Robust multidimensional pricing: separation without regret

Abstract

We study a robust monopoly pricing problem with a minimax regret objective, where a seller endeavors to sell multiple goods to a single buyer, only knowing that the buyer’s values for the goods range over a rectangular uncertainty set. We interpret this pricing problem as a zero-sum game between the seller, who chooses a selling mechanism, and a fictitious adversary or ‘nature’, who chooses the buyer’s values from within the uncertainty set. Using duality techniques rooted in robust optimization, we prove that this game admits a Nash equilibrium in mixed strategies that can be computed in closed form. The Nash strategy of the seller is a randomized posted price mechanism under which the goods are sold separately, while the Nash strategy of nature is a distribution on the uncertainty set under which the buyer’s values are comonotonic. We further show that the restriction of the pricing problem to deterministic mechanisms is solved by a deteministic posted price mechanism under which the goods are sold separately.

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References

  1. 1.

    Bandi, C., Bertsimas, D.: Optimal design for multi-item auctions: a robust optimization approach. Math. Oper. Res. 39(4), 1012–1038 (2014)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization. Princeton University Press, Princeton (2009)

    Google Scholar 

  3. 3.

    Bergemann, D., Schlag, K.: Pricing without priors. J. Eur. Econ. Assoc. 6(2–3), 560–569 (2008)

    Article  Google Scholar 

  4. 4.

    Bergemann, D., Schlag, K.: Robust monopoly pricing. J. Econ. Theory 146(6), 2527–2543 (2011)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Bertsimas, D., Brown, D.B., Caramanis, C.: Theory and applications of robust optimization. SIAM Rev. 53(3), 464–501 (2011)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bhargava, H.K.: Mixed bundling of two independently valued goods. Manag. Sci. 59(9), 2170–2185 (2013)

    Article  Google Scholar 

  7. 7.

    Bichler, M.: Market Design: A Linear Programming Approach to Auctions and Matching. Cambridge University Press, Cambridge (2017)

    Google Scholar 

  8. 8.

    Carrasco, V., Luz, V.F., Kos, N., Messner, M., Monteiro, P., Moreira, H.: Optimal selling mechanisms under moment conditions. J. Econ. Theory 177, 245–279 (2018)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Carroll, G.: Robustness and separation in multidimensional screening. Econometrica 85(2), 453–488 (2017)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Daskalakis, C., Deckelbaum, A., Tzamos, C.: Mechanism design via optimal transport. In: Proceedings of the 14th ACM Conference on Electronic Commerce, pp. 269–286 (2013)

  11. 11.

    Daskalakis, C., Deckelbaum, A., Tzamos, C.: The complexity of optimal mechanism design. In: Proceedings of the 25th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 1302–1318 (2014)

  12. 12.

    Daskalakis, C., Deckelbaum, A., Tzamos, C.: Strong duality for a multiple-good monopolist. Econometrica 85(3), 735–767 (2017)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Fanzeres, B., Ahmed, S., Street, A.: Robust strategic bidding in auction-based markets. Eur. J. Oper. Res. 272(3), 1158–1172 (2019)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Giannakopoulos, Y., Koutsoupias, E.: Duality and optimality of auctions for uniform distributions. In: Proceedings of the 15th ACM Conference on Economics and Computation, pp. 259–276 (2014)

  15. 15.

    Gravin, N., Lu, P.: Separation in correlation-robust monopolist problem with budget. In: Proceedings of the Twenty-Ninth Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 2069–2080. SIAM (2018)

  16. 16.

    Guslitser, E.: Uncertainty-Immunized Solutions in Linear Programming. Master’s thesis, Technion-Israel Institute of Technology (2002)

  17. 17.

    Hart, S., Nisan, N.: Approximate revenue maximization with multiple items. J. Econ. Theory 172, 313–347 (2017a)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Hart, S., Nisan, N.: The Menu-size Complexity of Auctions. Technical report, The Hebrew University of Jerusalem (2017)

  19. 19.

    Lagarias, J.: Euler’s constant: Euler’s work and modern developments. Bull. Am. Math. Soc. 50(4), 527–628 (2013)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Li, X., Yao, A.C.C.: On revenue maximization for selling multiple independently distributed items. Proc. Natl. Acad. Sci. 110(28), 11232–11237 (2013)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Lopomo, G., Marx, L.M., Sun, P.: Bidder collusion at first-price auctions. Rev. Econ. Des. 15(3), 177–211 (2011)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Manelli, A.M., Vincent, D.R.: Bundling as an optimal selling mechanism for a multiple-good monopolist. J. Econ. Theory 127(1), 1–35 (2006)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Myerson, R.B.: Optimal auction design. Math. Oper. Res. 6(1), 58–73 (1981)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Myerson, R.B., Satterthwaite, M.A.: Efficient mechanisms for bilateral trading. J. Econ. Theory 29(2), 265–281 (1983)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Pınar, M.Ç., Kızılkale, C.: Robust screening under ambiguity. Math. Program. 163(1–2), 273–299 (2017)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Riley, J., Zeckhauser, R.: Optimal selling strategies: when to haggle, when to hold firm. Q. J. Econ. 98(2), 267–289 (1983)

    Article  Google Scholar 

  27. 27.

    Savage, L.J.: The theory of statistical decision. J. Am. Stat. Assoc. 46(253), 55–67 (1951)

    Article  Google Scholar 

  28. 28.

    Thanassoulis, J.: Haggling over substitutes. J. Econ. Theory 117(2), 217–245 (2004)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Vohra, R.V.: Optimization and mechanism design. Math. Program. 134(1), 283–303 (2012)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Wiesemann, W., Tsoukalas, A., Kleniati, P.M., Rustem, B.: Pessimistic bilevel optimization. SIAM J. Optim. 23(1), 353–380 (2013)

    MathSciNet  Article  Google Scholar 

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Acknowledgements

This research was funded by the SNSF grant BSCGI0_157733 and by the NUS Start-up Grant R-266-000-131-133.

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Correspondence to Çağıl Koçyiğit.

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Koçyiğit, Ç., Rujeerapaiboon, N. & Kuhn, D. Robust multidimensional pricing: separation without regret. Math. Program. (2021). https://doi.org/10.1007/s10107-021-01615-4

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Mathematics Subject Classification

  • 91B03
  • 90C47
  • 90C17