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Prophet Inequalities vs. Approximating Optimum Online

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Web and Internet Economics (WINE 2018)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 11316))

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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.

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Notes

  1. 1.

    Although the focus of this paper is not on truthful mechanism design, all of our mechanisms are pricing and hence truthful.

  2. 2.

    We drop lower order terms by using \(\approx \).

  3. 3.

    In the case of non-atomic distributions, this LP is essentially a continuous program with uncountably many variables. In the case of discrete distributions, the LP has finitely many variables.

  4. 4.

    Note that we assume the true \(\pi \) is the unitary ordering, but that is without loss of generality.

References

  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. Alaei, S.: Bayesian combinatorial auctions: expanding single buyer mechanisms to many buyers. SIAM J. Comput. 43(2), 930–972 (2014)

    Article  MathSciNet  Google Scholar 

  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. 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. 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. 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. 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)

    Article  Google Scholar 

  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. 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. 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)

    Article  Google Scholar 

  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. 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. 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. Correa, J., Saona, R., Ziliotto, B.: Prophet secretary through blind strategies. arXiv preprint arXiv:1807.07483 (2018)

  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. Dütting, P., Fischer, F., Klimm, M.: Revenue gaps for discriminatory and anonymous sequential posted pricing. arXiv preprint arXiv:1607.07105 (2016)

  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)

    Chapter  Google Scholar 

  18. Esfandiari, H., Hajiaghayi, M., Liaghat, V., Monemizadeh, M.: Prophet secretary. SIAM J. Discrete Math. 31(3), 1685–1701 (2017)

    Article  MathSciNet  Google Scholar 

  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. 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. 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_43

    Chapter  MATH  Google Scholar 

  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. Hartline, J.D.: Approximation in mechanism design. Am. Econ. Rev. 102(3), 330–336 (2012)

    Article  Google Scholar 

  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. Hill, T.P., Kertz, R.P.: Comparisons of stop rule and supremum expectations of IID random variables. Ann. Probab. 10(2), 336–345 (1982)

    Article  MathSciNet  Google Scholar 

  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. Krengel, U., Sucheston, L.: On semiamarts, amarts, and processes with finite value. Probab. Banach Spaces 4, 197–266 (1978)

    MathSciNet  Google Scholar 

  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. Lucier, B.: An economic view of prophet inequalities. ACM SIGecom Exch. 16(1), 24–47 (2017)

    Article  Google Scholar 

  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. Samuel-Cahn, E.: Comparison of threshold stop rules and maximum for independent nonnegative random variables. Ann. Probab. 12(4), 1213–1216 (1984)

    Article  MathSciNet  Google Scholar 

  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 

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Author information

Authors and Affiliations

  1. Computer Science Department, Stanford University, Stanford, CA, 94305, USA

    Rad Niazadeh

  2. Management Science and Engineering, Stanford University, Stanford, CA, 94305, USA

    Amin Saberi & Ali Shameli

Authors
  1. Rad Niazadeh
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  2. Amin Saberi
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  3. Ali Shameli
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Correspondence to Rad Niazadeh .

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Editors and Affiliations

  1. University of Liverpool, Liverpool, UK

    George Christodoulou

  2. Universität Augsburg, Augsburg, Germany

    Tobias Harks

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Niazadeh, R., Saberi, A., Shameli, A. (2018). Prophet Inequalities vs. Approximating Optimum Online. In: Christodoulou, G., Harks, T. (eds) Web and Internet Economics. WINE 2018. Lecture Notes in Computer Science(), vol 11316. Springer, Cham. https://doi.org/10.1007/978-3-030-04612-5_24

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  • DOI: https://doi.org/10.1007/978-3-030-04612-5_24

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