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A Branch-and-Price Algorithm and New Test Problems for Spectrum Auctions

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  • Oktay Günlük

    () (IBM Research Division, T. J. Watson Research Center, Yorktown Heights, New York 10598)

  • Lászlo Ladányi

    () (IBM Research Division, T. J. Watson Research Center, Yorktown Heights, New York 10598)

  • Sven de Vries

    () (Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, D-85747 Garching bei München, Germany)

Abstract

When combinatorial bidding is permitted in auctions, such as the proposed FCC Auction #31, the resulting full valuations and winner-determination problem can be computationally challenging. We present a branch-and-price algorithm based on a set-packing formulation originally proposed by Dietrich and Forrest (2002, "A column generation approach for combinatorial auctions," in Mathematics of the Internet: E-Auction and Markets. The IMA Volumes in Mathematics and Its Applications, Vol. 127, Springer-Verlag, New York, 15--26). This formulation has a variable for every possible combination of winning bids for each bidder. Our algorithm exploits the structure of the XOR-of-OR bidding language used by the FCC. We also present a new methodology to produce realistic test problems based on the round-by-round results of FCC Auction #4. We generate 2,639 test problems, which involve 99 items and are substantially larger than most of the previously used benchmark problems. Because there are no real-life test problems for combinatorial spectrum auctions with the XOR-of-OR language, we used these test problems to observe the computational behavior of our algorithm. Our algorithm can solve all but one test problem within 10 minutes, appears to be very robust, and for difficult instances compares favorably to the natural formulation solved using a commercial optimization package with default settings. Although spectrum auctions are used as the guiding example to describe the merits of branch and price for combinatorial auctions, our approach applies to auctions of multiple goods in other scenarios similarly.

Suggested Citation

  • 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.
  • Handle: RePEc:inm:ormnsc:v:51:y:2005:i:3:p:391-406
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    File URL: http://dx.doi.org/10.1287/mnsc.1040.0332
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. 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.
    2. 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.
    3. Drexl, Andreas & Jörnsten, Kurt & Knof, Diether, 2005. "Non-linear anonymous pricing in combinatorial auctions," Discussion Papers 2005/6, Norwegian School of Economics, Department of Business and Management Science.
    4. repec:pal:jorsoc:v:58:y:2007:i:12:d:10.1057_palgrave.jors.2602299 is not listed on IDEAS
    5. G. Anandalingam & Robert W. Day & S. Raghavan, 2005. "The Landscape of Electronic Market Design," Management Science, INFORMS, vol. 51(3), pages 316-327, March.
    6. Drexl, Andreas & Jørnsten, Kurt & Knof, Diether, 2005. "Non-linear anonymous pricing in combinatorial auctions," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 598, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    7. Gediminas Adomavicius & Shawn P. Curley & Alok Gupta & Pallab Sanyal, 2012. "Effect of Information Feedback on Bidder Behavior in Continuous Combinatorial Auctions," Management Science, INFORMS, vol. 58(4), pages 811-830, April.
    8. Robert W. Day & S. Raghavan, 2007. "Fair Payments for Efficient Allocations in Public Sector Combinatorial Auctions," Management Science, INFORMS, vol. 53(9), pages 1389-1406, September.
    9. Ertem, Mustafa A. & Buyurgan, Nebil & Pohl, Edward A., 2012. "Using announcement options in the bid construction phase for disaster relief procurement," Socio-Economic Planning Sciences, Elsevier, vol. 46(4), pages 306-314.
    10. 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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