Advertisement

Sealed Bid Multi-object Auctions with Necessary Bundles and its Application to Spectrum Auctions

  • Tomomi Matsui
  • Takahiro Watanabe
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2132)

Abstract

In this paper, we consider multi-object auctions in which each bidder has a positive reservation value for only one special subset of objects, called a necessary bundle. In the auction, each bidder reports its necessary bundle and its reservation value. The seller solves the assignment problem of objects which maximizes its revenue and decides the winning bidders who can purchase their necessary bundles for their reporting prices. We show that this auction leads to an efficient allocation through Nash equilibria under complete information when the bid-grid size is sufficiently small. We apply our results to spectrum auctions satisfying the conditions that necessary bundles are intervals of discretized radio spectrum. We show that the revenue maximization problem for the seller can be solved in polynomial time for the above auctions. The algorithm also indicates a method to choose an accepted bidder randomly when the revenue maximization problem has multiple optimal solutions. Lastly, we introduce a linear inequality system which characterizes the set of Nash equilibria.

Keywords

Nash Equilibrium Combinatorial Auction Walrasian Equilibrium Pure Strategy Nash Equilibrium Auction Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahuja, R. K., Magnanti, T. L., and Orlin, J. B. (1993), Network Flows, Theory Algorithms and Applications, Princeton Hall.Google Scholar
  2. 2.
    A. Andersson, Tenhunen, M., and Ygge, F. (2000), “Integer Programming for Combinatorial Auction Winner Determination,” Proc. of the Fourth International Conference on Multiagent Systems (ICMAS-00).Google Scholar
  3. 3.
    Bikhchandani, S. (1999), “Auctions of heterogeneous objects,” Games and Economic Behavior, vol. 26, 193–220.MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bikhchandani, S. and Mamer, J. W. (1997), “Competitive equilibrium in an exchange economy with indivisibilities,” Journal of Economic Theory, vol. 74, 385–413.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    de Vries, S. and Vohra, R. “Combinatorial auctions; a survey,” Kellog School of Management, technical report.Google Scholar
  6. 6.
    Elmaghraby, S. E. (1979), Activity Networks, John Wiley and Sons.Google Scholar
  7. 7.
    Engelbrecht-Wiggans, R. and Kahn, C. M. (1998), “Multi-unit auctions with uniform prices,” Economic Theory, vol. 12, 227–258.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Gale, I. (1990), “A multiple-object auction with superadditive values,” Economic Letters, vol. 34, 323–328.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kashima, H. and Kajiwara, Y. (2000), “Optimal winner determination algorithms for E-procurement auction,” Technical Report of IEICE, COMP, vol. 59, 17–23.Google Scholar
  10. 10.
    Krishna, V. and Rosenthal, R. W. (1996), “Simultaneous auctions with synergies,” Games and Economic Behavior, vol. 17, 1–31.CrossRefMathSciNetGoogle Scholar
  11. 11.
    Lawler, E. L. (1976), Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston, New York.MATHGoogle Scholar
  12. 12.
    Levin, J. (1997), “An optimal auction for complements,” Games and Economic Behavior, vol. 18, 176–192.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    MacMillan, J. (1994), “Selling spectrum rights,” Journal of Economic Perspectives, vol. 8, 145–162.Google Scholar
  14. 14.
    Noussair, C. (1995), “Equilibria in a multi-object uniform price sealed bid auction with multi-unit demands,” Economic Theory, vol. 5, 337–351.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Rassenti, S. J., Smith, V. L., and Bulfin, R. L. (1982), “A combinatorial auction mechanism for airport time slot allocation,” Bell Journal of Economics, vol. 13, 402–417.CrossRefGoogle Scholar
  16. 16.
    Rosenthal, R. W. and Wang R. (1996), “Simultaneous auctions with synergies and common values,” Games and economic Behavior, vol. 17, 32–55.CrossRefMathSciNetGoogle Scholar
  17. 17.
    Rothkopf, M. H., Pekeć, A. and Harstad, R. M. (1998), “Computationally manageable combinatorial auctions”, Management Science, vol. 44, 1131–1147.MATHCrossRefGoogle Scholar
  18. 18.
    Schrijver, A. (1986), Theory of Linear and Integer Programming, John Wiley and Sons, New York.MATHGoogle Scholar
  19. 19.
    Tarjan, R. (1972), “Depth-first search and linear graph algorithms”, SIAM Journal on Computing, vol. 1, 146–160.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Weber, R. J. (1983), “Multi-object auctions,” in Auctions, Bidding and Contracting, Engelbrecht-Wiggans, R., Shubik, M. and Stark R. eds., New York University Press.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Tomomi Matsui
    • 1
  • Takahiro Watanabe
    • 2
  1. 1.Department of Mathematical Informatics, Graduate School of Information Science and TechnologyUniversity of TokyoTokyoJapan
  2. 2.Department of Policy StudiesIwate Prefectural UniversityUSA

Personalised recommendations