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Selling Multiple Items to a Unit-Demand Buyer via Automated Mechanism Design

Author

Listed:
  • Kento Hashimoto
  • Keita Kuwahara
  • Reo Nonaka

Abstract

Finding the optimal (revenue-maximizing) mechanism to sell multiple items has been a prominent and notoriously difficult open problem. Existing work has mainly focused on deriving analytical results tailored to a particular class of problems (for example, Giannakopoulos (2015) and Yang (2023)). The present paper explores the possibility of a generally applicable methodology of the Automated Mechanism Design (AMD). We first employ the deep learning algorithm developed by D\"utting et al. (2023) to numerically solve small-sized problems, and the results are then generalized by educated guesswork and finally rigorously verified through duality. By focusing on a single buyer who can consume one item, our approach leads to two key contributions: establishing a much simpler way to verify the optimality of a wide range of problems and discovering a completely new result about the optimality of grand bundling. First, we show that selling each item at an identical price (or equivalently, selling the grand bundle of all items) is optimal for any number of items when the value distributions belong to a class that includes the uniform distribution as a special case. Different items are allowed to have different distributions. Second, for each number of items, we established necessary and sufficient conditions that $c$ must satisfy for grand bundling to be optimal when the value distribution is uniform over an interval $[c, c + 1]$. This latter model does not satisfy the previously known sufficient conditions for the optimality of grand bundling Haghpanah and Hartline (2021). Our results are in contrast to the only known results for $n$ items (for any $n$), Giannakopoulos (2015) and Daskalakis et al. (2017), which consider a single buyer with additive preferences, where the values of items are narrowly restricted to i.i.d. according to a uniform or exponential distribution.

Suggested Citation

  • Kento Hashimoto & Keita Kuwahara & Reo Nonaka, 2025. "Selling Multiple Items to a Unit-Demand Buyer via Automated Mechanism Design," Papers 2502.10086, arXiv.org, revised Feb 2025.
  • Handle: RePEc:arx:papers:2502.10086
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    References listed on IDEAS

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    1. Pavlov Gregory, 2011. "Optimal Mechanism for Selling Two Goods," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 11(1), pages 1-35, February.
    2. Nima Haghpanah & Jason Hartline, 2021. "When Is Pure Bundling Optimal? [Commodity Bundling and the Burden of Monopoly]," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 88(3), pages 1127-1156.
    3. Palfrey, Thomas R, 1983. "Bundling Decisions by a Multiproduct Monopolist with Incomplete Information," Econometrica, Econometric Society, vol. 51(2), pages 463-483, March.
    4. Rochet, J. C., 1985. "The taxation principle and multi-time Hamilton-Jacobi equations," Journal of Mathematical Economics, Elsevier, vol. 14(2), pages 113-128, April.
    5. Manelli, Alejandro M. & Vincent, Daniel R., 2007. "Multidimensional mechanism design: Revenue maximization and the multiple-good monopoly," Journal of Economic Theory, Elsevier, vol. 137(1), pages 153-185, November.
    6. Roger B. Myerson, 1981. "Optimal Auction Design," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 58-73, February.
    7. Thirumulanathan, D. & Sundaresan, Rajesh & Narahari, Y., 2019. "On optimal mechanisms in the two-item single-buyer unit-demand setting," Journal of Mathematical Economics, Elsevier, vol. 82(C), pages 31-60.
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