Advertisement

Prophet Inequalities vs. Approximating Optimum Online

  • Rad Niazadeh
  • Amin Saberi
  • Ali Shameli
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11316)

Abstract

We revisit the classic prophet inequality problem, where the goal is selling a single item to an arriving sequence of buyers whose values are drawn from independent distributions, to maximize the expected allocated value. The usual benchmark is the expected value that an omniscient prophet who knows the future can attain. We diverge from this framework and compare the performance of the best single pricing mechanism with the best optimum online mechanism.

Somewhat surprisingly, we show that the original tight prophet inequality bounds comparing the single-pricing with the optimum offline are tight even when we use the optimum online as a benchmark, both for the identical and non-identical distributions. Moreover, we incorporate linear programming to characterize this benchmark, and show how this approach leads to a modular way of designing prophet inequalities, hence reconstructing the results of [31] and [13] with somewhat simpler proofs.

Keywords

Prophet inequality Optimal stopping Online algorithm 

References

  1. 1.
    Abolhassani, M., Ehsani, S., Esfandiari, H., HajiAghayi, M., Kleinberg, R., Lucier, B.: Beating \(1-1\)/e for ordered prophets. In: Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing, pp. 61–71. ACM (2017)Google Scholar
  2. 2.
    Alaei, S.: Bayesian combinatorial auctions: expanding single buyer mechanisms to many buyers. SIAM J. Comput. 43(2), 930–972 (2014)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Alaei, S., Hartline, J., Niazadeh, R., Pountourakis, E., Yuan, Y.: Optimal auctions vs. anonymous pricing. In: 2015 IEEE 56th Annual Symposium on Foundations of Computer Science (FOCS), pp. 1446–1463. IEEE (2015)Google Scholar
  4. 4.
    Anari, N., Niazadeh, R., Saberi, A., Shameli, A.: Nearly optimal pricing algorithms for production constrained and laminar bayesian selection. arXiv preprint arXiv:1807.05477 (2018)
  5. 5.
    Azar, P.D., Kleinberg, R., Weinberg, S.M.: Prophet inequalities with limited information. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1358–1377. Society for Industrial and Applied Mathematics (2014)Google Scholar
  6. 6.
    Azar, Y., Chiplunkar, A., Kaplan, H.: Prophet secretary: Surpassing the \(1-1\)/e barrier. In: Proceedings of the 2018 ACM Conference on Economics and Computation, pp. 303–318. ACM (2018)Google Scholar
  7. 7.
    Babaioff, M., Immorlica, N., Lucier, B., Weinberg, S.M.: A simple and approximately optimal mechanism for an additive buyer. ACM SIGecom Exch. 13(2), 31–35 (2015)CrossRefGoogle Scholar
  8. 8.
    Beyhaghi, H., Golrezaei, N., Leme, R.P., Pal, M., Siva, B.: Improved approximations for free-order prophets and second-price auctions. arXiv preprint arXiv:1807.03435 (2018)
  9. 9.
    Cai, Y., Daskalakis, C., Weinberg, S.M.: Optimal multi-dimensional mechanism design: reducing revenue to welfare maximization. In: 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 130–139. IEEE (2012)Google Scholar
  10. 10.
    Cai, Y., Devanur, N.R., Weinberg, S.M.: A duality based unified approach to Bayesian mechanism design. In: Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, pp. 926–939. ACM (2016)CrossRefGoogle Scholar
  11. 11.
    Chawla, S., Hartline, J.D., Malec, D.L., Sivan, B.: Multi-parameter mechanism design and sequential posted pricing. In: Proceedings of the Forty-Second ACM Symposium on Theory of Computing, pp. 311–320. ACM (2010)Google Scholar
  12. 12.
    Chawla, S., Miller, J.B.: Mechanism design for subadditive agents via an ex-ante relaxation. In: Proceedings of the 2016 ACM Conference on Economics and Computation, pp. 579–596. ACM (2016)Google Scholar
  13. 13.
    Correa, J., Foncea, P., Hoeksma, R., Oosterwijk, T., Vredeveld, T.: Posted price mechanisms for a random stream of customers. In: Proceedings of the 2017 ACM Conference on Economics and Computation, pp. 169–186. ACM (2017)Google Scholar
