Advertisement

Robust Revenue Maximization Under Minimal Statistical Information

  • Yiannis Giannakopoulos
  • Diogo PoçasEmail author
  • Alexandros Tsigonias-Dimitriadis
Conference paper
First Online:
  • 64 Downloads
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12495)

Abstract

We study the problem of multi-dimensional revenue maximization when selling m items to a buyer that has additive valuations for them, drawn from a (possibly correlated) prior distribution. Unlike traditional Bayesian auction design, we assume that the seller has a very restricted knowledge of this prior: they only know the mean \(\mu _j\) and an upper bound \(\sigma _j\) on the standard deviation of each item’s marginal distribution. Our goal is to design mechanisms that achieve good revenue against an ideal optimal auction that has full knowledge of the distribution in advance. Informally, our main contribution is a tight quantification of the interplay between the dispersity of the priors and the aforementioned robust approximation ratio. Furthermore, this can be achieved by very simple selling mechanisms.

More precisely, we show that selling the items via separate price lotteries achieves an \(O(\log r)\) approximation ratio where \(r=\max _j(\sigma _j/\mu _j)\) is the maximum coefficient of variation across the items. If forced to restrict ourselves to deterministic mechanisms, this guarantee degrades to \(O(r^2)\). Assuming independence of the item valuations, these ratios can be further improved by pricing the full bundle. For the case of identical means and variances, in particular, we get a guarantee of \(O(\log (r/m))\) which converges to optimality as the number of items grows large. We demonstrate the optimality of the above mechanisms by providing matching lower bounds. Our tight analysis for the deterministic case resolves an open gap from the work of Azar and Micali [ITCS’13].

Keywords

Optimal auctions Revenue maximization Parametric auctions Robust optimization 
This is a preview of subscription content, log in to check access.

