  We revisit the well-studied problem of designing mechanisms for one-sided
matching markets, where a set of $n$ agents needs to be matched to a set of $n$
heterogeneous items. Each agent $i$ has a value $v_{i,j}$ for each item $j$,
and these values are private information that the agents may misreport if doing
so leads to a preferred outcome. Ensuring that the agents have no incentive to
misreport requires a careful design of the matching mechanism, and mechanisms
proposed in the literature mitigate this issue by eliciting only the
\emph{ordinal} preferences of the agents, i.e., their ranking of the items from
most to least preferred. However, the efficiency guarantees of these mechanisms
are based only on weak measures that are oblivious to the underlying values. In
this paper we achieve stronger performance guarantees by introducing a
mechanism that truthfully elicits the full \emph{cardinal} preferences of the
agents, i.e., all of the $v_{i,j}$ values. We evaluate the performance of this
mechanism using the much more demanding Nash bargaining solution as a
benchmark, and we prove that our mechanism significantly outperforms all
ordinal mechanisms (even non-truthful ones). To prove our approximation bounds,
we also study the population monotonicity of the Nash bargaining solution in
the context of matching markets, providing both upper and lower bounds which
are of independent interest.
