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.We compare the following three auctions: (1) Full-revelation auctions. Auctions where each bidder reveals his exact private value to the seller. An optimal full-revelation auction maximizes the expected revenue in equilibrium; (2) Symmetric posted-price auctions. The seller publishes a price p, and the item is sold to one of the bidders that is willing to buy the item at the price p. In an optimal posted-price auction, the seller determines the price p such that his expected revenue is maximized. (3) Discriminatory posted-price auctions. Auctions where the seller publishes an individual price p(i) for each bidder, and the item is sold to the bidder with the highest price among those who accepted their offer. Note that although the distributions are ex-ante identical, the seller still may gain more power by publishing discriminatory prices.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 tip to terms with lower asymptotic order, that is, tip 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.Bounded supports.Theorem: (informal) Discriminatory posted-price auctions with the right choice of prices can be almost equivalent in terms of revenue to optimal full-revelation auctions. On the other hand, symmetric posted-price auctions incur an asymptotically greater loss by a logarithmic factor in the number of bidders.For example, in the case where the values of the bidders are distributed uniformly on [0, 1], the optimal expected revenue in the full revelation auction (i.e., the Myerson auction) is around 1 - 2/n. The optimal revenue with a symmetric posted price is around 1 - log n/n. With discriminatory prices, however, the expected revenue becomes very close to the full-revelation result, i.e., 1 - 4/n.Unbounded supports.Theorem: (informal) Optimal full revelation auctions, symmetric posted-price auctions and discriminatory posted-price auctions do not converge to the same revenue when the number of bidders grow. (We present an exact characterization of the ratio between the expected revenue in these auctions.)For example, consider a power-law distribution on the open interval [1, infinity), i.e., where the cdf is F(x) = 1 - 1/x(2) The expected revenue on the optimal full-revelation auction turns out to be 0.88 root n, in symmetric posted-price auctions it is 0.64 root n and in discriminatory posted-price auctions it is 0.7 root n (always omitting lower order terms). Note that the ratio between those revenues is not converging to 1.