A seller with one unit of a good faces N >= 3 buyers and a single competitor. Buyers who do not get the good from the seller will compete in a second-price auction for one other identical unit. We characterize the optimal mechanism for the seller in this setting and show that it cannot be implemented by a standard auction. Instead, it can be implemented by a modified third-price or first-price auction with transfers between the seller and the two highest bidders. The optimal mechanism features allocation to the buyer with the second-highest valuation and a withholding rule that depends on the second- and third-highest valuations. We show that this withholding rule raises significantly more revenue than would a standard reserve price. We also consider the novel implications of sequential cross-mechanism spillovers for competition between sellers.
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