Private Private Information in Second-Price Auction
For auction designers, this work clarifies the limits of information independence in achieving efficient outcomes, showing that while full surplus extraction is possible in equilibrium, it may require non-strict equilibria and exact efficiency is generally unattainable.
The paper shows that in a second-price auction, a seller can achieve full surplus extraction using a private private information structure where bidders' signals are independent ex ante but valuations are correlated. However, exact maximal welfare is generally impossible, and the paper characterizes conditions for approximate efficiency and achievable surplus-revenue pairs.
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.