Private Private Information in Second-Price Auction

TL;DR

This paper investigates how private private information structures in second-price auctions affect revenue and surplus, showing that optimal outcomes are achievable but may lack stability, with specific conditions for near-optimal welfare.

Contribution

It demonstrates that private private information can implement optimal outcomes with equilibrium, characterizes conditions for near-maximal welfare, and fully characterizes achievable surplus-revenue pairs.

Findings

Optimal efficient outcomes can be implemented with private private information.

Equilibrium may not be stable under private private information.

Conditions are identified where bidder surplus approaches maximum welfare.

Abstract

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.