Probability Surveys

A unified approach for solving sequential selection problems

Alexander Goldenshluger, Yaakov Malinovsky, and Assaf Zeevi

Full-text: Open access

Abstract

In this paper we develop a unified approach for solving a wide class of sequential selection problems. This class includes, but is not limited to, selection problems with no–information, rank–dependent rewards, and considers both fixed as well as random problem horizons. The proposed framework is based on a reduction of the original selection problem to one of optimal stopping for a sequence of judiciously constructed independent random variables. We demonstrate that our approach allows exact and efficient computation of optimal policies and various performance metrics thereof for a variety of sequential selection problems, several of which have not been solved to date.

Article information

Source
Probab. Surveys, Volume 17 (2020), 214-256.

Dates
Received: May 2019
First available in Project Euclid: 27 April 2020

Permanent link to this document
https://projecteuclid.org/euclid.ps/1587974426

Digital Object Identifier
doi:10.1214/19-PS333

Subjects
Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]
Secondary: 62L15: Optimal stopping [See also 60G40, 91A60]

Keywords
Sequential selection optimal stopping secretary problems relative ranks full information problems no–information problems

Rights
Creative Commons Attribution 4.0 International License.

Citation

Goldenshluger, Alexander; Malinovsky, Yaakov; Zeevi, Assaf. A unified approach for solving sequential selection problems. Probab. Surveys 17 (2020), 214--256. doi:10.1214/19-PS333. https://projecteuclid.org/euclid.ps/1587974426


