Market Pricing for Matroid Rank Valuations

Authors Kristóf Bérczi , Naonori Kakimura , Yusuke Kobayashi



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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

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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

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.

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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References

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