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Selling Multiple Correlated Goods: Revenue Maximization and Menu-Size Complexity (old title: "The Menu-Size Complexity of Auctions")

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  • Sergiu Hart
  • Noam Nisan

Abstract

We consider the well known, and notoriously difficult, problem of a single revenue-maximizing seller selling two or more heterogeneous goods to a single buyer whose private values for the goods are drawn from a (possibly correlated) known distribution, and whose valuation is additive over the goods. We show that when there are two (or more) goods, _simple mechanisms_ -- such as selling the goods separately or as a bundle -- _may yield only a negligible fraction of the optimal revenue_. This resolves the open problem of Briest, Chawla, Kleinberg, and Weinberg (JET 2015) who prove the result for at least three goods in the related setup of a unit-demand buyer. We also introduce the menu size as a simple measure of the complexity of mechanisms, and show that the revenue may increase polynomially with _menu size_ and that no bounded menu size can ensure any positive fraction of the optimal revenue. The menu size also turns out to "pin down" the revenue properties of deterministic mechanisms.

Suggested Citation

  • Sergiu Hart & Noam Nisan, 2013. "Selling Multiple Correlated Goods: Revenue Maximization and Menu-Size Complexity (old title: "The Menu-Size Complexity of Auctions")," Papers 1304.6116, arXiv.org, revised Nov 2018.
    • Handle: RePEc:arx:papers:1304.6116
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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. Hart, Sergiu & Nisan, Noam, 2017. "Approximate revenue maximization with multiple items," Journal of Economic Theory, Elsevier, vol. 172(C), pages 313-347.
    3. Sergiu Hart & Philip J. Reny, 2017. "The Better Half of Selling Separately," Papers 1712.08973, arXiv.org, revised Oct 2018.
    4. , & , J., 2015. "Maximal revenue with multiple goods: nonmonotonicity and other observations," Theoretical Economics, Econometric Society, vol. 10(3), September.
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    7. Hanming Fang & Peter Norman, 2006. "To bundle or not to bundle," RAND Journal of Economics, RAND Corporation, vol. 37(4), pages 946-963, December.
    8. 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.
    9. Roger B. Myerson, 1981. "Optimal Auction Design," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 58-73, February.
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    13. Gabriel Carroll, 2017. "Robustness and Separation in Multidimensional Screening," Econometrica, Econometric Society, vol. 85, pages 453-488, March.
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