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Convex Optimization for Bundle Size Pricing Problem

Author

Listed:
  • Xiaobo Li

    (Department of Industrial Systems Engineering and Management, National University of Singapore, Singapore 117576; Institute of Operations Research and Analytics, National University of Singapore, Singapore 117576)

  • Hailong Sun

    (Institute of Operations Research and Analytics, National University of Singapore, Singapore 117576)

  • Chung Piaw Teo

    (Department of Analytics and Operations, National University of Singapore, Singapore 117576)

Abstract

We study the bundle size pricing (BSP) problem in which 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 because it involves characterizing distributions of the maximum partial sums of order statistics. In this paper, we propose to solve the BSP problem under a discrete choice model using only the first and second moments of customer valuations. Correlations between valuations of bundles are captured by the covariance matrix. 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 prices for any given bundle size. Numerical results show that it performs very well compared with state-of-the-art heuristics. This provides a unified and efficient approach to solve the BSP problem under various distributions and dimensions.

Suggested Citation

  • Xiaobo Li & Hailong Sun & Chung Piaw Teo, 2022. "Convex Optimization for Bundle Size Pricing Problem," Management Science, INFORMS, vol. 68(2), pages 1095-1106, February.
    • Handle: RePEc:inm:ormnsc:v:68:y:2022:i:2:p:1095-1106
    DOI: 10.1287/mnsc.2021.4148
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    References listed on IDEAS

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    1. Chenghuan Sean Chu & Phillip Leslie & Alan Sorensen, 2011. "Bundle-Size Pricing as an Approximation to Mixed Bundling," American Economic Review, American Economic Association, vol. 101(1), pages 263-303, February.
    2. Shin-yi Wu & Lorin M. Hitt & Pei-yu Chen & G. Anandalingam, 2008. "Customized Bundle Pricing for Information Goods: A Nonlinear Mixed-Integer Programming Approach," Management Science, INFORMS, vol. 54(3), pages 608-622, March.
    3. Jianqing “Fisher” Wu & Mohit Tawarmalani & Karthik N. Kannan, 2019. "Cardinality Bundling with Spence–Mirrlees Reservation Prices," Management Science, INFORMS, vol. 65(4), pages 1891-1908, April.
    4. Hongmin Li & Scott Webster & Nicholas Mason & Karl Kempf, 2019. "Product-Line Pricing Under Discrete Mixed Multinomial Logit Demand," Service Science, INFORMS, vol. 21(1), pages 14-28, January.
    5. A. Michael Spence, 1980. "Multi-Product Quantity-Dependent Prices and Profitability Constraints," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 47(5), pages 821-841.
    6. Gabriel Carroll, 2017. "Robustness and Separation in Multidimensional Screening," Econometrica, Econometric Society, vol. 85, pages 453-488, March.
    7. Ahipaşaoğlu, Selin Damla & Meskarian, Rudabeh & Magnanti, Thomas L. & Natarajan, Karthik, 2015. "Beyond normality: A cross moment-stochastic user equilibrium model," Transportation Research Part B: Methodological, Elsevier, vol. 81(P2), pages 333-354.
    8. Lorin M. Hitt & Pei-yu Chen, 2005. "Bundling with Customer Self-Selection: A Simple Approach to Bundling Low-Marginal-Cost Goods," Management Science, INFORMS, vol. 51(10), pages 1481-1493, October.
    9. Dirk Bergemann & Karl Schlag, 2012. "Robust Monopoly Pricing," World Scientific Book Chapters, in: Robust Mechanism Design The Role of Private Information and Higher Order Beliefs, chapter 13, pages 417-441, World Scientific Publishing Co. Pte. Ltd..
    10. Lingxiu Dong & Panos Kouvelis & Zhongjun Tian, 2009. "Dynamic Pricing and Inventory Control of Substitute Products," Manufacturing & Service Operations Management, INFORMS, vol. 11(2), pages 317-339, December.
    11. Yannis Bakos & Erik Brynjolfsson, 1999. "Bundling Information Goods: Pricing, Profits, and Efficiency," Management Science, INFORMS, vol. 45(12), pages 1613-1630, December.
    12. Karthik Natarajan & Miao Song & Chung-Piaw Teo, 2009. "Persistency Model and Its Applications in Choice Modeling," Management Science, INFORMS, vol. 55(3), pages 453-469, March.
    13. Hongmin Li & Scott Webster, 2017. "Optimal Pricing of Correlated Product Options Under the Paired Combinatorial Logit Model," Operations Research, INFORMS, vol. 65(5), pages 1215-1230, October.
    14. Guang Li & Paat Rusmevichientong & Huseyin Topaloglu, 2015. "The d -Level Nested Logit Model: Assortment and Price Optimization Problems," Operations Research, INFORMS, vol. 63(2), pages 325-342, April.
    15. Train,Kenneth E., 2009. "Discrete Choice Methods with Simulation," Cambridge Books, Cambridge University Press, number 9780521747387.
    16. Woonghee Tim Huh & Hongmin Li, 2015. "Technical Note—Pricing Under the Nested Attraction Model with a Multistage Choice Structure," Operations Research, INFORMS, vol. 63(4), pages 840-850, August.
    17. Hongmin Li & Woonghee Tim Huh, 2011. "Pricing Multiple Products with the Multinomial Logit and Nested Logit Models: Concavity and Implications," Manufacturing & Service Operations Management, INFORMS, vol. 13(4), pages 549-563, October.
    18. R. Preston McAfee & John McMillan & Michael D. Whinston, 1989. "Multiproduct Monopoly, Commodity Bundling, and Correlation of Values," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 104(2), pages 371-383.
    19. Guiyun Feng & Xiaobo Li & Zizhuo Wang, 2017. "Technical Note—On the Relation Between Several Discrete Choice Models," Operations Research, INFORMS, vol. 65(6), pages 1516-1525, December.
    20. Guillermo Gallego & Ruxian Wang, 2014. "Multiproduct Price Optimization and Competition Under the Nested Logit Model with Product-Differentiated Price Sensitivities," Operations Research, INFORMS, vol. 62(2), pages 450-461, April.
    21. William James Adams & Janet L. Yellen, 1976. "Commodity Bundling and the Burden of Monopoly," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 90(3), pages 475-498.
    22. Ward Hanson & Kipp Martin, 1996. "Optimizing Multinomial Logit Profit Functions," Management Science, INFORMS, vol. 42(7), pages 992-1003, July.
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