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Column aggregation-based pricing combinatorial auctions

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  • Drexl, Andreas
  • Jørnsten, Kurt
  • Knof, Diether

Abstract

Combinatorial auctions permitting bids on bundles of items have been developed to remedy the exposure problem associated with single-item auctions. Given winning bundle prices a set of item prices is called market clearing or equilibrium if all the winning bids are greater than or equal and if all the losing bids are less than or equal to the total price of the bundle items. However, the prices for individual items are not readily computed once the winner determination problem is solved. This is due to the duality gap of integer programming caused by the indivisibility of the items. In this paper we propose a family of linear programming models the optimal solution of which is integral "almost always", producing linear prices at the expense of having reduced cost zero for the aggregate winning bids only. We provide a computational proof of this conjecture by an in-depth experimental study of 18,000 instances from the combinatoriaI auction test suite (CATS; see [13]). Summarizing this analysis we have linear prices for all but five of the whole bunch of instances and, hence, there exists a linear price function that supports the optimal allocation of winning bundles.

Suggested Citation

  • Drexl, Andreas & Jørnsten, Kurt & Knof, Diether, 2007. "Column aggregation-based pricing combinatorial auctions," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 624, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
  • Handle: RePEc:zbw:cauman:624
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    References listed on IDEAS

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    1. Drexl, Andreas & Jörnsten, Kurt, 2005. "Reflections about pseudo-dual prices in combinatorial auctions," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 590, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    2. Peter R. Wurman & Michael P. Wellman, 1999. "Equilibrium Prices in Bundle Auctions," Working Papers 99-09-064, Santa Fe Institute.
    3. Aleksandar Pekev{c} & Michael H. Rothkopf, 2003. "Combinatorial Auction Design," Management Science, INFORMS, vol. 49(11), pages 1485-1503, November.
    4. Anthony M. Kwasnica & John O. Ledyard & Dave Porter & Christine DeMartini, 2005. "A New and Improved Design for Multiobject Iterative Auctions," Management Science, INFORMS, vol. 51(3), pages 419-434, March.
    5. Paul Milgrom, 2000. "Putting Auction Theory to Work: The Simultaneous Ascending Auction," Journal of Political Economy, University of Chicago Press, vol. 108(2), pages 245-272, April.
    6. Xia, Mu & Koehler, Gary J. & Whinston, Andrew B., 2004. "Pricing combinatorial auctions," European Journal of Operational Research, Elsevier, vol. 154(1), pages 251-270, April.
    7. Delorme, Xavier & Gandibleux, Xavier & Rodriguez, Joaquin, 2004. "GRASP for set packing problems," European Journal of Operational Research, Elsevier, vol. 153(3), pages 564-580, March.
    8. Bikhchandani, Sushil & Ostroy, Joseph M., 2002. "The Package Assignment Model," Journal of Economic Theory, Elsevier, vol. 107(2), pages 377-406, December.
    9. John McMillan, 1994. "Selling Spectrum Rights," Journal of Economic Perspectives, American Economic Association, vol. 8(3), pages 145-162, Summer.
    10. Michael H. Rothkopf & Aleksandar Pekev{c} & Ronald M. Harstad, 1998. "Computationally Manageable Combinational Auctions," Management Science, INFORMS, vol. 44(8), pages 1131-1147, August.
    11. Oktay Günlük & Lászlo Ladányi & Sven de Vries, 2005. "A Branch-and-Price Algorithm and New Test Problems for Spectrum Auctions," Management Science, INFORMS, vol. 51(3), pages 391-406, March.
    12. S.J. Rassenti & V.L. Smith & R.L. Bulfin, 1982. "A Combinatorial Auction Mechanism for Airport Time Slot Allocation," Bell Journal of Economics, The RAND Corporation, vol. 13(2), pages 402-417, Autumn.
    13. R. H. Kwon & G. Anandalingam & L. H. Ungar, 2005. "Iterative Combinatorial Auctions with Bidder-Determined Combinations," Management Science, INFORMS, vol. 51(3), pages 407-418, March.
    14. Drexl, Andreas & Jörnsten, Kurt, 2005. "Reflections about pseudo-dual prices in combinatorial auctions," Discussion Papers 2005/1, Norwegian School of Economics, Department of Business and Management Science.
    15. Tuomas Sandholm & Subhash Suri & Andrew Gilpin & David Levine, 2005. "CABOB: A Fast Optimal Algorithm for Winner Determination in Combinatorial Auctions," Management Science, INFORMS, vol. 51(3), pages 374-390, March.
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    Cited by:

    1. Drexl, Andreas & Jørnsten, Kurt & Knof, Diether, 2007. "Non-linear anonymous pricing combinatorial auctions," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 625, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    2. Drexl, Andreas & Jørnsten, Kurt & Knof, Diether, 2009. "Non-linear anonymous pricing combinatorial auctions," European Journal of Operational Research, Elsevier, vol. 199(1), pages 296-302, November.

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