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.

中文翻译：

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一种单边匹配的真实基数机制

我们重新审视了为单边匹配市场设计机制的经过充分研究的问题，其中一组 $n$ 代理需要与一组 $n$ 异构项目匹配。每个代理 $i$ 对于每个项目 $j$ 都有一个值 $v_{i,j}$，这些值是私人信息，如果这样做会导致首选结果，代理可能会误报。确保代理没有误报的动机需要仔细设计匹配机制，文献中提出的机制通过仅引出代理的 \emph{ordinal} 偏好来缓解这个问题，即他们对大多数项目的排名最不喜欢的。然而，这些机制的效率保证仅基于忽略潜在价值的薄弱措施。在本文中，我们通过引入一种机制来真实地引出代理的完整 \emph{cardinal} 偏好，即所有 $v_{i,j}$ 值，从而实现更强的性能保证。我们使用要求更高的 Nash 讨价还价解决方案作为基准来评估该机制的性能，并且我们证明我们的机制显着优于所有有序机制（即使是非真实的）。为了证明我们的近似界限，我们还在匹配市场的背景下研究了纳什讨价还价解决方案的总体单调性，提供了独立感兴趣的上限和下限。我们使用要求更高的 Nash 讨价还价解决方案作为基准来评估该机制的性能，并且我们证明我们的机制显着优于所有有序机制（即使是非真实的）。为了证明我们的近似界限，我们还在匹配市场的背景下研究了纳什讨价还价解决方案的总体单调性，提供了独立感兴趣的上限和下限。我们使用要求更高的 Nash 讨价还价解决方案作为基准来评估该机制的性能，并且我们证明我们的机制显着优于所有有序机制（即使是非真实的）。为了证明我们的近似界限，我们还在匹配市场的背景下研究了纳什讨价还价解决方案的总体单调性，提供了独立感兴趣的上限和下限。