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References

  • Ajtai, M., Megiddo, N. and Waarts, O. (2001). Improved algorithms and analysis for secretary problems and generalizations. SIAM J. Discrete Math. 14, 1–27.
  • Albright, S. C., Jr. (1972). Stochastic sequential assignment problems. Technical report No. 147, Department of Statistics, Stanford University.
  • Arlotto, A. and Gurvich, I. (2019). Uniformly bounded regret in the multi-secretary problem. Stochastic Systems 9, 231–260.
  • Arlotto, A., Nguyen, Vinh V. and Steele, J. M. (2015). Optimal online selection of a monotone subsequence: a central limit theorem. Stochastic Process. Appl. 125, 3596–3622.
  • Berezovsky, B. A. and Gnedin, A. B. (1984). The Problem of Optimal Choice. Nauka, Moscow (in Russian).
  • Bruss, F. T. (2000). Sum the odds and stop. Ann. Probab. 28, 1384–1391.
  • Bruss, F. T. (2019). Odds–theorem and monotonicity. Mathematica Applicanda 47, 25–43.
  • Bruss, F. T. and Louchard, G. (2016). Sequential selection of the ${\kappa }$ best out of $n$ rankable objects. Discrete Math. Theor. Comput. Sci. 18, no. 3, Paper No. 13, 1–12.
  • Bruss, F. T. and Ferguson, T. (1996). Half–prophets and Robbins’ problem of minimising the expected rank with i.i.d. random variables. Athens Conference on Applied Probability and Time Series Analysis, Vol. I (1995), 1–17, Lect. Notes Stat., 114, Springer, New York.
  • Chow, Y. S., Moriguti, S., Robbins, H. and Samuels, S. M. (1964). Optimal selection based on relative rank (the “Secretary problem”). Israel J. Math. 2, 81–90.
  • Chow, Y. S., Robbins, H. and Siegmung, D. (1971). Great Expectations: The Theory of Optimal Stopping. Houghton Mifflin Company, Boston.
  • Coffman, E. G., Jr., Flatto, L. and Weber, R. R. (1987). Optimal selection of stochastic intervals under a sum constraint. Adv. in Appl. Probab. 19, 454–473.
  • Derman, C., Lieberman, G. J. and Ross, S. (1972). A sequential stochastic assignment problem. Management Science 18, 349–355.
  • Dietz, C., van der Laan, D. and Ridder, A. (2011). Approximate results for a generalized secretary problem. Probab. Engrg. Inform. Sci. 25, 157–169.
  • Dynkin, E. B. (1963). Optimal choice of the stopping moment of a Markov process. (Russian) Dokl. Akad. Nauk SSSR 150, 238–240. Also in: Selected papers of E. B. Dynkin with commentary, 485–488. Edited by A. A. Yushkevich, G. M. Seitz and A. L. Onishchik. American Mathematical Society, Providence, RI; International Press, Cambridge.
  • Ferguson, T. S. (1989). Who solved the secretary problem? Statist. Science 4, 282–296.
  • Ferguson, T.S. (2008). Optimal Stopping and Applications. https://www.math.ucla.edu/~tom/Stopping/Contents.html.
  • Frank, A. Q. and Samuels, S. M. (1980). On an optimal stopping problem of Gusein–Zade. Stoch. Proc. Appl. 10, 299–311.
  • Freeman, P. R. (1983). The secretary problem and its extensions: a review. Int. Statist. Review 51, 189–206.
  • Gianini-Pettitt, J. (1979). Optimal selection based on relative ranks with a random number of individuals. Adv. Appl. Probab. 11, 720–736.
  • Gilbert, J. and Mosteller, F. (1966). Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.
  • Goldenshluger, A. and Zeevi, A. (2018). Optimal stopping of a random sequence with unknown distribution. Preprint.
  • Gnedin, A. V. (1999). Sequential selection of an increasing subsequence from a sample of random size. J. Appl. Probab. 36, 1074–1085.
  • Gnedin, A. V. (2007). Optimal stopping with rank-dependent loss. J. Appl. Probab. 44, 996–1011.
  • Gnedin, A. V. and Krengel, U. (1996). Optimal selection problems based on exchangeable trials. Ann. Appl. Probab. 6, 862–882.
  • Gusein–Zade, S. M. (1966). The problem of choice and the optimal stopping rule for a sequence of independent trials. Theory Probab. Appl. 11, 472–476.
  • Haggstrom, G. W. (1967). Optimal sequential procedures when more than one stop is required. Ann. Math. Statist. 38, 1618–1626.
  • Irle, A. (1980). On the best choice problem with random population size. Zeitschrift für Operations Research 24, 177–190.
  • Kawai, M. and Tamaki, M. (2003). Choosing either the best or the second best when the number of applicants is random. Comput. Math. Appl. 46, 1065–1071.
  • Krieger, A. M. and Samuel-Cahn, E. (2009). The secretary problem of minimizing the expected rank: a simple suboptimal approach with generalizations. Adv. in Appl. Probab. 41, 1041–1058
  • Krieger, A. M., Pollak, M. and Samuel-Cahn, E. (2007). Select sets: rank and file. Ann. Appl. Probab. 17, 360–385.
  • Krieger, A. M., Pollak, M. and Samuel-Cahn, E. (2008). Beat the mean: sequential selection by better than average rules. J. Appl. Probab. 45, 244–259.
  • Lin, Y.-S., Hsiau, S.-R. and Yao, Y.-C. (2019). Optimal selection of the $k$-th best candidate. Probab. Engrg. Inform. Sci. 33, 327–347.
  • Lindley, D. V. (1961). Dynamic programming and decision theory. Appl. Statist. 10, 39–51.
  • Matsui T. and Ano, K. (2016). Lower bounds for Bruss’ Odds Theorem with multiple stoppings. Math. Oper. Res. 41, 700–714.
  • Moser, L. (1956). On a problem of Cayley. Scripta Math. 22, 289–292.
  • Mucci, A. G. (1973). Differential equations and optimal choice problems. Ann. Statist. 1, 104–113.
  • Nikolaev, A. G. and Jacobson, S. H. (2010). Stochastic sequential decision–making with a random number of jobs. Oper. Res. 58, 1023–1027.
  • Nikolaev, M. L. and Sofronov, G. (2007). A multiple optimal stopping rule for sums of independent random variables. Diskr. Mat. 17, 42–51.
  • Presman, E. L. and Sonin, I. M. (1972). The best choice problem for a random number of objects. Teor. Veroyatnost. i Primenen. 17, 695–706.
  • Quine, M. P. and Law, J. S. (1996). Exact results for a secretary problem. J. Appl. Probab. 33, 630–639.
  • Robbins, H. (1970). Optimal stopping. Amer. Math. Monthly 77, 333–343.
  • Robbins, H. (1991). Remarks on the secretary problem. Amer. J. Math. and Management Sci. 11, 25–37.
  • Rose, J.S (1982a). A problem of optimal choice and assignment. Oper. Res. 30, 172–181.
  • Rose, J.S. (1982b). Selection of nonextremal candidates from a sequence. J. Optimization Theory Applic. 38, 207–219.
  • Sakaguchi, M. (1984). A sequential stochastic assignment problem with an unknown number of jobs. Math. Japonica 29, 141–152.
  • Samuels, S. (1991). Secretary problems. In: Handbook of Sequential Analysis, edited by B. K. Ghosh and P. K. Sen, Marcel Dekker Inc., New York.
  • Samuels, S. M. and Steele, J. M. (1981). Optimal sequential selection of a monotone sequence from a random sample. Ann. Probab. 9, 937–947.
  • Szajowski, K. (1982). Optimal choice of an object with $a$th rank (Polish). Mat. Stos. 19, 51-65.
  • Woryna, A. (2017). The solution of a generalized secretary problem via analytic expressions. J. Comb. Optim. 33, 1469–1491.
  • Vanderbei, R.J. (1980). The optimal choice of a subset of a population. Math. Oper. Res. 5, 481–486.
  • Vanderbei, R.J. (2012). The postdoc variant of the secretary problem. Tech. Report.
  • Yeo, G. F. (1997). Duration of a secretary problem. J. Appl. Probab. 34, 556–558.