On-line auctions are arguably one of the most important and distinctly new applications of the Internet. The predominant p l a yer in on-line auctions, eBay, has over 42 million users, and it was the host of over $9.3 billion worth of goods sold in the year 2001. Using methods from approximate dynamic programming and integer programming, we design algorithms for optimally bidding for a single item in an on-line auction, and in simultaneous or overlapping multiple on-line auctions. We report computational evidence using data from eBay's web site from 1772 completed auctions for personal digital assistants and from 4208 completed auctions for stamp collections that shows that (a) the optimal dynamic policy outperforms simple but widely used static heuristic rules for a single auction, and (b) a new approach f o r t h e m ultiple auctions problem that uses the value functions of single auctions found by dynamic programming in an integer programming framework produces high quality solutions fast and reliably.
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