Auction Design with Private Priors and Costly Learning

Abstract

We study two-stage auctions in which bidders privately know their valuation priors and learning costs, while the seller jointly designs information acquisition, continuation allocation, and payments. Revenue maximization requires screening learning costs while satisfying cross-prior incentives, potentially distorting both experimentation and continuation allocation. We derive a revenue representation involving allocated posterior values, virtual learning costs, and prior-dependent rents, and characterize cross-prior incentives through difference inequalities. We establish existence of optimal mechanisms for finite experiment menus and arbitrary prior distributions. With finite prior support, bounded finite-category linear programs approximate optimal revenue at a linear rate while preserving exact dynamic incentive compatibility and interim individual rationality. Their dual yields joint optimality conditions for experimentation, allocation, and payments. A two-prior example demonstrates that efficient continuation can be revenue-suboptimal and that cost-contingent experimentation can strictly increase revenue.