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Optimal Stopping with Rank-Dependent Loss

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Alexander V. Gnedin
Alexander V. Gnedin*
Affiliation:
Utrecht University
*
∗Postal address: Department of Mathematics, Utrecht University, Postbus 80010, 3508 TA Utrecht, The Netherlands. Email address: gnedin@math.uu.nl
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Abstract

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For τ, a stopping rule adapted to a sequence of n independent and identically distributed observations, we define the loss to be E[q(Rτ)], where Rj is the rank of the jth observation and q is a nondecreasing function of the rank. This setting covers both the best-choice problem, with q(r) = 1(r > 1), and Robbins' problem, with q(r) = r. As n tends to ∞, the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit; thus answering a question asked by Bruss (2005) in the context of Robbins' problem.

Keywords

Optimal stoppingRobbins' problembest-choice problemplanar Poisson process

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 60G70: Extreme value theory; extremal processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 44 , Issue 4 , December 2007 , pp. 996 - 1011
Copyright
Copyright © Applied Probability Trust 2007 

References

[1] Assaf, D. and Samuel-Cahn, E. (1996). The secretary problem: minimizing the expected rank with i.i.d. random variables. Adv. Appl. Prob. 28, 828–852.Google Scholar
[2] Berezovsky, B. A. and Gnedin, A. V. (1984). The Best Choice Problem. Nauka, Moscow.Google Scholar
[3] Bruss, F. T. (2005). What is known about Robbins' problem? J. Appl. Prob. 42, 108–120.Google Scholar
[4] Bruss, F. T. and Ferguson, T. S. (1993). Minimizing the expected rank with full information. J. Appl. Prob. 30, 616–626.Google Scholar
[5] Bruss, F. T. and Ferguson, T. S. (1996). Half-prophets and Robbins' problem of minimizing the expected rank. In Athens Conf. Appl. Prob. Time Series Anal. (Lecture Notes Statist. 114), Springer, New York, pp. 1–17.Google Scholar
[6] Bruss, F. T. and Rogers, L. C. G. (1991). Embedding optimal selection problems in a Poisson process. Stoch. Process. Appl. 38, 267–278.Google Scholar
[7] 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.Google Scholar
[8] Frank, A. and Samuels, S. M. (1980). On an optimal stopping problem of Gusein-Zade. Stoch. Process. Appl. 10, 299–311.Google Scholar
[9] Gianini, J. (1977). The infinite secretary problem as the limit of the finite problem. Ann. Prob. 5, 636–644.Google Scholar
[10] Gianini, J. and Samuels, S. M. (1976). The infinite secretary problem. Ann. Prob. 4, 418–432.Google Scholar
[11] Gnedin, A. V. (1996). On the full-information best-choice problem. J. Appl. Prob. 33, 678–687.Google Scholar
[12] Gnedin, A. V. (2004). Best choice from the planar Poisson process. Stoch. Process. Appl. 111, 317–354.Google Scholar
[13] Gnedin, A. V. (2007). Recognising the last record of a sequence. Stochastics 79, 199–210.CrossRefGoogle Scholar
[14] Gnedin, A. V. and Miretskiy, D. I. (2007). Winning rate in the full-information best-choice problem. J. Appl. Prob. 44, 560–565.Google Scholar
[15] Hill, T. and Kennedy, D. (1992). Sharp inequalities for optimal stopping with rewards based on ranks. Ann. Appl. Prob. 2, 503–517.Google Scholar
[16] Kühne, R. and Rüschendorf, L. (2000). Approximation of optimal stopping problems. Stoch. Process. Appl. 90, 301–325.Google Scholar
[17] Mucci, A. (1973). Differential equations and optimal choice problems. Ann. Statist. 1, 104–113.Google Scholar
[18] Mucci, A. (1973). On a class of best-choice problems. Ann. Prob. 1, 417–427.Google Scholar
[19] Rubin, H. and Samuels, S. M. (1977). The finite-memory secretary problem. Ann. Prob. 5, 627–635.CrossRefGoogle Scholar
[20] Samuels, S. M. (1991). Secretary problems. In Handbook of Sequential Analysis, eds Ghosh, B. K. and Sen, P. K., Marcel Dekker, New York, pp. 381–405.Google Scholar
[21] Shiryaev, A. N. (1978). Optimal Stopping Rules, Springer, New York.Google Scholar