Pricing Problems with Buyer Preselection

Authors Vittorio Bilò, Michele Flammini, Gianpiero Monaco, Luca Moscardelli

Thumbnail PDF


  • Filesize: 0.49 MB
  • 16 pages

Document Identifiers

Author Details

Vittorio Bilò
  • Univ. of Salento, Lecce, Italy
Michele Flammini
  • GSSI Institute, L'Aquila, Italy & Univ. of L'Aquila, L'Aquila, Italy
Gianpiero Monaco
  • Univ. of L'Aquila, L'Aquila Italy
Luca Moscardelli
  • Univ. of Chieti-Pescara, Pescara, Italy

Cite AsGet BibTex

Vittorio Bilò, Michele Flammini, Gianpiero Monaco, and Luca Moscardelli. Pricing Problems with Buyer Preselection. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 47:1-47:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


We investigate the problem of preselecting a subset of buyers participating in a market so as to optimize the performance of stable outcomes. We consider four scenarios arising from the combination of two stability notions, item and bundle envy-freeness, with the two classical objective functions, i.e., the social welfare and the seller's revenue. When adopting the notion of item envy-freeness, we prove that, for both the two objective functions, the problem cannot be approximated within n^(1-epsilon) for any epsilon >0, and provide tight or nearly tight approximation algorithms. We also prove that maximizing the seller's revenue is NP-hard even for a single buyer, thus closing an open question. Under bundle envy-freeness, instead, we show how to transform in polynomial time any stable outcome for a market involving only a subset of buyers to a stable one for the whole market without worsening its performance, both for the social welfare and the seller's revenue. Finally, we consider multi-unit markets, where all items are of the same type and are assigned the same price. For this specific case, we show that buyer preselection can improve the performance of stable outcomes in all of the four considered scenarios, and we design corresponding approximation algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational pricing and auctions
  • Pricing problems
  • Envy-freeness
  • Revenue maximization
  • Social Welfare maximization


