Market Pricing for Matroid Rank Valuations

TL;DR

This paper investigates pricing schemes in combinatorial markets with matroid rank valuations, providing polynomial algorithms for certain cases and connecting the problem to matroid intersection and gross substitutes functions.

Contribution

It offers polynomial-time algorithms for finding optimal prices in specific matroid valuation cases and links the problem to weighted matroid intersection and M-natural concavity.

Findings

Polynomial algorithms for two buyers with rank valuations and specific matroid types.

Reduction of weighted to unweighted problems using weight-splitting techniques.

Extension of results to M-natural concave functions and gross substitutes.

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 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 D\"uetting and V\'egh in 2017. We further formalize a weighted variant of the conjecture of D\"uetting 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, gross substitutes functions.