Envy-Free Pricing with General Supply Constraints

  • Sungjin Im
  • Pinyan Lu
  • Yajun Wang
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6484)


The envy-free pricing problem can be stated as finding a pricing and allocation scheme in which each consumer is allocated a set of items that maximize her utility under the pricing. The goal is to maximize seller revenue. We study the problem with general supply constraints which are given as an independence system defined over the items. The constraints, for example, can be a number of linear constraints or matroids. This captures the situation where items do not pre-exist, but are produced in reflection of consumer valuation of the items under the limit of resources.

This paper focuses on the case of unit-demand consumers. In the setting, there are n consumers and m items; each item may be produced in multiple copies. Each consumer i ∈ [n] has a valuation vij on item j in the set Si in which she is interested. She must be allocated (if any) an item which gives the maximum (non-negative) utility. Suppose we are given an α-approximation oracle for finding the maximum weight independent set for the given independence system (or a slightly stronger oracle); for a large number of natural and interesting supply constraints, constant approximations are available. We obtain the following results.
  • O(αlogn)-approximation for the general case.

  • O(αk)-approximation when each consumer is interested in at most k distinct types of items.

  • O(αf)-approximation when each item is interesting to at most f consumers.

Note that the final two results were previously unknown even without the independence system constraint.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Balcan, M.-F., Blum, A.: Approximation algorithms and online mechanisms for item pricing. In: EC 2006, pp. 29–35. ACM, New York (2006)Google Scholar
  2. 2.
    Briest, P.: Uniform budgets and the envy-free pricing problem. In: ICALP (1), pp. 808–819 (2008)Google Scholar
  3. 3.
    Briest, P., Krysta, P.: Single-minded unlimited supply pricing on sparse instances. In: SODA 2006, pp. 1093–1102 (2006)Google Scholar
  4. 4.
    Chekuri, C.: Personal Communication (2010)Google Scholar
  5. 5.
    Chekuri, C., Vondrák, J., Zenklusen, R.: Multi-budgeted matchings and matroid intersection via dependent rounding (2010) (manuscript)Google Scholar
  6. 6.
    Chen, N., Ghosh, A., Vassilvitskii, S.: Optimal envy-free pricing with metric substitutability. In: EC 2008, pp. 60–69 (2008)Google Scholar
  7. 7.
    Frieze, A.M., Clarke, M.R.B.: Approximation algorithms for the m-dimensional 0-1 knapsack problem: Worst-case and probabilistic analyses. European Journal of Operational Research 15(1), 100–109 (1984)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Goldberg, A.V., Hartline, J.D.: Competitive auctions for multiple digital goods. In: Meyer auf der Heide, F. (ed.) ESA 2001. LNCS, vol. 2161, pp. 416–427. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Goldberg, A.V., Hartline, J.D., Wright, A.: Competitive auctions and digital goods. In: SODA, pp. 735–744 (2001)Google Scholar
  10. 10.
    Gul, F., Stacchetti, E.: Walrasian equilibrium with gross substitues. Journal of Economic Theory 87, 95–124 (1999)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Guruswami, V., Hartline, J.D., Karlin, A.R., Kempe, D., Kenyon, C., McSherry, F.: On profit-maximizing envy-free pricing. In: SODA, pp. 1164–1173 (2005)Google Scholar
  12. 12.
    Hartline, J.D., Koltun, V.: Near-optimal pricing in near-linear time. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 422–431. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  13. 13.
    Koopmans, T., Beckmann, M.: Assignment problems and the location of economic activities. Econometrica 25, 53–76 (1957)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lee, J., Mirrokni, V.S., Nagarajan, V., Sviridenko, M.: Non-monotone submodular maximization under matroid and knapsack constraints. In: STOC, pp. 323–332 (2009)Google Scholar
  15. 15.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. 24. Springer, Heidelberg (2003)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Sungjin Im
    • 1
  • Pinyan Lu
    • 2
  • Yajun Wang
    • 2
  1. 1.Dept. of Computer ScienceUniversity of IllinoisUrbana
  2. 2.Microsoft Research AsiaChina

Personalised recommendations