Screening with Limited Information : The Minimax Theorem and A Geometric Approach

A seller seeks a selling mechanism to maximize the worst-case revenue obtained from a buyer whose valuation distribution lies in a certain ambiguity set. For a generic convex ambiguity set, we show via the minimax theorem that strong duality holds between the problem of finding the optimal robust mechanism and a minimax pricing problem where the adversary first chooses a worst-case distribution and then the seller decides the best posted price mechanism. This observation connects prior literature that separately studies the primal (robust mechanism design) and problems related to the dual (e.g., robust pricing, buyer-optimal pricing and personalized pricing). We provide a geometric approach to analytically solving the minimax pricing problem (and the robust pricing problem) for several important ambiguity sets such as the ones with mean and various dispersion measures, and with the Wasserstein metric. The solutions are then used to construct the optimal robust mechanism and to compare with the solutions to the robust pricing problem.