Screening with Limited Information : A Dual Perspective and A Geometric Approach

Consider a seller seeking 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 implies that the extra value of optimizing over more sophisticated mechanisms exactly amounts to the value of eliminating distributional ambiguity under a posted price mechanism. The duality result also 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 further provide a geometric approach to analytically solving the minimax pricing problem (as well as 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. Uniqueness of the worst-case distribution can also be established for some cases