TY - JOUR
T1 - MARKET PRICING FOR MATROID RANK VALUATIONS
AU - Bérczi, Kristóf
AU - Kakimura, Naonori
AU - Kobayashi, Yusuke
N1 - Funding Information:
https://doi.org/10.1137/20M1386335 Funding: This work was supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center located in Kyoto University. The first author was supported by the J\a'nos Bolyai Research Fellowship and the Momentum Programme of the Hungarian Academy of Sciences, grant LP2021-1/2021, and by the Hungarian National Research, Development and Innovation Office--NKFIH, grants FK128673, UNKP-20-5,\' and TKP2020-NKA-06. The second author was supported by JSPS KAKENHI grants JP17K00028, JP18H05291, 20H05795, and 21H03397. The third author was supported by JSPS KAKENHI grants JP18H05291, JP19H05485, and JP20K11692.
Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics.
PY - 2021
Y1 - 2021
N2 - 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 of achieving 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 partition matroid or both matroids are strongly base orderable. This result partially answers a question raised by D\" utting and V\'egh [Private communication, 2017]. We further formalize a weighted variant of the conjecture of D\" utting and V\'egh, 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\natural -concave functions or, equivalently, for gross substitutes functions.
AB - 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 of achieving 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 partition matroid or both matroids are strongly base orderable. This result partially answers a question raised by D\" utting and V\'egh [Private communication, 2017]. We further formalize a weighted variant of the conjecture of D\" utting and V\'egh, 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\natural -concave functions or, equivalently, for gross substitutes functions.
KW - Walrasian equilibrium
KW - gross substitutes valuation
KW - matroid rank function
KW - pricing scheme
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U2 - 10.1137/20M1386335
DO - 10.1137/20M1386335
M3 - Article
AN - SCOPUS:85123415156
SN - 0895-4801
VL - 35
SP - 2662
EP - 2678
JO - SIAM Journal on Discrete Mathematics
JF - SIAM Journal on Discrete Mathematics
IS - 4
ER -