We initiate the study of distortion in stable matching. Concretely, we aim to design algorithms that have limited access to the agents' cardinal preferences and compute stable matchings of high quality with respect to some aggregate objective, e.g., the social welfare. Our first result is a strong impossibility: the classic Deferred Acceptance (DA) algorithm of Gale and Shapley [1962], as well as any deterministic algorithm that relies solely on ordinal information about the agents' preferences, has unbounded distortion. To circumvent this impossibility, we consider algorithms that either (a) use randomization or (b) perform a small number of value queries to the agents' cardinal preferences. In the former case, we prove that a simple randomized version of the DA algorithm achieves a distortion of $2$, and that this is optimal among all randomized stable matching algorithms. For the latter case, we prove that the same bound of $2$ can be achieved with only $1$ query per agent, and improving upon this bound requires $Ω(\log n)$ queries per agent. We further show that this query bound is asymptotically optimal for any constant approximation: for any $\varepsilon >0$, there exists an algorithm which uses $O(\log n /\varepsilon^2)$ queries, and achieves a distortion of $1+\varepsilon$. Moreover, under natural structural restrictions on the instances of the problem, we provide improved upper bounds on the number of queries required for a $(1+\varepsilon)$-approximation. We complement our main findings above with theoretical and empirical results on the average-case performance of stable matching algorithms, when the preferences of the agents are drawn i.i.d. from a given distribution.

翻译：我们首次对稳定匹配中的失真问题展开研究。具体而言，我们的目标是设计算法，这些算法仅能有限地访问代理人的基数偏好，并计算出在某种聚合目标（例如社会福利）下高质量的稳定匹配。我们的第一个结果是强有力的不可能性定理：经典的Gale和Shapley [1962]的延迟接受（DA）算法，以及任何仅依赖代理人偏好序数信息的确定性算法，都具有无界失真。为了规避这一不可能性，我们考虑以下两类算法：(a) 使用随机化的算法，或 (b) 对代理人的基数偏好执行少量价值查询。对于前者，我们证明了一个简单的随机化DA算法可以达到$2$的失真度，并且这在所有随机化稳定匹配算法中是最优的。对于后者，我们证明每个代理人仅需$1$次查询即可达到相同的$2$失真度界限，而要改进这一界限则需要对每个代理人进行$Ω(\log n)$次查询。我们进一步证明，对于任何常数近似，这个查询界限是渐近最优的：对于任意$\varepsilon >0$，存在一种算法，它使用$O(\log n /\varepsilon^2)$次查询，并实现$1+\varepsilon$的失真度。此外，在问题实例具有自然结构限制的条件下，我们为达到$(1+\varepsilon)$近似所需的查询次数提供了改进的上界。除了上述主要发现，我们还通过理论和实证结果，补充研究了当代理人的偏好从给定分布中独立同分布抽取时，稳定匹配算法的平均性能。