# A Truthful Cardinal Mechanism for One-Sided Matching

## TL;DR

This paper introduces a truthful mechanism for one-sided matching that elicits full cardinal preferences, achieving better efficiency guarantees than ordinal-based mechanisms by benchmarking against the Nash bargaining solution.

## Contribution

The paper presents a novel mechanism that truthfully captures full cardinal preferences in one-sided matching, surpassing previous ordinal-only mechanisms in performance guarantees.

## Key findings

- Mechanism outperforms all ordinal mechanisms in efficiency.
- Significant improvements in approximation bounds for the Nash bargaining solution.
- Analysis of population monotonicity of Nash bargaining in matching markets.

## Abstract

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.