Market Pricing for Matroid Rank Valuations

Authors Kristóf Bérczi , Naonori Kakimura , Yusuke Kobayashi

Thumbnail PDF


  • Filesize: 462 kB
  • 15 pages

Document Identifiers

Author Details

Kristóf Bérczi
  • MTA-ELTE Egerváry Research Group, Department of Operations Research, Eötvös Loránd University, Budapest, Hungary
Naonori Kakimura
  • Department of Mathematics, Faculty of Science and Technology, Keio University, Yokohama, Japan
Yusuke Kobayashi
  • Research Institute for Mathematical Sciences, Kyoto University, Japan

Cite AsGet BibTex

Kristóf Bérczi, Naonori Kakimura, and Yusuke Kobayashi. Market Pricing for Matroid Rank Valuations. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


In this paper, we study the problem of maximizing social welfare in combinatorial markets through pricing schemes. We consider the existence of prices that are capable to achieve optimal social welfare without a central tie-breaking coordinator. In the case of two buyers with matroid rank valuations, we give polynomial-time algorithms that always find such prices when one of the matroids is a simple partition matroid or both matroids are strongly base orderable. This result partially answers a question raised by Düetting and Végh in 2017. We further formalize a weighted variant of the conjecture of Düetting and Végh, and show that the weighted variant can be reduced to the unweighted one based on the weight-splitting theorem for weighted matroid intersection by Frank. We also show that a similar reduction technique works for M^♮-concave functions, or equivalently, for gross substitutes functions.

Subject Classification

ACM Subject Classification
  • Theory of computation → Algorithmic mechanism design
  • Theory of computation → Computational pricing and auctions
  • Pricing schemes
  • Walrasian equilibrium
  • gross substitutes valuations
  • matroid rank functions


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


  1. Kristóf Bérczi, Naonori Kakimura, and Yusuke Kobayashi. Market pricing for matroid rank valuations. arXiv preprint, 2020. URL:
  2. Ben Berger, Alon Eden, and Michal Feldman. On the power and limits of dynamic pricing in combinatorial markets. arXiv preprint, 2020. URL:
  3. Liad Blumrosen and Thomas Holenstein. Posted prices vs. negotiations: an asymptotic analysis. EC, 10:1386790-1386801, 2008. Google Scholar
  4. Shuchi Chawla, Jason D. Hartline, David L. Malec, and Balasubramanian Sivan. Multi-parameter mechanism design and sequential posted pricing. In Proceedings of the Forty-Second ACM Symposium on Theory of Computing, pages 311-320, 2010. Google Scholar
  5. Shuchi Chawla, David L. Malec, and Balasubramanian Sivan. The power of randomness in Bayesian optimal mechanism design. In Proceedings of the 11th ACM Conference on Electronic Commerce, pages 149-158, 2010. Google Scholar
  6. Shuchi Chawla, J Benjamin Miller, and Yifeng Teng. Pricing for online resource allocation: intervals and paths. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1962-1981. SIAM, 2019. Google Scholar
  7. Edward H Clarke. Multipart pricing of public goods. Public choice, pages 17-33, 1971. Google Scholar
  8. Vincent Cohen-Addad, Alon Eden, Michal Feldman, and Amos Fiat. The invisible hand of dynamic market pricing. In Proceedings of the 2016 ACM Conference on Economics and Computation, pages 383-400, 2016. Google Scholar
  9. Joan Davies and Colin McDiarmid. Disjoint common transversals and exchange structures. Journal of the London Mathematical Society, 2(1):55-62, 1976. Google Scholar
  10. Paul Dütting, Michal Feldman, Thomas Kesselheim, and Brendan Lucier. Posted prices, smoothness, and combinatorial prophet inequalities. arXiv preprint, 2016. URL:
  11. Paul Dütting, Michal Feldman, Thomas Kesselheim, and Brendan Lucier. Prophet inequalities made easy: Stochastic optimization by pricing non-stochastic inputs. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 540-551. IEEE, 2017. Google Scholar
  12. Paul Dütting and Lśzló A. Végh. Private Communication, 2017. Google Scholar
  13. Alon Eden, Uriel Feige, and Michal Feldman. Max-min greedy matching. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019), 145:7, 2019. Google Scholar
  14. Jack Edmonds. Matroids and the greedy algorithm. Mathematical Programming, 1(1):127-136, 1971. Google Scholar
  15. Tomer Ezra, Michal Feldman, Tim Roughgarden, and Warut Suksompong. Pricing multi-unit markets. In International Conference on Web and Internet Economics, pages 140-153. Springer, 2018. Google Scholar
  16. Michal Feldman, Nick Gravin, and Brendan Lucier. Combinatorial auctions via posted prices. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 123-135. SIAM, 2014. Google Scholar
  17. Michal Feldman, Nick Gravin, and Brendan Lucier. Combinatorial Walrasian equilibrium. SIAM Journal on Computing, 45(1):29-48, 2016. Google Scholar
  18. András Frank. A weighted matroid intersection algorithm. Journal of Algorithms, 2(4):328-336, 1981. Google Scholar
  19. Theodore Groves. Incentives in teams. Econometrica: Journal of the Econometric Society, pages 617-631, 1973. Google Scholar
  20. Faruk Gul and Ennio Stacchetti. Walrasian equilibrium with gross substitutes. Journal of Economic theory, 87(1):95-124, 1999. Google Scholar
  21. Justin Hsu, Jamie Morgenstern, Ryan Rogers, Aaron Roth, and Rakesh Vohra. Do prices coordinate markets? In Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, pages 440-453, 2016. Google Scholar
  22. Rosa Huang and Gian-Carlo Rota. On the relations of various conjectures on latin squares and straightening coefficients. Discrete Mathematics, 128(1-3):225-236, 1994. Google Scholar
  23. Alexander S. Kelso Jr and Vincent P. Crawford. Job matching, coalition formation, and gross substitutes. Econometrica: Journal of the Econometric Society, pages 1483-1504, 1982. Google Scholar
  24. Stein Krogdahl. A combinatorial base for some optimal matroid intersection algorithms. Technical Report STAN-CS-74-468, Computer Science Department, Stanford University, Stanford, CA, U.S., 1974. Google Scholar
  25. Stein Krogdahl. A combinatorial proof for a weighted matroid intersection algorithm. Technical Report Computer Science Report 17, Institute of Mathematical and Physical Sciences, University of Tromso, Tromso, Norway, 1976. Google Scholar
  26. Stein Krogdahl. The dependence graph for bases in matroids. Discrete Mathematics, 19(1):47-59, 1977. Google Scholar
  27. Noam Nisan and Ilya Segal. The communication requirements of efficient allocations and supporting prices. Journal of Economic Theory, 129(1):192-224, 2006. Google Scholar
  28. Hirokazu Nishimura and Susumu Kuroda. A Lost Mathematician, Takeo Nakasawa: The Forgotten Father of Matroid Theory. Springer Science & Business Media, 2009. Google Scholar
  29. James Oxley. Matroid Theory. Oxford University Press, 2011. Google Scholar
  30. Alexander Schrijver. Combinatorial Optimization: Polyhedra and Efficiency, volume 24. Springer Science & Business Media, 2003. Google Scholar
  31. William Vickrey. Counterspeculation, auctions, and competitive sealed tenders. The Journal of Finance, 16(1):8-37, 1961. Google Scholar
  32. Léon Walras. Éléments d'économie politique pure, ou, Théorie de la richesse sociale. F. Rouge, 1896. Google Scholar
  33. Hassler Whitney. On the abstract properties of linear dependence. In Hassler Whitney Collected Papers, pages 147-171. Springer, 1992. 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