The selection of products and prices offered by a firm significantly impacts its profits. Existing approaches do not provide flexible models that capture the joint effect of assortment and price. We propose a nonparametric framework in which each customer is represented by a particular price threshold and a particular preference list over the alternatives. The customers follow a two-stage choice process; they consider the set of products with prices less than the threshold and choose the most

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... eferred product from the set considered. We develop a tractable nonparametric expectation-maximization (EM) algorithm to fit the model to the aggregate transaction data and design an efficient algorithm to determine the profit-maximizing combination of offer set and price. We also identify classes of pricing structures of increasing complexity, which determine the computational complexity of the estimation and decision problems. Our pricing structures are naturally expressed as business constraints, allowing a manager to trade off pricing flexibility with computational burden. Parametric models are parsimonious and therefore computationally tractable. However, the choice structures must be pre-specified, increasing the risk of model misspecification and leading to inaccuracies in decision making. The alternative is to adopt a nonparametric approach, which removes the need for explicit specification of choice structures and instead 'learns' the appropriate structure from data 1 . However, the lack of parsimonious structures makes model estimation, and particularly optimization, computationally difficult. Existing work (e.g., Farias and Jagabathula [15] , Haensel and Koole [18] , and van Ryzin and Vulcano [38]) proposes computationally efficient techniques to estimate model parameters when data consist of only assortment changes and not price changes. The literature is, however, silent on capturing the joint assortment and price changes and solving the joint assortment and price decision problem using nonparametric approaches. The key contribution of this paper is a nonparametric approach for joint assortment and price optimization. Despite its flexibility, our model allows for both tractable estimation and optimization. Our framework allows managers to make a continuous trade-off between decision complexity and computational burden. We follow the general model, fit, and optimize procedure described above. We focus on a canonical retailer selling a universe of n products from a specific category or sub-category. There are frequent changes in prices and more frequent changes in the offer sets, either due to stock-outs or deliberate screening, which is common in the online environment. Product features, other than prices, remain fixed. The retailer has collected historical data in the form of sales transactions, product availabilities, and offered prices. The retailer must utilize available data to determine the assortment and price combination that maximizes the expected revenue. The above setting is broad and includes the classical revenue management setting for airlines, hotels, and cruises, in which the available bookings and prices change frequently. Overview of our approach. Model. We extend the prevailing nonparametric rank-based choice model to capture the impact of price changes on demand. In a rank-based choice model, customers make choices from an offer set according to a preference list so that if the most preferred option is unavailable, they go down the list to pick an available option, as long as it is preferred over the no-purchase option. We extend the rank-based model by supposing that customers follow a two-stage procedure. In the first stage, the customer forms a consideration set by selecting the subset of products whose prices are less than or equal to a price threshold. In the second stage, she chooses the most preferred product from the chosen consideration set. Existing literature allows the customer to form a consideration set by applying broader threshold-based screening rules Jagabathula and Rusmevichientong: Nonparametric Joint Assortment and Price Model