References

  1. 1.
    Azar, P., Daskalakis, C., Micali, S., Weinberg, S.M.: Optimal and efficient parametric auctions. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 596–604 (2013).  https://doi.org/10.1137/1.9781611973105.43
  2. 2.
    Azar, P., Micali, S.: Optimal parametric auctions. Technical report, MIT-CSAIL-TR-2012-015 (2012). http://hdl.handle.net/1721.1/70556
  3. 3.
    Azar, P.D., Micali, S.: Parametric digital auctions. In: Proceedings of the 4th Conference on Innovations in Theoretical Computer Science (ITCS), pp. 231–232 (2013).  https://doi.org/10.1145/2422436.2422464
  4. 4.
    Babaioff, M., Immorlica, N., Lucier, B., Weinberg, S.M.: A simple and approximately optimal mechanism for an additive buyer. In: Proceedings of the 55th Annual Symposium on Foundations of Computer Science (FOCS), pp. 21–30 (2014).  https://doi.org/10.1109/FOCS.2014.11
  5. 5.
    Bandi, C., Bertsimas, D.: Optimal design for multi-item auctions: a robust optimization approach. Math. Oper. Res. 39(4), 1012–1038 (2014).  https://doi.org/10.1287/moor.2014.0645MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bei, X., Gravin, N., Lu, P., Tang, Z.G.: Correlation-robust analysis of single item auction. In: Proceedings of the 30th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 193–208 (2019).  https://doi.org/10.1137/1.9781611975482.13
  7. 7.
    Bergemann, D., Schlag, K.: Robust monopoly pricing. J. Econ. Theory 146(6), 2527–2543 (2011).  https://doi.org/10.1016/j.jet.2011.10.018MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998)zbMATHGoogle Scholar
  9. 9.
    Bulow, J., Klemperer, P.: Auctions versus negotiations. Am. Econ. Rev. 86(1), 180–194 (1996)Google Scholar
  10. 10.
    Cai, Y., Daskalakis, C.: Learning multi-item auctions with (or without) samples. In: Proceedings of the 58th Annual Symposium on Foundations of Computer Science (FOCS), pp. 516–527 (2017).  https://doi.org/10.1109/FOCS.2017.54
  11. 11.
    Cai, Y., Devanur, N.R., Weinberg, S.M.: A duality based unified approach to Bayesian mechanism design. In: Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pp. 926–939. ACM Press (2016).  https://doi.org/10.1145/2897518.2897645
  12. 12.
    Cai, Y., Zhao, M.: Simple mechanisms for subadditive buyers via duality. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing (STOC), pp. 170–183 (2017).  https://doi.org/10.1145/3055399.3055465
  13. 13.
    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).  https://doi.org/10.1016/j.jet.2018.05.005MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Carroll, G.: Robustness and separation in multidimensional screening. Econometrica 85(2), 453–488 (2017).  https://doi.org/10.3982/ECTA14165MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Carroll, G.: Robustness in mechanism design and contracting. Annu. Rev. Econ. 11(1), 139–166 (2019).  https://doi.org/10.1146/annurev-economics-080218-025616CrossRefGoogle Scholar
  16. 16.
    Chawla, S., Fu, H., Karlin, A.R.: Approximate revenue maximization in interdependent value settings. CoRR abs/1408.4424 (2014)Google Scholar
  17. 17.
    Chawla, S., Hartline, J.D., Malec, D.L., Sivan, B.: Multi-parameter mechanism design and sequential posted pricing. In: Proceedings of the 42nd ACM Symposium on Theory of Computing (STOC), pp. 311–320 (2010).  https://doi.org/10.1145/1806689.1806733
  18. 18.
    Chen, J., Li, B., Li, Y., Lu, P.: Bayesian auctions with efficient queries. CoRR abs/1804.07451 (2018)Google Scholar
  19. 19.
    Chen, X., Diakonikolas, I., Orfanou, A., Paparas, D., Sun, X., Yannakakis, M.: On the complexity of optimal lottery pricing and randomized mechanisms. In: Proceedings of 56th Annual Symposium on Foundations of Computer Science (FOCS), pp. 1464–1479 (2015).  https://doi.org/10.1109/FOCS.2015.93
  20. 20.
    Chen, X., Matikas, G., Paparas, D., Yannakakis, M.: On the complexity of simple and optimal deterministic mechanisms for an additive buyer. In: Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2036–2049 (2018).  https://doi.org/10.1137/1.9781611975031.133
  21. 21.
    Daskalakis, C., Deckelbaum, A., Tzamos, C.: The complexity of optimal mechanism design. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1302–1318 (2013).  https://doi.org/10.1137/1.9781611973402.96
  22. 22.
    Daskalakis, C., Deckelbaum, A., Tzamos, C.: Strong duality for a multiple-good monopolist. Econometrica 85(3), 735–767 (2017).  https://doi.org/10.3982/ECTA12618MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Dhangwatnotai, P., Roughgarden, T., Yan, Q.: Revenue maximization with a single sample. Games Econ. Behav. 91(C), 318–333 (2014)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Dütting, P., Roughgarden, T., Talgam-Cohen, I.: Simple versus optimal contracts. In: Proceedings of the 20th ACM Conference on Economics and Computation (EC), pp. 369–387 (2019).  https://doi.org/10.1145/3328526.3329591
