Offering Free Upgrades Even Before Stocks Run Out: The Value of Proactive Upgrades

Abstract

Problem definition: When selling multiple products with different feature combinations over a short selling season, a seller often adopts a “reactive” upgrade policy by offering a free upgrade to the next-price-level product only after a customer’s preferred product is out of stock. However, when customers’ preferences are heterogeneous for different feature combinations, some unyielding customers may reject free upgrades. In this paper, we consider a new “proactive” upgrade policy under which the seller may offer free upgrades even before a product is out of stock. Academic/practical relevance: The proactive upgrade policy enables the seller to strategically keep some units of a product in reserve to secure future sales of this product for those unyielding customers. However, the value of the proactive upgrade policy over the traditional reactive upgrade policy remains unclear. Methodology: Given the product choice probability the “upgrade acceptance probability” of each arriving customer, we formulate the problem of how to offer proactive upgrades as a finite horizon dynamic program with an embedded Markov decision process, and we determine the optimal proactive upgrade policy. Results: By exploiting the underlying mathematical structure, we prove that the optimal value function possesses the “anti-multimodularity” property such that the optimal upgrade strategy under the proactive upgrade policy is governed by two state-dependent thresholds: one threshold dictates when to offer proactive upgrades, and the other threshold dictates when to offer reactive upgrades. We also show that the proactive upgrade policy can create significant value over the reactive upgrade policy when the next-price-level product has similar consumer utility or when the price sensitivity is intermediate. Managerial implications: We identify the conditions under which the proactive upgrade policy provides significant value over the traditional reactive upgrade policy. These results can be useful for sellers who sell variants of similar products with different feature combinations to customers with heterogeneous feature preferences.