  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 indicates that robust mechanism design provides a stronger
theoretical guarantee than robust deterministic pricing. Meanwhile, the
superior performance of robust mechanism design comes at the expense of
implementation complexity given that the seller offers a menu with an infinite
number of options, each coupled with a lottery and a payment for the buyer's
selection. In view of this, the primary focus of our research is to find simple
selling mechanisms that can effectively hedge against market ambiguity. We show
that a selling mechanism with a small menu size (or limited randomization
across a finite number of prices) is already capable of deriving significant
benefits achieved by the optimal robust mechanism with infinite options. In
particular, we develop a general framework to study the robust selling
mechanism problem where the seller only offers a finite number of options in
the menu. Then we propose a tractable reformulation that addresses a variety of
ambiguity sets of the buyer's valuation distribution. Our formulation further
enables us to characterize the optimal selling mechanisms and the corresponding
competitive ratio for different menu sizes and various ambiguity sets,
including support, mean, and quantile information. In light of the closed-form
competitive ratios associated with different menu sizes, we provide managerial
implications that incorporating a modest menu size already yields a competitive
ratio comparable to the optimal robust mechanism with infinite options, which
establishes a favorable trade-off between theoretical performance and
implementation simplicity. Remarkably, a menu size of merely two can
significantly enhance the competitive ratio, compared to the deterministic
pricing scheme.
