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Description

Combinatorial Auctions are a central problem in Algorithmic Mechanism Design: pricing and allocating goods to buyers with complex preferences in order to maximize some desired objective (e.g., social welfare, revenue, or profit). The problem has been well-studied in the case of limited supply (one copy of each item), and in the case of digital goods (the seller can produce additional copies at no cost). Yet the case of resources---think oil, labor, computing cycles, etc.---neither of these abstractions is just right: additional supplies of these resources can be found, but only at a cost (where the marginal cost is an increasing function). In this work, we initiate the study of the algorithmic mechanism design problem of combinatorial pricing under increasing marginal cost. The goal is to sell these goods to buyers with unknown and arbitrary combinatorial valuation functions to maximize either the social welfare, or our own profit; specifically we focus on the setting of \emph{posted item prices} with buyers arriving online. We give algorithms that achieve constant factor approximations for a class of natural cost functions---linear, low-degree polynomial, logarithmic---and that give logarithmic approximations for all convex marginal cost functions (along with a necessary additive loss). We show that these bounds are essentially best possible for these settings

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