Computing a Walrasian Equilibrium in Iterative Auctions with Multiple Differentiated Items

  • Kazuo Murota
  • Akiyoshi Shioura
  • Zaifu Yang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8283)


We address the problem of computing a Walrasian equilibrium price in an ascending auction with gross substitutes valuations. In particular, an auction market is considered where there are multiple differentiated items and each item may have multiple units. Although the ascending auction is known to find an equilibrium price vector in finite time, little is known about its time complexity. The main aim of this paper is to analyze the time complexity of the ascending auction globally and locally, by utilizing the theory of discrete convex analysis. An exact bound on the number of iterations is given in terms of the ℓ ∞  distance between the initial price vector and an equilibrium, and an efficient algorithm to update a price vector is designedbased on a min-max theorem for submodular function minimization.


Lyapunov Function Price Vector Valuation Function Combinatorial Auction Submodular Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ausubel, L.M.: An efficient dynamic auction for heterogeneous commodities. American Economic Review 96, 602–629 (2006)CrossRefGoogle Scholar
  2. 2.
    Ausubel, L.M., Milgrom, P.: Ascending auctions with package bidding. Front. Theor. Econ. 1, Article 1 (2002)Google Scholar
  3. 3.
    Bichler, M., Shabalin, P., Pikovsky, A.: A computational analysis of linear price iterative combinatorial auction formats. Inform. Syst. Res. 20, 33–59 (2009)CrossRefGoogle Scholar
  4. 4.
    Bing, M., Lehmann, D., Milgrom, P.: Presentation and structure of substitutes valuations. In: Proc. EC 2004, pp. 238–239 (2004)Google Scholar
  5. 5.
    Blumrosen, L., Nisan, N.: Combinatorial auction. In: Nisan, N., et al. (eds.) Algorithmic Game Theory, pp. 267–299. Cambridge Univ. Press (2007)Google Scholar
  6. 6.
    Cramton, P., Shoham, Y., Steinberg, R.: Combinatorial Auctions. MIT Press (2006)Google Scholar
  7. 7.
    Fujishige, S.: Submodular Functions and Optimization, 2nd edn. Elsevier (2005)Google Scholar
  8. 8.
    Fujishige, S., Yang, Z.: A note on Kelso and Crawford’s gross substitutes condition. Math. Oper. Res. 28, 463–469 (2003)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gul, F., Stacchetti, E.: Walrasian equilibrium with gross substitutes. J. Econ. Theory 87, 95–124 (1999)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gul, F., Stacchetti, E.: The English auction with differentiated commodities. J. Economic Theory 92, 66–95 (2000)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Iwata, S.: A faster scaling algorithm for minimizing submodular functions. SIAM J. Comp. 32, 833–840 (2002)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kelso, A.S., Crawford, V.P.: Job matching, coalition formation and gross substitutes. Econometrica 50, 1483–1504 (1982)CrossRefMATHGoogle Scholar
  13. 13.
    Kolmogorov, V., Shioura, A.: New algorithms for convex cost tension problem with application to computer vision. Discrete Optimization 6, 378–393 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lehmann, B., Lehmann, D., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. Games Econom. Behav. 55, 270–296 (2006)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Milgrom, P., Strulovici, B.: Substitute goods, auctions, and equilibrium. J. Economic Theory 144, 212–247 (2009)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Murota, K.: Convexity and Steinitz’s exchange property. Adv. Math. 124, 272–311 (1996)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Murota, K.: Discrete Convex Analysis. SIAM, Philadelphia (2003)CrossRefMATHGoogle Scholar
  18. 18.
    Murota, K., Shioura, A.: M-convex function on generalized polymatroid. Math. Oper. Res. 24, 95–105 (1999)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Murota, K., Tamura, A.: New characterizations of M-convex functions and their applications to economic equilibrium models with indivisibilities. Discrete Appl. Math. 131, 495–512 (2003)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Parkes, D.C., Ungar, L.H.: Iterative combinatorial auctions: theory and practice. In: Proc. 17th National Conference on Artificial Intelligence (AAAI 2000), pp. 74–81 (2000)Google Scholar
  21. 21.
    Sun, N., Yang, Z.: A double-track adjustment process for discrete markets with substitutes and complements. Econometrica 77, 933–952 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Kazuo Murota
    • 1
  • Akiyoshi Shioura
    • 2
  • Zaifu Yang
    • 3
  1. 1.University of TokyoTokyoJapan
  2. 2.Tohoku UniversitySendaiJapan
  3. 3.University of YorkYorkUK

Personalised recommendations