Classic results show that even an arbitrarily small correlation across bidders' information can enable full surplus extraction in auctions and related mechanism design settings. Motivated by this fragility, we study the information independence in a second-price auction when the seller commits to a private private information structure, meaning bidders' signals are independent ex ante, while bidders share a symmetric and arbitrarily correlated prior distribution over their valuations. We first show that the seller optimal efficient outcome with full surplus extraction can always be implemented by a private private information structure that admits a Bayes Nash equilibrium. However, this equilibrium may not be stable. We then further construct a private private information structure that achieves revenue arbitrarily close to maximum welfare while admitting a strict equilibrium. At the same time, we establish an impossibility result: under private private information, in general, bidder surplus cannot achieve maximal welfare exactly, and we characterize necessary and sufficient conditions on the prior distribution under which bidder surplus can be made arbitrarily close to maximal welfare. We then explore which other efficient outcomes are achievable under private private information. Finally, moving beyond private private information, we provide a complete characterization of the achievable pairs (bidder surplus, seller revenue) under general information structures.

翻译：经典结论表明，即使竞拍者信息之间存在任意微小的相关性，在拍卖及相关机制设计场景中也能实现完全剩余提取。受这一脆弱性启发，我们研究了在卖家承诺采用私有隐私信息结构（即竞拍者信号事先独立，但竞拍者对自身估值共享对称且任意相关的先验分布）时第二价格拍卖中的信息独立性。我们首先证明，总能通过一个允许贝叶斯纳什均衡的私有隐私信息结构实现具有完全剩余提取的卖家最优有效结果。然而，该均衡可能存在不稳定性。我们进一步构造了一种私有隐私信息结构，在实现严格均衡的同时使收益任意接近最大福利。同时，我们建立了一个不可能性结果：在私有隐私信息条件下，竞拍者剩余通常无法精确实现最大福利，并刻画了先验分布下竞拍者剩余可被任意接近最大福利的充要条件。随后，我们探讨了私有隐私信息条件下可实现的其它有效结果。最终，在超越私有隐私信息框架的基础上，我们对一般信息结构下可实现的（竞拍者剩余，卖家收益）组合给出了完整刻画。