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.

Given data on choices made by consumers for different assortments, a key challenge is to develop parsimonious models that describe and predict consumer choice behavior. One such choice model is the marginal distribution model (MDM), which requires only the specification of the marginal distributions of the random utilities of the alternatives to explain choice data.

We study the bundle size pricing (BSP) problem where a monopolist sells bundles of products to customers, and the price of each bundle depends only on the size (number of items) of the bundle. Although this pricing mechanism is attractive in practice, finding optimal bundle prices is difficult since it involves characterizing distributions of the maximum partial sums of order statistics. In this paper, we propose to solve the BSP problem under a class of choice model using only the first and second moments of customer valuations. We show that the BSP problem under this model is convex and can be efficiently solved using off-the-shelf solvers. Our approach is flexible in optimizing any given bundle sizes, and numerical results show that it performs very well compared with state-of-the-art heuristics.

In this paper, we consider online learning with non-convex loss functions. Similar to Besbes et al. [2015] we apply non-stationary regret as the performance metric. In particular, we study the regret bounds under different assumptions on the information available regarding the loss functions. When the gradient of the loss function at the decision point is available, we propose an online normalized gradient descent algorithm (ONGD) to solve the online learning problem. In another situation, when only the value of the loss function is available, we propose a bandit online normalized gradient descent algorithm (BONGD). Under a condition to be called weak pseudo-convexity (WPC), we show that both algorithms achieve a cumulative regret bound of O($\sqrt{T+V_T T}$), where $V_T$ is the total temporal variations of the loss functions, thus establishing a sublinear regret bound for online learning with non-convex loss functions and non-stationary regret measure.