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.