Partition and Prosper: Design and Pricing of Single Bundle
Abstract
Product bundling is a widely used selling strategy among multiproduct firms, yet designing and pricing bundles optimally remain a complex challenge. This paper addresses this fundamental issue by exploring the selection and pricing of a single bundle from a range of products. For instance, in the single bundle with the rest (SBR) framework, the bundle is optimally chosen and priced, whereas the remaining products are offered individually, collectively maximizing profit. We show that the SBR optimization problem under multivariate normal valuations is polynomial-time solvable, provided that the associated covariance matrix can be decomposed into a positive diagonal matrix minus a positive semidefinite matrix of (small) fixed rank. Interestingly, we also show that the subproblem of SBR optimization, where individual product prices are predetermined, is -hard, even if customer valuations are independent. Building on these results, we use a Bayesian optimization (BO) algorithm combined with a conic integer programming reformulation to solve the general SBR optimization problem under correlated valuations. We further show that SBR is a constant approximation to more complex mechanisms in terms of profit performance. Extensive numerical results demonstrate that our BO algorithm has superior performance over baseline heuristics, and SBR achieves significantly higher profit than separate selling and grand bundling. Interestingly, simulation studies reveal that allowing customers the additional option to purchase products either as part of a bundle or individually enhances social welfare (i.e., increases both profit and customer surplus) compared with SBR, separate selling, and grand bundling. These findings highlight the potential benefits of bundle pricing strategies in achieving improved outcomes for both firms and customers.
Funding: H. Sun is supported by the National Natural Science Foundation of China [Grant 72301168] and the Shanghai Pujiang Programme [Grant 23PJC062]. X. Li is supported by the Singapore Ministry of Education [Tier 1 Grant 23-0619-P0001 and Tier 1 Grant 24-0500-A0001] and the National Natural Science Foundation of China [Grant 72171156]. C.-P. Teo is supported by the National Research Foundation Singapore [Grant I2001E0059] and the Singapore Ministry of Education [Grant MOE-2019-T3-1-010]. C.-P. Teo is also supported by the Natural Science Foundation of Chongqing, China [Grant CSTB2022NSCQ-MSX1667].
Supplemental Material: All supplemental materials, including the computer code and data that support the findings of this study, are available at https://doi.org/10.1287/opre.2022.0465.
