Opaque sales are a selling mechanism in which a seller offers multiple products but the buyer does not know which specific product they will receive until after the purchase. This mechanism is widely used in the tourism industry, such as staycation packages with undisclosed hotel options or tours to different destinations, and in e-commerce through mystery boxes. In many such settings, buyers operate under a unit demand constraint, meaning they derive benefit from only one item despite multiple options being available. In this paper, we formulate and solve a multi-item mechanism design problem for opaque sales. Our goal is to identify a distribution-free mechanism that maximizes the competitive ratio, defined as the ratio between the revenue generated by the mechanism and the maximum revenue attainable with full knowledge of the buyer's valuations. Despite the problem's infinite dimensionality, we show that the optimal mechanism admits a semi-analytical form, characterized by the solution of an auxiliary convex optimization problem whose size scales linearly with the number of items. Furthermore, we demonstrate that this mechanism can be equivalently implemented through a menu of infinite level-access lotteries, where the buyer pays an amount to access a randomly assigned subset of items, with higher payments granting access to a larger selection. We then analyze the impact of menu size limitations on the competitive ratio and conclude with a prototypical example illustrating our approach.

We study a robust auction design problem with a minimax regret objective, in which a seller seeks a mechanism for selling multiple items to multiple bidders with additive values. The seller knows that the bidders’ values range over a box uncertainty set but has no information on their probability distribution. The robust auction design model we study requires no distributional information except for upper bounds on the bidders’ values for each item. This model is relevant if there is no trustworthy distributional information or if any distributional information is costly or time-consuming to acquire. We propose a mechanism that sells each item separately via a second price auction with a random reserve price and prove that this mechanism is optimal using duality techniques from robust optimization. We then interpret the auction design problem as a zero-sum game between the seller, who chooses a mechanism, and a fictitious adversary or “nature,” who chooses the bidders’ values from within the uncertainty set with the aim to maximize the seller’s regret. We characterize the Nash equilibrium of this game analytically when the bidders are symmetric. The Nash strategy of the seller coincides with the optimal separable second price auction, whereas the Nash strategy of nature is mixed and constitutes a probability distribution on the uncertainty set under which each bidder’s values for the items are comonotonic. We also study a restricted auction design problem over deterministic mechanisms. In this setting, we characterize the suboptimality of a separable second price auction with deterministic reserve prices and show that this auction becomes optimal if the bidders are symmetric. The optimal mechanism is derived in closed form and can easily be implemented by practitioners. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Funding: This research was funded by the Swiss National Science Foundation [Grant BSCGI0_157733] and by the Ministry of Education, Singapore, under its Academic Research Fund Tier 2 [Grant MOE-T2EP20222-0003]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0275 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0275 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .