Description:
We characterize an optimal mechanism for a seller with one unit of a good facing N ≥ 3 buyers and a single competitor who sells another identical unit in a second-price auction. Buyers who do not get the seller's good compete in the competitor's subsequent auction. The mechanism features transfers from buyers with the two highest valuations, allocation to the buyer with the second-highest valuation, and an allocation rule that depends on the two highest valuations. It can be implemented by a modified third-price auction, and it raises significantly more revenue than would a standard second- or first-price auction with a reserve price.