S. Dobzinski and N. Nisan (2010) "Mechanisms for Multi-Unit Auctions", Volume 37, pages 85-98

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We present an incentive-compatible polynomial-time approximation scheme for multi-unit auctions with general k-minded player valuations. The mechanism fully optimizes over an appropriately chosen sub-range of possible allocations and then uses VCG payments over this sub-range. We show that obtaining a fully polynomial-time incentive-compatible approximation scheme, at least using VCG payments, is NP-hard. For the case of valuations given by black boxes, we give a polynomial-time incentive-compatible 2-approximation mechanism and show that no better is possible, at least using VCG payments.

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