Inefficiency of Standard Multi-unit Auctions

  • Bart de Keijzer
  • Evangelos Markakis
  • Guido Schäfer
  • Orestis Telelis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8125)


We study two standard multi-unit auction formats for allocating multiple units of a single good to multi-demand bidders. The first one is the Discriminatory Auction, which charges every winner his winning bids. The second is the Uniform Price Auction, which determines a uniform price to be paid per unit. Variants of both formats find applications ranging from the allocation of state bonds to investors, to online sales over the internet. For these formats, we consider two bidding interfaces: (i) standard bidding, which is most prevalent in the scientific literature, and (ii) uniform bidding, which is more popular in practice. In this work, we evaluate the economic inefficiency of both multi-unit auction formats for both bidding interfaces, by means of upper and lower bounds on the Price of Anarchy for pure Nash equilibria and mixed Bayes-Nash equilibria. Our developments improve significantly upon bounds that have been obtained recently for submodular valuation functions. Also, for the first time, we consider bidders with subadditive valuation functions under these auction formats. Our results signify near-efficiency of these auctions, which provides further justification for their use in practice.


Valuation Function Combinatorial Auction Pure Nash Equilibrium Auction Format Auction Theory 
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., Cramton, P.: Demand Reduction and Inefficiency in Multi-Unit Auctions. Tech. rep., University of Maryland (2002)Google Scholar
  2. 2.
    Bhawalkar, K., Roughgarden, T.: Welfare Guarantees for Combinatorial Auctions with Item Bidding. In: Proc. of the 22nd ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 700–709 (2011)Google Scholar
  3. 3.
    Binmore, K., Swierzbinski, J.: Treasury auctions: Uniform or discriminatory? Review of Economic Design 5(4), 387–410 (2000)CrossRefGoogle Scholar
  4. 4.
    Bresky, M.: Pure Equilibrium Strategies in Multi-unit Auctions with Private Value Bidders. Tech. Rep. 376, CERGE Economics Institute (2008)Google Scholar
  5. 5.
    Christodoulou, G., Kovács, A., Schapira, M.: Bayesian Combinatorial Auctions. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 820–832. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Engelbrecht-Wiggans, R., Kahn, C.M.: Multi-unit auctions with uniform prices. Economic Theory 12(2), 227–258 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Feldman, M., Fu, H., Gravin, N., Lucier, B.: Simultaneous Auctions are (almost) Efficient. In: Proc. of the 45th ACM Symposium on Theory of Computing (STOC), pp. 201–210 (2013)Google Scholar
  8. 8.
    Friedman, M.: A Program for Monetary Stability. Fordham University Press, New York (1960)Google Scholar
  9. 9.
    Fu, H., Kleinberg, R., Lavi, R.: Conditional equilibrium outcomes via ascending price processes with applications to combinatorial auctions with item bidding. In: Proc. of the 13th ACM Conference on Electronic Commerce, p. 586 (2012)Google Scholar
  10. 10.
    Hassidim, A., Kaplan, H., Mansour, Y., Nisan, N.: Non-price equilibria in markets of discrete goods. In: Proc. of the 12th ACM Conference on Electronic Commerce, pp. 295–296 (2011)Google Scholar
  11. 11.
    de Keijzer, B., Markakis, E., Schäfer, G., Telelis, O.: On the Inefficiency of Standard Multi-Unit Auctions. arXiv:1303.1646 [cs.GT] (2013)Google Scholar
  12. 12.
    Krishna, V.: Auction Theory. Academic Press (2002)Google Scholar
  13. 13.
    Lehmann, B., Lehmann, D.J., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. Games and Economic Behavior 55(2), 270–296 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Markakis, E., Telelis, O.: Uniform Price Auctions: Equilibria and Efficiency. In: Serna, M. (ed.) SAGT 2012. LNCS, vol. 7615, pp. 227–238. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Milgrom, P.: Putting Auction Theory to Work. Cambridge University Press (2004)Google Scholar
  16. 16.
    Noussair, C.: Equilibria in a multi-object uniform price sealed bid auction with multi-unit demands. Economic Theory 5, 337–351 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Ockenfels, A., Reiley, D.H., Sadrieh, A.: Economics and Information Systems. In: Handbooks in Information Systems, ch. 12. Online Auctions, vol. 1, pp. 571–628. Elsevier (2006)Google Scholar
  18. 18.
    Roughgarden, T.: The price of anarchy in games of incomplete information. In: Proc. of the 13th ACM Conference on Electronic Commerce, pp. 862–879 (2012)Google Scholar
  19. 19.
    Syrgkanis, V., Tardos, E.: Composable and efficient mechanisms. In: Proc. of the 45th ACM Symposium on Theory of Computing (STOC), pp. 211–220 (2013)Google Scholar
  20. 20.
    Syrgkanis, V.: Bayesian games and the smoothness framework. arXiv:1203.5155 [cs.GT] (2012)Google Scholar
  21. 21.
    U.S. Department of Treasury: Uniform-Price Auctions: Update of the Treasury Experience (1998),
  22. 22.
    Vickrey, W.: Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance 16(1), 8–37 (1961)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Bart de Keijzer
    • 1
  • Evangelos Markakis
    • 3
  • Guido Schäfer
    • 1
    • 2
  • Orestis Telelis
    • 3
  1. 1.CWI AmsterdamThe Netherlands
  2. 2.VU AmsterdamThe Netherlands
  3. 3.Athens University of Economics and BusinessGreece

Personalised recommendations