Problem Definition
This problem studies the one round, sealed-bid auction model where an auctioneer would like to sell an idiosyncratic commodity with unlimited copies to n bidders and each bidder \( { i \in \{1, \dots,n\} } \) will get at most one item.
First, for any i, bidder i bids a value b i representing the price he is willing to pay for the item. They submit the bids simultaneously. After receiving the bidding vector \( { \mathbf{b}=(b_1, \dots, b_n) } \), the auctioneer computes and outputs the allocation vector \( { \mathbf{x}=(x_1, \dots, x_n) \in \{0,1\}^n } \) and the price vector \( { \mathbf{p}=(p_1, \dots, p_n) } \). If for any i, \( { x_i=1 } \), then bidder i gets the item and pays p i for it. Otherwise, bidder i loses and pays nothing. In the auction, the auctioneer's revenue is \( { \sum_{i=1}^n \mathbf{x}\mathbf{p}^\mathrm{T} } \).
Definition 1 (Optimal Single Price Omniscient Auction \( { \mathcal{F} } \))
Given a bidding vector \( { \mathbf{b} } \)sorted in...
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Bu, TM. (2008). Competitive Auction. In: Kao, MY. (eds) Encyclopedia of Algorithms. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-30162-4_78
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