This paper studies auctions in a setting where the different bidders arrive at different times and the auction mechanism is required to make decisions about each bid as it is received. Such settings occur in computerized auctions of computational resources as well as in other settings. We call such auctions, on-line auctions. We first characterize exactly on-line auctions that are incentive compatible, i.e. where rational bidders are always motivated to bid their true valuation. We then embark on a competitive worst-case analysis of incentive compatible on-line auctions. We obtain several results, the cleanest of which is an incentive compatible on-line auction for a large number of identical items. This auction has an optimal competitive ratio, both in terms of seller's revenue and in terms of the total social efficiency obtained.
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