Abstract
Optimal stopping theory is a powerful tool for analyzing scenarios such as online auctions in which we generally require optimizing an objective function over the space of stopping rules for an allocation process under uncertainty. Perhaps the most classic problems of stopping theory are the prophet inequality problem and the secretary problem. The classical prophet inequality states that by choosing the same threshold OPT/2 for every step, one can achieve the tight competitive ratio of 0.5. On the other hand, for the basic secretary problem, the optimal strategy achieves the tight competitive ratio of 1/e ≈ 0.36
In this paper, we introduce prophet secretary, a natural combination of the prophet inequality and the secretary problems. We show that by using a single uniform threshold one cannot break the 0.5 barrier of the prophet inequality for the prophet secretary problem. However, we show that
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using n distinct non-adaptive thresholds one can obtain a competitive ratio that goes to (1 − 1/e ≈ 0.63) as n grows; and
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no online algorithm can achieve a competitive ratio better than 0.73.
Our results improve the (asymptotic) approximation guarantee of single-item sequential posted pricing mechanisms from 0.5 to (1 − 1/e) when the order of agents (customers) is chosen randomly.
We also consider the minimization variants of stopping theory problems and in particular the prophet secretary problem. Interestingly, we show that, even for the simple case in which the input elements are drawn from identical and independent distributions (i.i.d.), there is no constant competitive online algorithm for the minimization variant of the prophet secretary problems. We extend this hardness result to the minimization variants of both the prophet inequality and the secretary problem as well.
The full version of this paper is available on http://arxiv.org/abs/1507.01155
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References
Ajtai, M., Megiddo, N., Waarts, O.: Improved algorithms and analysis for secretary problems and generalizations. SIAM J. Discrete Math. 14(1), 1–27 (2001)
Alaei, S., Hajiaghayi, M., Liaghat, V.: Online prophet-inequality matching with applications to ad allocation. In: EC (2012)
Alaei, S., Hajiaghayi, M., Liaghat, V.: The online stochastic generalized assignment problem. In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds.) RANDOM 2013 and APPROX 2013. LNCS, vol. 8096, pp. 11–25. Springer, Heidelberg (2013)
Babaioff, M., Immorlica, N., Kempe, D., Kleinberg, R.: A knapsack secretary problem with applications. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 16–28. Springer, Heidelberg (2007)
Babaioff, M., Immorlica, N., Kempe, D., Kleinberg, R.: Online auctions and generalized secretary problems. SIGecom Exch. 7(2), 1–11 (2008)
Babaioff, M., Immorlica, N., Kleinberg, R.: Matroids, secretary problems, and online mechanisms. In: SODA, pp. 434–443 (2007)
Chawla, S., Hartline, J., Malec, D., Sivan, B.: Multi-parameter mechanism design and sequential posted pricing (2010)
Dynkin, E.B.: The optimum choice of the instant for stopping a markov process. Sov. Math. Dokl. 4, 627–629 (1963)
Glasser, K.S., Holzsager, R., Barron, A.: The d choice secretary problem. Comm. Statist. C—Sequential Anal. 2(3), 177–199 (1983)
Hajiaghayi, M.T., Kleinberg, R., Parkes, D.C.: Adaptive limited-supply online auctions. In: EC, pp. 71–80 (2004)
Hajiaghayi, M.T., Kleinberg, R., Sandholm, T.: Automated online mechanism design and prophet inequalities. In: AAAI, pp. 58–65 (2007)
Immorlica, N., Kleinberg, R.D., Mahdian, M.: Secretary problems with competing employers. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 389–400. Springer, Heidelberg (2006)
Kennedy, D.P.: Prophet-type inequalities for multi-choice optimal stopping. In: Stoch. Proc. Applic. (1978)
Kleinberg, R.: A multiple-choice secretary algorithm with applications to online auctions. In: SODA, pp. 630–631 (2005)
Kleinberg, R., Weinberg, S.M.: Matroid prophet inequalities. In: STOC (2012)
Krengel, U., Sucheston, L.: Semiamarts and finite values. Bull. Am. Math. Soc. (1977)
Krengel, U., Sucheston, L.: On semiamarts, amarts, and processes with finite value. In: Kuelbs, J. (ed.) Probability on Banach Spaces (1978)
Myerson, R.B.: Optimal auction design. Mathematics of Operations Research 6(1), 58–73 (1981)
Vanderbei, R.J.: The optimal choice of a subset of a population. Math. Oper. Res. 5(4), 481–486 (1980)
Wilson, J.G.: Optimal choice and assignment of the best m of n randomly arriving items. Stochastic Process. Appl. 39(2), 325–343 (1991)
Yan, Q.: Mechanism design via correlation gap. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, pp. 710–719 (2011)
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Esfandiari, H., Hajiaghayi, M., Liaghat, V., Monemizadeh, M. (2015). Prophet Secretary. In: Bansal, N., Finocchi, I. (eds) Algorithms - ESA 2015. Lecture Notes in Computer Science(), vol 9294. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48350-3_42
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DOI: https://doi.org/10.1007/978-3-662-48350-3_42
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