AI 中文总结
研究非对称估值下稳定匹配机制中稳定性与效率权衡，提出α稳定性概念，刻画其与不对称程度关系，给出多项式时间算法计算α稳定匹配并获效率保证，还证明计算最优α稳定匹配是NP难的。
AI 中文摘要
稳定匹配机制是市场设计的基础，但在稳定性和社会福利最优性之间存在内在矛盾。我们研究了一种自然的稳定性松弛，称为α稳定性，它将参与者建模为仅在潜在改进足够大时才愿意偏离。在α稳定性下，没有一对参与者可以偏离并将其估值提高超过1/α倍，其中α∈(0,1]。我们提供了非对称估值下稳定性-效率权衡的完整刻画。这种权衡取决于不对称程度μ∈(0,1]，它限制了任何一对参与者估值之间的比率。我们的结果表明，放宽稳定性可以显著提高可实现的效率保证。我们还提出了一种多项式时间算法，该算法计算出一个α稳定匹配，实现了最佳的效率保证。对于α≤μ/(μ+1)，我们的算法实现了1-效率；对于更大的α，它计算出一个α稳定匹配，实现了至少(1/α)·μ/(μ+1)的最优社会福利。值得注意的是，我们的算法对最优匹配的值进行膨胀，然后将Gale-Shapley算法应用于修改后的实例。最后，我们表明，计算最优α稳定匹配是NP难的，即使在稳定性稍有松弛的情况下，即对于接近1的α。
英文摘要
Stable matching mechanisms are fundamental to market design but face an inherent tension between stability and social welfare optimality. We study a natural relaxation of stability, termed $α$-stability, which models agents as willing to deviate only when the potential improvement is sufficiently large. Under $α$-stability, no pair of agents can deviate and improve their valuations by more than a factor of $1/α$, with $α\in (0,1]$. We provide a complete characterization of the stability-efficiency tradeoff under asymmetric valuations. This tradeoff depends on the degree of asymmetry $μ\in (0,1]$, which bounds the ratio between agents' valuations for any pair. Our results show that relaxing stability can substantially improve achievable efficiency guarantees. We further present a polynomial-time algorithm that computes an $α$-stable matching attaining the best possible efficiency guarantee. For $α\le μ/(μ+1)$, our algorithm achieves 1-efficiency; for larger $α$, it computes an $α$-stable matching achieving at least $(1/α)\cdot μ/(μ+1)$ of the optimal social welfare. Remarkably, our algorithm inflates the values of an optimal matching and then applies the Gale-Shapley algorithm to the modified instance. Finally, we show that computing an optimal $α$-stable matching is NP-hard, even under slight relaxations of stability, i.e., for $α$ close to 1.
Comments14 pages, 2 figures
