Optimal Auction Design Under Non-Commitment

Abstract

First (or second) price auctions with optimally chosen reserve prices maximize revenue among all possible selling procedures when buyers are risk-neutral, ex-ante identical, and the seller commits to throw away the object for sale if no one bids above the reserve price. However, sellers seldom remove unsold items from the market: Real estate, used cars and art reappear in later auctions. This paper derives the profit-maximizing selling procedure when the seller, after each unsuccessful attempt to sell the item, updates her information about the buyers’ willingness to pay and proposes an optimal selling procedure given the updated information. We show that first- (or second-) price auctions with optimally chosen reserve prices are revenue-maximizing when buyers are ex-ante identical. When buyers’ valuations are drawn from different distributions, the seller maximizes revenue by assigning the good to the buyer with the highest virtual valuation if it is above a buyer-specific reserve price. Reserve prices drop over time. How much the optimal reserve prices drop depends on how the seller discounts the future. Inability to commit is costly for the seller. The revenue loss is highest for intermediate values of the discount factor and when the number of buyers is small.

Keywords: mechanism design, optimal auctions, limited commitment