supports. In the first case we require a mild assumption on the way the distribution function approaches the end of the support, called the first von Mises condition; the latter case requires a similar mild condition called the second von Mises condition. These very weak conditions are taken from works on extreme-value theory and highest-order statistics.