The Power of Simple Menus in Robust Selling Mechanisms

We study a robust selling problem where a seller attempts to sell one item to a buyer but is uncertain about the buyer's valuation distribution. Existing literature shows that robust screening provides a stronger theoretical guarantee than robust deterministic pricing, but at the expense of implementation complexity, as it requires a menu of infinite options. Our research aims to find simple mechanisms to hedge against market ambiguity effectively. We develop a general framework for robust selling mechanisms with a finite menu (or randomization across finite prices). We propose a tractable reformulation that addresses various ambiguity sets of the buyer's valuation distribution, including support, mean, and quantile ambiguity sets. We derive optimal selling mechanisms and corresponding performance ratios for different menu sizes, showing that even a modest menu size can deliver benefits similar to those achieved by the optimal robust mechanism with infinite options, establishing a favorable trade-off between theoretical performance and implementation simplicity. Remarkably, a menu size of merely two can significantly enhance the performance ratio compared to deterministic pricing.