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Market Pricing for Matroid Rank Valuations

Authors Kristóf Bérczi , Naonori Kakimura , Yusuke Kobayashi

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Subject Classification

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

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Abstract

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.

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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) https://doi.org/10.4230/LIPIcs.ISAAC.2020.39

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

Funding

This work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University.
  • Bérczi, Kristóf: Supported by the János Bolyai Research Fellowship of the Hungarian Academy of Sciences, by the ÚNKP-19-4 New National Excellence Program of the Ministry for Innovation and Technology, and by projects no. NKFI-128673 and TKP2020-NKA-06 (National Challenges Subprogramme, "Application Domain Specific Highly Reliable IT Solutions") provided by the National Research, Development and Innovation Fund of Hungary.
  • Kakimura, Naonori: Supported by JSPS KAKENHI grant numbers JP17K00028 and JP18H05291, Japan.
  • Kobayashi, Yusuke: Supported by JSPS, KAKENHI grant numbers JP18H05291, JP19H05485, and JP20K11692, Japan.

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