  We consider design of monetary mechanisms for two-sided matching. Mechanisms
in the tradition of the deferred acceptance algorithm, even in variants
incorporating money, tend to focus on the criterion of stability. Instead, in
this work we seek a simple auction-inspired mechanism with social welfare
guarantees. We consider a descending-price mechanism called the Marshallian
Match, proposed (but not analyzed) by Waggoner and Weyl (2019). When all values
for potential matches are positive, we show the Marshallian Match with a
"rebate" payment rule achieves constant price of anarchy. This result extends
to models with costs for acquiring information about one's values, and also to
matching on hypergraphs. With possibly-negative valuations, which capture e.g.
job markets, the problem becomes harder. We introduce notions of approximate
stability and show that they have beneficial welfare implications. However, the
main problem of proving constant factor welfare guarantees in "ex ante stable
equilibrium" remains open.
