  A seller with one unit of a good faces N\geq3 buyers and a single competitor
who sells one other identical unit in a second-price auction with a reserve
price. Buyers who do not get the seller's good will compete in the competitor's
subsequent auction. We characterize the optimal mechanism for the seller in
this setting. The first-order approach typically fails, so we develop new
techniques. The optimal mechanism features transfers from buyers with the two
highest valuations, allocation to the buyer with the second-highest valuation,
and a withholding rule that depends on the highest two or three valuations. It
can be implemented by a modified third-price auction or a pay-your-bid auction
with a rebate. This optimal withholding rule raises significantly more revenue
than would a standard reserve price. Our analysis also applies to procurement
auctions. Our results have implications for sequential competition in
mechanisms.

Theoretical Economics