  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.
