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1. The Power of Opaque Products in Pricing

   https://doi.org/10.1287/mnsc.2020.3750  

   Elmachtoub, Adam N.; Hamilton, Michael L. (August 2021, Management Science)

   null (Ed.)

   We study the power of selling opaque products, that is, products where a feature (such as color) is hidden from the customer until after purchase. Opaque products, which are sold with a price discount, have emerged as a powerful vehicle to increase revenue for many online retailers and service providers that offer horizontally differentiated items. In the opaque selling models we consider, all of the items are sold at a single common price alongside opaque products that may correspond to various subsets of the items. We consider two types of customers, risk-neutral ones, who assume they will receive a truly random item of the opaque product, and pessimistic ones, who assume they will receive their least favorite item of the opaque product. We benchmark opaque selling against two common selling strategies: discriminatory pricing, where one explicitly charges different prices for each item, and single pricing, where a single price is charged for all the items. We give a sharp characterization of when opaque selling outperforms discriminatory pricing; namely, this result holds for situations where all customers are pessimistic or the item valuations are supported on two points. In the latter case, we also show that opaque selling with just one opaque product guarantees at least 71.9% of the revenue from discriminatory pricing. We then provide upper bounds on the potential revenue increase from opaque selling strategies over single pricing and describe cases where the increase can be significantly more than that of discriminatory pricing. Finally, we provide pricing algorithms and conduct an extensive numerical study to assess the power of opaque selling for a variety valuation distributions and model extensions. This paper was accepted by Gabriel Weintraub, revenue management and market analytics. 

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2. Technical Note—Static Pricing: Universal Guarantees for Reusable Resources

   https://doi.org/10.1287/opre.2020.2054  

   Besbes, Omar; Elmachtoub, Adam N.; Sun, Yunjie (March 2022, Operations Research)

   We consider a fundamental pricing model in which a fixed number of units of a reusable resource are used to serve customers. Customers arrive to the system according to a stochastic process and, upon arrival, decide whether to purchase the service, depending on their willingness to pay and the current price. The service time during which the resource is used by the customer is stochastic, and the firm may incur a service cost. This model represents various markets for reusable resources, such as cloud computing, shared vehicles, rotable parts, and hotel rooms. In the present paper, we analyze this pricing problem when the firm attempts to maximize a weighted combination of three central metrics: profit, market share, and service level. Under Poisson arrivals, exponential service times, and standard assumptions on the willingness-to-pay distribution, we establish a series of results that characterize the performance of static pricing in such environments. In particular, although an optimal policy is fully dynamic in such a context, we prove that a static pricing policy simultaneously guarantees 78.9% of the profit, market share, and service level from the optimal policy. Notably, this result holds for any service rate and number of units the firm operates. Our proof technique relies on a judicious construction of a static price that is derived directly from the optimal dynamic pricing policy. In the special case in which there are two units and the induced demand is linear, we also prove that the static policy guarantees 95.5% of the profit from the optimal policy. Our numerical findings on a large test bed of instances suggest that the latter result is quite indicative of the profit obtained by the static pricing policy across all parameters. 

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3. Robust Auctions for Revenue via Enhanced Competition

   https://doi.org/10.1287/opre.2019.1929  

   Roughgarden, Tim; Talgam-Cohen, Inbal; Yan, Qiqi (July 2020, Operations Research)

