Private Private Information in Second-Price Auction

A new paper on arXiv by Boyu Liu, Wei Tang, Zihe Wang, and Shuo Zhang tackles a foundational puzzle in auction theory: how to design auctions that extract maximum bidder surplus without relying on correlated information. Classic results show that even tiny correlations across bidders' signals enable sellers to extract full surplus, but this relies on fragile assumptions. The authors introduce a 'private private information' structure where bidders' signals are independent ex ante, but they share a symmetric, arbitrarily correlated prior over valuations.

The paper proves that such a structure can always implement the seller-optimal efficient outcome with full surplus extraction in a Bayes-Nash equilibrium. However, that equilibrium may not be stable. The authors then construct a stricter equilibrium that achieves revenue arbitrarily close to maximum welfare. Yet they also prove an impossibility result: under private private information, bidder surplus cannot exactly hit maximal welfare in general. They characterize necessary and sufficient conditions on the prior distribution for approximate maximal welfare. The work extends to general information structures, providing a complete characterization of achievable (bidder surplus, seller revenue) pairs. This has implications for mechanism design and digital advertising markets.