We study auction design within the widely acclaimed model of interdependent values, introduced by Milgrom and Weber [1982]. In this model, every bidder $i$ has a private signal $s_i$ for the item for sale, and a public valuation function $v_i(s_1,\ldots,s_n)$ which maps every vector of private signals (of all bidders) into a real value. A recent line of work established the existence of approximately-optimal mechanisms within this framework, even in the more challenging scenario where each bidder's valuation function $v_i$ is also private. This body of work has primarily focused on single-item auctions with two natural classes of valuations: those exhibiting submodularity over signals (SOS) and $d$-critical valuations. In this work we advance the state of the art on interdependent values with private valuation functions, with respect to both SOS and $d$-critical valuations. For SOS valuations, we devise a new mechanism that gives an improved approximation bound of $5$ for single-item auctions. This mechanism employs a novel variant of an "eating mechanism", leveraging LP-duality to achieve feasibility with reduced welfare loss. For $d$-critical valuations, we broaden the scope of existing results beyond single-item auctions, introducing a mechanism that gives a $(d+1)$-approximation for any environment with matroid feasibility constraints on the set of agents that can be simultaneously served. Notably, this approximation bound is tight, even with respect to single-item auctions.

翻译：我们在Milgrom和Weber [1982]提出的广受认可的相依估值模型框架内研究拍卖设计。在该模型中，每个投标者i对拍卖品拥有私人信号s_i，以及一个公开的估值函数v_i(s_1,...,s_n)，该函数将所有投标者的信号向量映射为实数值。近期一系列研究证明，在此框架下存在近似最优的机制，即使在每个投标者的估值函数v_i也为私有的更具挑战性的场景中依然成立。这些研究主要聚焦于具有两类自然估值函数的单品拍卖：信号子模性（SOS）估值和d临界估值。在本工作中，我们针对具有私有估值函数的相依估值问题，在SOS估值和d临界估值两方面均取得了最新进展。对于SOS估值，我们设计了一种新机制，在单品拍卖中实现了改进的5倍近似下界。该机制采用了一种新颖的"吃蛋糕机制"变体，利用线性规划对偶性在减少福利损失的同时实现可行性。对于d临界估值，我们将现有结果的应用范围从单品拍卖扩展到更广泛的环境，提出了一种在满足拟阵可行性约束（即可同时服务的投标者集合）的任何环境中实现(d+1)倍近似的机制。值得注意的是，该近似下界是紧的，即使对于单品拍卖也是如此。