Welfare and Revenue Guarantees for Competitive Bundling Equilibrium

  • Shahar Dobzinski
  • Michal Feldman
  • Inbal Talgam-Cohen
  • Omri Weinstein
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9470)


Competitive equilibrium, the central equilibrium notion in markets with indivisible goods, is based on pricing each good such that the demand for goods equals their supply and the market clears. This equilibrium notion is not guaranteed to exist beyond the narrow case of substitute goods, might result in zero revenue even when consumers value the goods highly, and overlooks the widespread practice of pricing bundles rather than individual goods. Alternative equilibrium notions proposed to address these shortcomings have either made a strong assumption on the ability to withhold supply in equilibrium, or have allowed an exponential number of prices.

In this paper we study the notion of competitive bundling equilibrium – a competitive equilibrium over the market induced by partitioning the goods into bundles. Such an equilibrium is guaranteed to exist, is succinct, and satisfies the fundamental economic condition of market clearance. We establish positive welfare and revenue guarantees for this solution concept: For welfare we show that in markets with homogeneous goods, there always exists a competitive bundling equilibrium that achieves a logarithmic fraction of the optimal welfare. We also extend this result to establish nontrivial welfare guarantees for markets with heterogeneous goods. For revenue we show that in a natural class of markets for which competitive equilibrium does not guarantee positive revenue, there always exists a competitive bundling equilibrium that extracts as revenue a logarithmic fraction of the optimal welfare. Both results are tight.


  1. 1.
    Ausubel, L., Milgrom, P.: Ascending auctions with package bidding. B.E. J. Theoret. Econ. 1(1), 1–44 (2002)MathSciNetGoogle Scholar
  2. 2.
    Ausubel, L.M.: An efficient ascending-bid auction for multiple objects. Am. Econ. Rev. 94, 1452–1475 (2004)CrossRefGoogle Scholar
  3. 3.
    Bikhchandani, S., Ostroy, J.M.: The package assignment model. J. Econ. Theory 107, 377–406 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Borgs, C., Chayes, J., Immorlica, N., Mahdian, M., Saberi, A.: Multi-unit auctions with budget-constrained bidders. In: Proceedings of 7th ACM Conference on Electronic Commerce (EC) (2005)Google Scholar
  5. 5.
    Coase, R.H.: Durability and monopoly. J. Law Econ. 15(1), 143–149 (1972)CrossRefGoogle Scholar
  6. 6.
    Dobzinski, S., Feldman, M., Talgam-Cohen, I., Weinstein, O.: Welfare and revenue guarantees for competitive bundling equilibrium. CoRR abs/1406.0576 (2014).
  7. 7.
    Dobzinski, S., Nisan, N.: Mechanisms for multi-unit auctions. J. Artif. Intell. Res. 37, 85–98 (2010)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Feldman, M., Gravin, N., Lucier, B.: Combinatorial walrasian equilibrium. In: Proceedings of 44th ACM Symposium on Theory of Computing (STOC), pp. 61–70 (2013)Google Scholar
  9. 9.
    Feldman, M., Lucier, B.: Clearing markets via bundles. In: Lavi, R. (ed.) SAGT 2014. LNCS, vol. 8768, pp. 158–169. Springer, Heidelberg (2014) Google Scholar
  10. 10.
    Gul, F., Stacchetti, E.: Walrasian equilibrium with gross substitutes. J. Econ. Theory 87(1), 95–124 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Lahaie, S., Parkes, D.C.: Fair package assignment. In: Das, S., Ostrovsky, M., Pennock, D., Szymanksi, B. (eds.) AMMA 2009. LNICST, vol. 14, pp. 92–92. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  12. 12.
    Lehmann, B., Lehmann, D.J., Nisan, N.: Combinatorial auctions with decreasing marginal utilities. Games Econ. Behav. 55(2), 270–296 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Manelli, A.M., Vincent, D.R.: Bundling as an optimal selling mechanism for a multiple-good monopolist. J. Econ. Theory 127, 1–35 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Mas-Collel, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, Oxford (1995) zbMATHGoogle Scholar
  15. 15.
    Maskin, E., Riley, J.: Optimal multi-unit auctions. In: Hahn, F. (ed.) The Economics of Missing Markets, Information, and Games, pp. 312–335. Oxford University Press, Oxford (1989) Google Scholar
  16. 16.
    Milgrom, P.: Putting Auction Theory to Work. Cambridge University Press, Cambridge (2004) CrossRefGoogle Scholar
  17. 17.
    Nisan, N.: Survey: algorithmic mechanism design (through the lens of multi-unit auctions). In: Young, P., Zamir, S. (eds.) Handbook of Game Theory, vol. 4, chap. 9. Elsevier, Amsterdam (2014)Google Scholar
  18. 18.
    Parkes, D.: Iterative combinatorial auctions. In: Cramton, P., Shoham, Y., Steinberg, R. (eds.) Combinatorial Auctions, chap. 2. MIT Press, Cambridge (2006) Google Scholar
  19. 19.
    Parkes, D.C.: iBundle: an efficient ascending price bundle auction. In: Proceedings of 1st ACM Conference on Electronic Commerce (EC), pp. 148–157 (1999)Google Scholar
  20. 20.
    Parkes, D.C., Ungar, L.H.: Iterative combinatorial auctions: theory and practice. In: Proceedings of the 17th National Conference on Artificial Intelligence, AAAI 2000, pp. 74–81 (2000)Google Scholar
  21. 21.
    Parkes, D.C., Ungar, L.H.: An ascending-price generalized Vickrey auction. In: Proceedings of Stanford Institute for Theoretical Economics Workshop on The Economics of the Internet, Stanford, CA (2002)Google Scholar
  22. 22.
    Sun, N., Yang, Z.: An efficient and incentive compatible dynamic auction for multiple complements. J. Polit. Econ. (2014, to appear)Google Scholar
  23. 23.
    Vickrey, W.: Counterspeculation, auctions, and competitive sealed tenders. J. Finance 16, 8–37 (1961)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Vohra, R.V.: Mechanism Design: A Linear Programming Approach (Econometric Society Monographs). Cambridge University Press, Cambridge (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Open Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Shahar Dobzinski
    • 1
  • Michal Feldman
    • 2
  • Inbal Talgam-Cohen
    • 3
  • Omri Weinstein
    • 4
  1. 1.Weizmann Institute of ScienceRehovotIsrael
  2. 2.Tel-Aviv UniversityTel AvivIsrael
  3. 3.Stanford UniversityStanfordUSA
  4. 4.Princeton UniversityPrincetonUSA

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