The Power of Uncertainty: Bundle-Pricing for Unit-Demand Customers

  • Patrick Briest
  • Heiko Röglin
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6534)


We study an extension of the unit-demand pricing problem in which the seller may offer bundles of items. If a customer buys such a bundle she is guaranteed to get one item out of it, but the seller does not make any promises of how this item is selected. This is motivated by the sales model of retailers like, which offers bundles of hotel rooms based on location and rating, and only identifies the booked hotel after the purchase has been made.

As the selected item is known only in hindsight, the buying decision depends on the customer’s belief about the allocation mechanism. We study strictly pessimistic and optimistic customers who always assume the worst-case or best-case allocation mechanism relative to their personal valuations, respectively. While the latter model turns out to be equivalent to the pure item pricing problem, the former is fundamentally different, and we prove the following results about it: (1) A revenue-maximizing pricing can be computed efficiently in the uniform version, in which every customer has a subset of items and the same non-zero value for all items in this subset and a value of zero for all other items. (2) For non-uniform customers computing a revenue-maximizing pricing is APX-hard. (3) For the case that any two values of a customer are either identical or differ by at least some constant factor, we present a polynomial time algorithm that obtains a constant approximation guarantee.


Polynomial Time Algorithm Price Problem Valuation Function Sales Model Constant Factor Approximation Algorithm 
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.
    Aggarwal, G., Feder, T., Motwani, R., Zhu, A.: Algorithms for Multi-product Pricing. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 72–83. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  2. 2.
    Balcan, M.-F., Blum, A.: Approximation Algorithms and Online Mechanisms for Item Pricing. In: Proc. of the 7th ACM Conference on Electronic Commerce, EC (2006)Google Scholar
  3. 3.
    Balcan, M.-F., Blum, A., Hartline, J.D., Mansour, Y.: Mechanism Design via Machine Learning. In: Proc. of the 46th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 605–614 (2005)Google Scholar
  4. 4.
    Briest, P.: Uniform Budgets and the Envy-Free Pricing Problem. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 808–819. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Briest, P., Chawla, S., Kleinberg, R., Weinberg, M.: Pricing Randomized Allocations. In: Proc. of the 21st ACM-SIAM Symposium on Discrete Algorithms, SODA (2010)Google Scholar
  6. 6.
    Briest, P., Krysta, P.: Buying Cheap is Expensive: Hardness of Non- Parametric Multi-Product Pricing. In: Proc. of the 18th ACM-SIAM Symposium on Discrete Algorithms, SODA (2007)Google Scholar
  7. 7.
    Chawla, S., Hartline, J.D., Kleinberg, R.: Algorithmic Pricing via Virtual Valuations. In: Proc. of the 8th ACM Conference on Electronic Commerce (EC), pp. 243–251 (2007)Google Scholar
  8. 8.
    Chen, N., Ghosh, A., Vassilvitskii, S.: Optimal Envy-Free Pricing with Metric Substitutability. In: Proc. of the 9th ACM Conference on Electronic Commerce (EC), pp. 60–69 (2008)Google Scholar
  9. 9.
    Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The Strong Perfect Graph Theorem. Annals of Mathematics 164, 51–229 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Crescenzi, P., Silvestri, R., Trevisan, L.: On Weighted vs Unweighted Versions of Combinatorial Optimization Problems. Inf. Comput. 167(1), 10–26 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Demaine, E., Feige, U., HajiAghayi, M.T., Salavatipour, M.: Combination Can Be Hard: Approximability of the Unique Coverage Problem. In: Proc. of the 17th ACM-SIAM Symposium on Discrete Algorithms, SODA (2006)Google Scholar
  12. 12.
    Goldberg, A.V., Hartline, J.D., Karlin, A.R., Saks, M., Wright, A.: Competitive Auctions. Games and Economic Behavior 55(2), 242–269 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grötschel, M., Lovász, L., Schrijver, A.: The Ellipsoid Method and its Consequences in Combinatorial Optimization. Combinatorica 1(2), 169–197 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Guruswami, V., Hartline, J.D., Karlin, A.R., Kempe, D., Kenyon, C., McSherry, F.: On Profit-Maximizing Envy-Free Pricing. In: Proc. of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1164–1173 (2005)Google Scholar
  15. 15.
    Myerson, R.: Optimal Auction Design. Mathematics of Operations Research 6, 58–73 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Riley, J., Zeckhauser, R.: Optimal Selling Strategies: When to Haggle, When to Hold Firm. Quarterly J. Economics 98(2), 267–289 (1983)CrossRefGoogle Scholar
  17. 17.
    Thanassoulis, J.: Haggling Over Substitutes. J. Economic Theory 117, 217–245 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Patrick Briest
    • 1
  • Heiko Röglin
    • 2
  1. 1.Department of Computer ScienceUniversity of PaderbornGermany
  2. 2.Department of Computer ScienceUniversity of BonnGermany

Personalised recommendations