Abstract
Given a seller with \(k\) types of items, \(m\) of each, a sequence of users \(\{u_1, u_2,\ldots \}\) arrive one by one. Each user is single-minded, i.e., each user is interested only in a particular bundle of items. The seller must set the price and assign some amount of bundles to each user upon his/her arrival. Bundles can be sold fractionally. Each \(u_i\) has his/her value function \(v_i(\cdot )\) such that \(v_i(x)\) is the highest unit price \(u_i\) is willing to pay for \(x\) bundles. The objective is to maximize the revenue of the seller by setting the price and amount of bundles for each user. In this paper, we first show that a lower bound of the competitive ratio for this problem is \(\Omega (\log h+\log k)\), where \(h\) is the highest unit price to be paid among all users. We then give a deterministic online algorithm, Pricing, whose competitive ratio is \(O(\sqrt{k}\cdot \log h\log k)\). When \(k=1\) the lower and upper bounds asymptotically match the optimal result \(O(\log h)\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Babaioff, M., Dughmi, S., Kleinberg, R., Slivkins A.: Dynamic pricing with limited supply. In: Proceedings of the 13th ACM Conference on Electronic Commerce (EC12), pp. 74–91. Valencia, Spain, June 4–8 (2012)
Balcan, M.-F., Blum, A., Mansour, Y.: Item pricing for revenue maximization. In: Proceedings of the 9th ACM Conference on Electronic Commerce (EC’08), pp. 50–59 (2008)
Blum, A., Gupta, A., Mansour Y., Sharma, A.: Welfare and profit maximization with production costs. In: Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science (FOCS 2011), pp. 77–86, Palm Springs (2011)
Blum, A., Hartline J.D.: Near-optimal online auctions. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2005), pp. 1156–1163. Vancouver, British Columbia, January 23–25, 2005 (2005)
Bansal, N., Chen, N., Cherniavsky, N., Rurda, A., Schieber, B., Sviridenko, M.: Dynamic pricing for impatient bidders. ACM Trans. Algorithm. 6(2), 35 (2010)
Briest, P.: Uniform budgets and the envy-free pricing problem. In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming, Part I, Reykjavik, Iceland, July 07–11, 2008 (2008)
Briest, P., Krysta, P.: Buying cheap is expensive: hardness of non-parametric multi-product pricing. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 716–725. New Orleans, Louisiana, January 07–09, 2007 (2007)
Chen, N., Deng, X.: Envy-free pricing in multi-item markets In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming (ICALP 2010), pp. 418–429 (2012)
Chen, N., Ghosh, A., Vassilvitskii, S.: Optimal envy-free pricing with metric substitutability. In: Proceedings of the 9th ACM Conference on Electronic Commerce (EC’08), pp. 60–69 (2008)
Chawla, S., Hartline, J.D., Kleinberg, R.: Algorithmic pricing via virtual valuations. In: Proceedings of the 8th ACM Conference on Electronic Commerce (EC’07), pp. 243–251. San Diego, California, USA, June, 13–16 (2007)
Chakraborty, T., Huang, Z., Khanna, S.: Dynamic and non-uniform pricing strategies for revenue maximization. In: Proceedings of the 2009 IEEE 50th annual symposium on foundations of computer science (FOCS 2009), 495 C504, Atlanta (2009)
Cheung, M., Swamy, C.: Approximation algorithms for single-minded envy-free profit-maximization problems with limited supply. In: Proceedings of 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2008), pp. 35–44 (2008)
Elbassioni, K., Raman, R., Ray, S., Sitters, R.: On profit-maximizing pricing for the highway and tollbooth problems. In: Proceedings of the 2nd International Symposium on Algorithmic Game Theory, p. 275–286. Paphos, Cyprus, October 18–20, 2009 (2009)
Elbassioni, K., Sitters, R., Zhang, Y.: A quasi-PTAS for profit-maximizing pricing on line graphs. In: Proceedings of the 15th Annual European Conference on Algorithms (ESA’07), pp. 451–462 (2007)
Fiat, A., Wingarten A.: Envy, multi envy and revenue maximization. In: Proceedings of the 5th International Workshop on Internet and Network Economics (WINE 2009), pp. 498–504 (2009)
Grigoriev, A., van Loon, J., Sitters, R., Uetz, M.: How to sell a graph: guidelines for graph retailers. In: Proceedings of the 32nd International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2006), pp. 125–136 (2006)
Guruswami, V., Hartline, J., Karlin, A., Kempe, D., Kenyon, C., McSherry F.: On profit-maximizing envy-free pricing. In: Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 05), pp. 1164–1173 (2005)
Im S., Lu , P., Wang Y.: Envy-free pricing with general supply constraints. In: Proceedings of the 6th International Workshop on Internet and Network Economics (WINE 2010), pp. 483–491 (2010)
Krauthgamer, Robert, Mehta, Aranyak, Rudra, Atri: Pricing commodities. Theor. Comput. Sci. 412(7), 602–613 (2011)
Zhang, Y., Chin, F., Ting, H.-F.: Competitive algorithms for online pricing. In: Proceedings of the 17th Annual International Computing and Combinatorics Conference (COCOON 2011), LNCS 6842, pp. 391–401 (2011)
Acknowledgments
We thank Jack Snoeyink and the anonymous referees for their very helpful insights and comments on this paper. Y. Zhang was supported by NSFC 11171086. F.Y.L. Chin was supported by Hong Kong GRF grant. H.-F. Ting was supported by Hong Kong GRF grant HKU716412E.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, Y., Chin, F.Y.L. & Ting, HF. Online pricing for bundles of multiple items. J Glob Optim 58, 377–387 (2014). https://doi.org/10.1007/s10898-013-0043-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-013-0043-4