Motivated by recent research on combinatorial markets with endowed valuations by (Babaioff et al., EC 2018) and (Ezra et al., EC 2020), we introduce a notion of perturbation stability in Combinatorial Auctions (CAs) and study the extend to which stability helps in social welfare maximization and mechanism design. A CA is $\gamma\textit{-stable}$ if the optimal solution is resilient to inflation, by a factor of $\gamma \geq 1$, of any bidder's valuation for any single item. On the positive side, we show how to compute efficiently an optimal allocation for 2-stable subadditive valuations and that a Walrasian equilibrium exists for 2-stable submodular valuations. Moreover, we show that a Parallel 2nd Price Auction (P2A) followed by a demand query for each bidder is truthful for general subadditive valuations and results in the optimal allocation for 2-stable submodular valuations. To highlight the challenges behind optimization and mechanism design for stable CAs, we show that a Walrasian equilibrium may not exist for $\gamma$-stable XOS valuations for any $\gamma$, that a polynomial-time approximation scheme does not exist for $(2-\epsilon)$-stable submodular valuations, and that any DSIC mechanism that computes the optimal allocation for stable CAs and does not use demand queries must use exponentially many value queries. We conclude with analyzing the Price of Anarchy of P2A and Parallel 1st Price Auctions (P1A) for CAs with stable submodular and XOS valuations. Our results indicate that the quality of equilibria of simple non-truthful auctions improves only for $\gamma$-stable instances with $\gamma \geq 3$.

中文翻译：

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扰动稳定组合拍卖的机制设计

受 (Babaioff et al., EC 2018) 和 (Ezra et al., EC 2020) 赋予估值的组合市场的近期研究的启发，我们在组合拍卖 (CA) 中引入了扰动稳定性的概念，并研究了稳定性有助于社会福利最大化和机制设计。CA 是 $\gamma\textit{-stable}$，如果最佳解决方案对通货膨胀有弹性，任何投标人对任何单个项目的估值的系数为 $\gamma\geq 1$。从积极的方面来看，我们展示了如何有效地计算 2 稳定子可加性估值的最佳分配，以及 2 稳定子模估值存在瓦尔拉斯均衡。而且，我们表明，并行第二价格拍卖 (P2A) 后跟对每个投标人的需求查询对于一般次加法估值是真实的，并导致 2 稳定子模估值的最佳分配。为了强调稳定 CA 的优化和机制设计背后的挑战，我们表明对于任何 $\gamma$ 的 $\gamma$ 稳定 XOS 估值可能不存在瓦尔拉斯均衡，对于 $\gamma$ 不存在多项式时间近似方案(2-\epsilon)$-stable submodular 估值，并且任何为稳定 CA 计算最佳分配且不使用需求查询的 DSIC 机制都必须使用指数多的值查询。我们最后分析了具有稳定子模块和 XOS 估值的 CA 的 P2A 和并行第一次价格拍卖 (P1A) 的无政府状态的价格。