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Incentive Compatible Mulit-Unit Combinatorial Auctions: A Primal Dual Approach

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Abstract

We study multi-unit combinatorial auctions with multi-minded buyers. We provide two deterministic, efficient maximizing, incentive compatible mechanisms that improve upon the known algorithms for the problem (Bartal et al., Proceedings of the 9th Conference on Theoretical Aspects of Rationality and Knowledge TARK IX, pp. 72–87, 2003). The first mechanism is an online mechanism for a setting in which buyers arrive one-by-one in an online fashion. We then design an offline mechanism with better performance guarantees based on the online mechanism. We complement the results by lower bounds that show that the performance of our mechanisms is close to optimal. The results are based on an online primal-dual approach that was used extensively recently and reveals the underlying structure of the problem.

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Notes

  1. This is equivalent to the assumption of the value Θ in [8].

  2. It is hard to approximate within \(\min\{\frac{n}{k}, O(m^{\frac{1}{k}-\epsilon})\}\)-factor (e.g. [20, 23, 28, 37] in the case of single-unit items.

  3. Single-value buyers are buyers that have a single value for all bundles they desire.

  4. We do not assume that the value m′ is known to the algorithm.

  5. Note that s i might be the empty set.

  6. If v max is unknown to the algorithm, then any deterministic algorithm has an unbounded competitive ratio even if there is only a single item (with multiple copies). To see this, consider the following simple adversarial sequence. In each iteration the next bidder would like a single copy of the (single) item and her bid is the smallest value for which the next copy of item is allocated. If there is no such value then certainly the algorithm is not competitive. Otherwise, the algorithm always allocates all copies, and then the next bidder has value that is very large compared to all previous bids.

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Authors and Affiliations

  1. Statistics and Operations Research Dept., Tel Aviv University, P.O. Box 39040, Tel Aviv, 6997801, Israel

    Niv Buchbinder

  2. Dept. of Management and Economics, The Open University of Israel, The Dorothy de Rothschild Campus, 1 University Road, Raanana, 4353701, Israel

    Rica Gonen

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  1. Niv Buchbinder
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  2. Rica Gonen
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Correspondence to Rica Gonen.

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Supported by ISF grant 954/11 and BSF grant 2010426.

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Buchbinder, N., Gonen, R. Incentive Compatible Mulit-Unit Combinatorial Auctions: A Primal Dual Approach. Algorithmica 72, 167–190 (2015). https://doi.org/10.1007/s00453-013-9854-4

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