   Most results in revenue-maximizing mechanism design hinge on “getting the price right”—selling goods to bidders at prices low enough to encourage a sale but high enough to garner nontrivial revenue. This approach is difficult to implement when the seller has little or no a priori information about bidder valuations or when the setting is sufficiently complex, such as matching markets with heterogeneous goods. In this paper, we apply a robust approach to designing auctions for revenue. Instead of relying on prior knowledge regarding bidder valuations, we “let the market do the work” and let prices emerge from competition for scarce goods. We analyze the revenue guarantees of one of the simplest imaginable implementations of this idea: first, we enhance competition in the market by increasing demand (or alternatively, by limiting supply), and second, we run a standard second price (Vickrey) auction. In their renowned work from 1996 , Bulow and Klemperer [Bulow J, Klemperer P (1996) Auctions vs. negotiations. Amer. Econom. Rev. 86(1):180–194.] apply this method to markets with single goods. As our main result, we give the first application beyond single-parameter settings, proving that, simultaneously for many valuation distributions, this method achieves expected revenue at least as good as the optimal revenue in the original market. Our robust and simple approach provides a handle on the elusive optimal revenue in multiitem matching markets and shows when the use of welfare-maximizing Vickrey auctions is justified, even if revenue is a priority. By establishing quantitative tradeoffs, our work provides guidelines for a seller in choosing among two different revenue-extracting strategies: sophisticated pricing based on market research or advertising to draw additional bidders. 

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4. Dynamic Pricing of Wireless Internet Based on Usage and Stochastically Changing Capacity

   Batur, Demet; Ryan, Jennifer K.; Zhao, Zhongyuan; Vuran, Mehmet C. (January 2019, Manufacturing & service operations management)

   Problem definition: Inspired by new developments in dynamic spectrum access, we study the dynamic pricing of wireless Internet access when demand and capacity (bandwidth) are stochastic. Academic/practical relevance: The demand for wireless Internet access has increased enormously. However, the spectrum available to wireless service providers is limited. The industry has, thus, altered conventional license-based spectrum access policies through unlicensed spectrum operations. The additional spectrum obtained through these operations has stochastic capacity. Thus, the pricing of this service by the service provider has novel challenges. The problem considered in this paper is, therefore, of high practical relevance and new to the academic literature. Methodology: We study this pricing problem using a Markov decision process model in which customers are posted dynamic prices based on their bandwidth requirement and the available capacity. Results: We characterize the structure of the optimal pricing policy as a function of the system state and of the input parameters. Because it is impossible to solve this problem for practically large state spaces, we propose a heuristic dynamic pricing policy that performs very well, particularly when the ratio of capacity to demand rate is low. Managerial implications: We demonstrate the value of using a dynamic heuristic pricing policy compared with the myopic and optimal static policies. The previous literature has studied similar systems with fixed capacity and has characterized conditions under which myopic policies perform well. In contrast, our setting has dynamic (stochastic) capacity, and we find that identifying good state-dependent heuristic pricing policies is of greater importance. Our heuristic policy is computationally more tractable and easier to implement than the optimal dynamic and static pricing policies. It also provides a significant performance improvement relative to the myopic and optimal static policies when capacity is scarce, a condition that holds for the practical setting that motivated this research. 

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5. The I.I.D. Prophet Inequality with Limited Flexibility

   https://doi.org/10.1287/moor.2022.0345  

   Perez-Salazar, Sebastian; Singh, Mohit; Toriello, Alejandro (January 2025, Mathematics of Operations Research)

   In online sales, sellers usually offer each potential buyer a posted price in a take-it-or-leave fashion. Buyers can sometimes see posted prices faced by other buyers, and changing the price frequently could be considered unfair. The literature on posted-price mechanisms and prophet inequality problems has studied the two extremes of pricing policies, the fixed-price policy and fully dynamic pricing. The former is suboptimal in revenue but is perceived as fairer than the latter. This work examines the middle situation, where there are at most k distinct prices over the selling horizon. Using the framework of prophet inequalities with independent and identically distributed random variables, we propose a new prophet inequality for strategies that use at most k thresholds. We present asymptotic results in k and results for small values of k. For k = 2 prices, we show an improvement of at least 11% over the best fixed-price solution. Moreover, k = 5 prices suffice to guarantee almost 99% of the approximation factor obtained by a fully dynamic policy that uses an arbitrary number of prices. From a technical standpoint, we use an infinite-dimensional linear program in our analysis; this formulation could be of independent interest to other online selection problems. 

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