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.
ASJC Scopus subject areas
- 数学 (全般)