Algorithmic pricing via virtual valuations

ABSTRACT

Algorithmic pricing is the computational problem that sellers (e.g.,in supermarkets) face when trying to set prices for their items to maximize their profit in the presence of a known demand. Guruswami etal. (SODA, 2005) proposed this problem and gave logarithmic approximations (in the number of consumers) for the unit-demand and single-parameter cases where there is a specific set of consumers and their valuations for bundles are known precisely. Subsequently several versions of the problem have been shown to have poly-logarithmic in approximability. This problem has direct ties to the important open question of better understanding the Bayesian optimal mechanism in multi-parameter agent settings; however, for this purpose approximation factors logarithmic in the number of agents are inadequate. It is therefore of vital interest to consider special cases where constant approximations are possible. We consider the unit-demand variant of this pricing problem. Here a consumer has a valuation for each different item and their value for aset of items is simply the maximum value they have for any item in the set. Instead of considering a set of consumers with precisely known preferences, like the prior algorithmic pricing literature, we assume that the preferences of the consumers are drawn from a distribution. This is the standard assumption in economics; furthermore, the setting of a specific set of customers with specific preferences, which is employed in all of the prior work in algorithmic pricing, is a special case of this general Bayesian pricing problem, where there is a discrete Bayesian distribution for preferences specified by picking one consumer uniformly from the given set of consumers. Notice that the distribution over the valuations for the individual items that this generates is obviously correlated. Our work complements these existing works by considering the case where the consumer's valuations for the different items are independent random variables. Our main result is a constant approximation algorithm for this problem that makes use of an interesting connection between this problem and the concept of virtual valuations from the single-parameter Bayesian optimal mechanism design literature.

REFERENCES

• 1: G. Aggarwal, T. Feder, R. Motwani, and A. Zhu. Algorithms for multi-product pricing. In Proc. of ICALP, 2004.
• 2: Gagan Aggarwal , Jason D. Hartline, Knapsack auctions, Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, p.1083-1092, January 22-26, 2006, Miami, Florida [doi>10.1145/1109557.1109677]
• 3: Maria-Florina Balcan , Avrim Blum, Mechanism Design via Machine Learning, Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, p.605-614, October 23-25, 2005 [doi>10.1109/SFCS.2005.50]
• 4: Maria-Florina Balcan , Avrim Blum, Approximation algorithms and online mechanisms for item pricing, Proceedings of the 7th ACM conference on Electronic commerce, p.29-35, June 11-15, 2006, Ann Arbor, Michigan, USA [doi>10.1145/1134707.1134711]
• 5: P. Briest and P. Krysta. Buying cheap is expensive: Hardness of non-parametric multi-product pricing. In Proc. 18th ACM Symp. on Discrete Algorithms. ACM/SIAM, 2007. To appear.
• 6: J. Bulow and J. Roberts. The simple economics of optimal auctions. The Journal of Political Economy, 97:1060--90, 1989.
• 7: Erik D. Demaine , Mohammad Taghi Hajiaghayi , Uriel Feige , Mohammad R. Salavatipour, Combination can be hard: approximability of the unique coverage problem, Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm, p.162-171, January 22-26, 2006, Miami, Florida [doi>10.1145/1109557.1109577]
• 8: E. Elkind. Designing and learning optimal finite support auctions. In Proc. 18th ACM Symp. on Discrete Algorithms, 2007.
• 9: Venkatesan Guruswami , Jason D. Hartline , Anna R. Karlin , David Kempe , Claire Kenyon , Frank McSherry, On profit-maximizing envy-free pricing, Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms, January 23-25, 2005, Vancouver, British Columbia
• 10: J. Hartline and V. Koltun. Near-Optimal Pricing in Near-Linear Time. In Proc. of 9th Workshop on Algorithms and Data Structures. Springer-Verlag, 2005.
• 11: V. Krishna. Auction Theory. Academic Press, 2002.
• 12: R. Myerson. Optimal auction design. Mathematics of Operations Research, 6:58--73, 1981.

INDEX TERMS

Primary Classification:
  F. Theory of Computation
  F.0 GENERAL

General Terms:
Algorithms, Economics, Theory

Keywords:
approximation algorithms, pricing, virtual valuations