  14. 14.
    Correa, J., Saona, R., Ziliotto, B.: Prophet secretary through blind strategies. arXiv preprint arXiv:1807.07483 (2018)
  15. 15.
    Düetting, P., Feldman, M., Kesselheim, T., Lucier, B.: Prophet inequalities made easy: stochastic optimization by pricing non-stochastic inputs. In: 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pp. 540–551. IEEE (2017)Google Scholar
  16. 16.
    Dütting, P., Fischer, F., Klimm, M.: Revenue gaps for discriminatory and anonymous sequential posted pricing. arXiv preprint arXiv:1607.07105 (2016)
  17. 17.
    Ehsani, S., Hajiaghayi, M., Kesselheim, T., Singla, S.: Prophet secretary for combinatorial auctions and matroids. In: Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 700–714. SIAM (2018)CrossRefGoogle Scholar
  18. 18.
    Esfandiari, H., Hajiaghayi, M., Liaghat, V., Monemizadeh, M.: Prophet secretary. SIAM J. Discrete Math. 31(3), 1685–1701 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Feldman, M., Fu, H., Gravin, N., Lucier, B.: Simultaneous auctions are (almost) efficient. In: Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing, pp. 201–210. ACM (2013)Google Scholar
  20. 20.
    Feldman, M., Svensson, O., Zenklusen, R.: Online contention resolution schemes. In: Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1014–1033. Society for Industrial and Applied Mathematics (2016)Google Scholar
  21. 21.
    Göbel, O., Hoefer, M., Kesselheim, T., Schleiden, T., Vöcking, B.: Online independent set beyond the worst-case: secretaries, prophets, and periods. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8573, pp. 508–519. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-43951-7_43CrossRefzbMATHGoogle Scholar
  22. 22.
    Hajiaghayi, M.T., Kleinberg, R., Sandholm, T.: Automated online mechanism design and prophet inequalities. AAAI. vol. 7, pp. 58–65 (2007)Google Scholar
  23. 23.
    Hartline, J.D.: Approximation in mechanism design. Am. Econ. Rev. 102(3), 330–336 (2012)CrossRefGoogle Scholar
  24. 24.
    Hartline, J.D., Roughgarden, T.: Simple versus optimal mechanisms. In: Proceedings of the 10th ACM Conference on Electronic Commerce, pp. 225–234. ACM (2009)Google Scholar
  25. 25.
    Hill, T.P., Kertz, R.P.: Comparisons of stop rule and supremum expectations of IID random variables. Ann. Probab. 10(2), 336–345 (1982)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Kleinberg, R., Weinberg, S.M.: Matroid prophet inequalities. In: Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing, pp. 123–136. ACM (2012)Google Scholar
  27. 27.
    Krengel, U., Sucheston, L.: On semiamarts, amarts, and processes with finite value. Probab. Banach Spaces 4, 197–266 (1978)MathSciNetGoogle Scholar
  28. 28.
    Lee, E., Singla, S.: Optimal online contention resolution schemes via ex-ante prophet inequalities. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 112. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2018)Google Scholar
  29. 29.
    Lucier, B.: An economic view of prophet inequalities. ACM SIGecom Exch. 16(1), 24–47 (2017)CrossRefGoogle Scholar
  30. 30.
    Rubinstein, A.: Beyond matroids: secretary problem and prophet inequality with general constraints. In: Proceedings of the Forty-eighth Annual ACM Symposium on Theory of Computing, pp. 324–332. ACM (2016)Google Scholar
  31. 31.
    Samuel-Cahn, E.: Comparison of threshold stop rules and maximum for independent nonnegative random variables. Ann. Probab. 12(4), 1213–1216 (1984)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Yan, Q.: Mechanism design via correlation gap. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 710–719. Society for Industrial and Applied Mathematics (2011)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Computer Science DepartmentStanford UniversityStanfordUSA
  2. 2.Management Science and EngineeringStanford UniversityStanfordUSA

Personalised recommendations