  25. 25.
    Fu, H., Immorlica, N., Lucier, B., Strack, P.: Randomization beats second price as a prior-independent auction. In: Proceedings of the 16th ACM Conference on Economics and Computation (EC), p. 323 (2015).  https://doi.org/10.1145/2764468.2764489
  26. 26.
    Giannakopoulos, Y., Koutsoupias, E.: Duality and optimality of auctions for uniform distributions. SIAM J. Comput. 47(1), 121–165 (2018).  https://doi.org/10.1137/16M1072218MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Giannakopoulos, Y., Poças, D., Tsigonias-Dimitriadis, A.: Robust revenue maximization under minimal statistical information. CoRR abs/1907.04220 (2019)Google Scholar
  28. 28.
    Gravin, N., Lu, P.: Separation in correlation-robust monopolist problem with budget. In: Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 2069–2080 (2018).  https://doi.org/10.1137/1.9781611975031.135
  29. 29.
    Haghpanah, N., Hartline, J.: Reverse mechanism design. In: Proceedings of the 16th ACM Conference on Economics and Computation (EC), pp. 757–758 (2015).  https://doi.org/10.1145/2764468.2764498
  30. 30.
    Hart, S., Nisan, N.: The menu-size complexity of auctions. In: Proceedings of the 14th ACM Conference on Electronic Commerce (EC), pp. 565–566 (2013).  https://doi.org/10.1145/2482540.2482544
  31. 31.
    Hart, S., Nisan, N.: Approximate revenue maximization with multiple items. J. Econ. Theory 172, 313–347 (2017).  https://doi.org/10.1016/j.jet.2017.09.001MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Hartline, J.D.: Mechanism design and approximation (2013), manuscript. http://jasonhartline.com/MDnA/
  33. 33.
    Hartline, J.D., Roughgarden, T.: Simple versus optimal mechanisms. In: Proceedings of the 10th ACM Conference on Electronic Commerce (EC), pp. 225–234 (2009).  https://doi.org/10.1145/1566374.1566407
  34. 34.
    Kendall, M.G.: The Advanced Theory of Statistics, vol. I, 4th edn. Charles Griffin, Glasgow (1948)Google Scholar
  35. 35.
    Li, X., Yao, A.C.C.: On revenue maximization for selling multiple independently distributed items. Proc. Nat. Acad. Sci. 110(28), 11232–11237 (2013).  https://doi.org/10.1073/pnas.1309533110MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Li, Y., Lu, P., Ye, H.: Revenue maximization with imprecise distribution. In: Proceedings of the 18th International Conference on Autonomous Agents and MultiAgent Systems (AAMAS), pp. 1582–1590 (2019). http://dl.acm.org/citation.cfm?id=3306127.3331877
  37. 37.
    Manelli, A.M., Vincent, D.R.: Multidimensional mechanism design: revenue maximization and the multiple-good monopoly. J. Econ. Theory 137(1), 153–185 (2007).  https://doi.org/10.1016/j.jet.2006.12.007MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Milgrom, P.: Putting Auction Theory to Work. Cambridge University Press, Cambridge (2004).  https://doi.org/10.1017/CBO9780511813825.009CrossRefGoogle Scholar
  39. 39.
    Mood, A.M., Graybill, F.A., Boes, D.C.: Introduction to the Theory of Statistics, 3rd edn. McGraw-Hill, New York (1974)zbMATHGoogle Scholar
  40. 40.
    Moriguti, S.: Extremal properties of extreme value distributions. Ann. Math. Stat. 22(4), 523–536 (1951)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  42. 42.
    Myerson, R.B.: Optimal auction design. Math. Oper. Res. 6(1), 58–73 (1981).  https://doi.org/10.1287/moor.6.1.58MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Roughgarden, T., Talgam-Cohen, I.: Approximately optimal mechanism design. Annu. Rev. Econ. 11(1), 355–381 (2019).  https://doi.org/10.1146/annurev-economics-080218-025607CrossRefGoogle Scholar
  44. 44.
    Rubinstein, A., Weinberg, S.M.: Simple mechanisms for a subadditive buyer and applications to revenue monotonicity. ACM Trans. Econ. Comput. 6(3–4), 19:1–19:25 (2018).  https://doi.org/10.1145/3105448MathSciNetCrossRefGoogle Scholar
  45. 45.
    Wilson, R.: Game-theoretic analyses of trading processes. In: Advances in Economic Theory: Fifth World Congress, pp. 33–70. Econometric Society Monographs, Cambridge University Press (1987).  https://doi.org/10.1017/CCOL0521340446.002
  46. 46.
    Yao, A.C.C.: An \(n\)-to-\(1\) bidder reduction for multi-item auctions and its applications. In: Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 92–109 (2015).  https://doi.org/10.1137/1.9781611973730.8

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.TU MunichMunichGermany
  2. 2.LASIGE, Faculdade de CiênciasUniversidade de LisboaLisbonPortugal

Personalised recommendations

Cite paper

Buy options