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Noga Alon, Yishay Mansour, and Moshe Tennenholtz. Differential pricing with inequity aversion in social networks. In Proc. of EC, pages 9-24, 2013. Google Scholar
  2. Georgios Amanatidis, Evangelos Markakis, and Krzysztof Sornat. Inequity aversion pricing over social networks: Approximation algorithms and hardness results. In Proc. of MFCS, pages 9:1-9:13, 2016. Google Scholar
  3. E. Anshelevich, K. Kar, and S. Sekar. Envy-free pricing in large markets: Approximating revenue and welfare. In Proc. of ICALP, pages 52-64. Springer, 2015. Google Scholar
  4. A. Archer, C. H. Papadimitriou, K. Talwar, and É. Tardos. An approximate truthful mechanism for combinatorial auctions with single parameter agents. Internet Mathematics, 1(2):129-150, 2003. Google Scholar
  5. M. F. Balcan, A. Blum, and Y. Mansour. Item pricing for revenue maximization. In Proc. of EC, pages 50-59, 2008. Google Scholar
  6. S. Bikhchandani and J. W. Mamer. Competitive equilibrium in an exchange economy with indivisibilities. Journal of Economic Theory, 74(2):386-413, 1997. Google Scholar
  7. V. Bilò, M. Flammini, and G. Monaco. Approximating the revenue maximization problem with sharp demands. Theoretical Computer Science, 662:9-30, 2017. Google Scholar
  8. V. Bilò, M. Flammini, G. Monaco, and L. Moscardelli. On the impact of buyers preselection in pricing problems. In Proc. of AAMAS, 2018. Google Scholar
  9. P. Briest. Uniform budgets and the envy-free pricing problem. In Proc. of ICALP, pages 808-819. Springer, 2008. Google Scholar
  10. P. Briest and P. Krysta. Single-minded unlimited supply pricing on sparse instances. In Proc. of SODA, pages 1093-1102. ACM Press, 2006. Google Scholar
  11. P. Chalermsook, J. Chuzhoy, S. Kannan, and S. Khanna. Improved hardness results for profit maximization pricing problems with unlimited supply. In Proc. of APPROX, pages 73-84. Springer, 2012. Google Scholar
  12. P. Chalermsook, B. Laekhanukit, and D. Nanongkai. Graph products revisited: Tight approximation hardness of induced matching, poset dimension and more. In Proc. of SODA, pages 1557-1576. ACM Press, 2013. Google Scholar
  13. P. Chalermsook, B. Laekhanukit, and D. Nanongkai. Independent set, induced matching, and pricing: Connections and tight (subexponential time) approximation hardnesses. In Proc. of FOCS, pages 370-379. IEEE Computer Society, 2013. Google Scholar
  14. N. Chen and X. Deng. Envy-free pricing in multi-item markets. In Proc. of ICALP, pages 418-429. Springer, 2010. Google Scholar
  15. N. Chen, X. Deng, P. W. Goldberg, and J. Zhang. On revenue maximization with sharp multi-unit demands. Journal of Combinatorial Optimization, 31(3):1174-1205, 2016. Google Scholar
  16. N. Chen, A. Ghosh, and S. Vassilvitskii. Optimal envy-free pricing with metric substitutability. SIAM Journal on Computing, 40(3):623-645, 2011. Google Scholar
  17. Ning Chen and Atri Rudra. Walrasian equilibrium: Hardness, approximations and tractable instances. Algorithmica, 52(1):44-64, 2008. Google Scholar
  18. M. Cheung and C. Swamy. Approximation algorithms for single-minded envy-free profit-maximization problems with limited supply. In Proc. of FOCS, pages 35-44. IEEE Computer Society, 2008. Google Scholar
  19. E. H. Clarke. Multipart pricing of public goods. Public Choice, 11:17-33, 1971. Google Scholar
  20. E. D. Demaine, U. Feige, M. Hajiaghayi, and M. R. Salavatipour. Combination can be hard: Approximability of the unique coverage problem. SIAM Journal on Computing, 38(4):1464-1483, 2008. Google Scholar
  21. K. Dudzinski and S. Walukiewicz. Exact methods for the knapsack problem and its generalizations. European Journal of Operational Research, 28(1):3-21, 1987. Google Scholar
  22. M. Feldman, A. Fiat, S. Leonardi, and P. Sankowski. Revenue maximizing envy-free multi-unit auctions with budgets. In Proc. of EC, pages 532-549. ACM Press, 2012. Google Scholar
  23. M. Feldman, N. Gravin, and B. Lucier. Combinatorial walrasian equilibrium. SIAM Journal on Computing, 45(1):29-48, 2016. Google Scholar
  24. M. Flammini, M. Mauro, and M. Tonelli. On social envy-freeness in multi-unit markets. In Proc. of AAAI, 2018. Google Scholar
  25. D. Foley. Resource allocation and the public sector. Yale Economic Essays, 7:45-98, 1967. Google Scholar
  26. T. Groves. Incentives in teams. Econometrica, 41:617-631, 1973. Google Scholar
  27. F. Gul and E. Stacchetti. Walrasian equilibrium with gross substitutes. Journal of Economic Theory, 87:95-124, 1999. Google Scholar
  28. V. Guruswami, J. D. Hartline, A. R. Karlin, D. Kempe, C. Kenyon, and F. McSherry. On profit-maximizing envy-free pricing. In Proc. of SODA, pages 1164-1173. ACM Press, 2005. Google Scholar
  29. J. Hartline and Q. Yan. Envy, truth, and profit. In Proc. of EC, pages 243-252. ACM Press, 2011. Google Scholar
  30. E. L. Lawler. Fast approximation algorithms for knapsack problems. Mathematics of Operations Research, 44(4):339-356, 1979. Google Scholar
  31. G. Monaco, P. Sankowski, and Q. Zhang. Revenue maximization envy-free pricing for homogeneous resources. In Proc. of IJCAI, pages 90-96, 2015. Google Scholar
  32. H. R. Varian. Equity, envy, and efficiency. Journal of Economic Theory, 9:63-91, 1974. Google Scholar
  33. W. Vickrey. Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 16:8-37, 1961. Google Scholar
  34. L. Walras. Elements of Pure Economics. Allen and Unwin, 1954. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail