Abstract
In this paper, we study the problem of online pricing for bundles of items. Given a seller with k types of items, m of each, a sequence of users {u 1, u 2, ...} arrives 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 (·) 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 the lower bound of the competitive ratio for this problem is Ω(logh + logk), 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(logh).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Balcan, N., Blum, A., Mansour, Y.: Item pricing for revenue maximization. In: Proc. of the 9th ACM Conference on Electronic Commerce (EC 2008), pp. 50–59 (2008)
Bansal, N., Chen, N., Cherniavsky, N., Rurda, A., Schieber, B., Sviridenko, M.: Dynamic pricing for impatient bidders. ACM Transactions on Algorithms 6(2) (March 2010)
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)
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, New Orleans, Louisiana, January 07-09, pp. 716–725 (2007)
Chen, N., Deng, X.: Envy-Free Pricing in Multi-item Markets. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 418–429. Springer, Heidelberg (2010)
Chen, N., Ghosh, A., Vassilvitskii, S.: Optimal envy-free pricing with metric substitutability. In: Proc. of the 9th ACM Conference on Electronic Commerce (EC 2008), pp. 60–69 (2008)
Cheung, M., Swamy, C.: Approximation Algorithms for Single-minded Envy-free Profit-maximization Problems with Limited Supply. In: Proc. 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, Paphos, Cyprus, October 18-20, pp. 275–286 (2009)
Elbassioni, K., Sitters, R., Zhang, Y.: A Quasi-PTAS for Profit-Maximizing Pricing on Line Graphs. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 451–462. Springer, Heidelberg (2007)
Fiat, A., Wingarten, A.: Envy, Multi Envy, and Revenue Maximization. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 498–504. Springer, Heidelberg (2009)
Grigoriev, A., van Loon, J., Sitters, R.A., Uetz, M.: How to Sell a Graph: Guidelines for Graph Retailers. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 125–136. Springer, Heidelberg (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 2005), pp. 1164–1173 (2005)
Im, S., Lu, P., Wang, Y.: Envy-Free Pricing with General Supply Constraints. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 483–491. Springer, Heidelberg (2010)
Krauthgamer, R., Mehta, A., Rudra, A.: Pricing commodities. Theoretical Computer Science 412(7), 602–613 (2011)
Zhang, Y., Chin, F.Y.L., Ting, H.-F.: Competitive Algorithms for Online Pricing. In: Fu, B., Du, D.-Z. (eds.) COCOON 2011. LNCS, vol. 6842, pp. 391–401. Springer, Heidelberg (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Zhang, Y., Chin, F.Y.L., Ting, HF. (2012). Online Pricing for Multi-type of Items. In: Snoeyink, J., Lu, P., Su, K., Wang, L. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 7285. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29700-7_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-29700-7_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29699-4
Online ISBN: 978-3-642-29700-7
eBook Packages: Computer ScienceComputer Science (R0)