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Optimal Choice of the Best Available Applicant in Full-Information Models

Part of: Sequential methods Stochastic processes

Published online by Cambridge University Press:  14 July 2016

Mitsushi Tamaki
Mitsushi Tamaki*
Affiliation:
Aichi University
*
∗Postal address: Department of Business Administration, Aichi University, Nagoya Campus, 370 Kurozasa, Miyoshi, Nishikamo, Aichi 470-0296, Japan. Email address: [email protected]
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Abstract

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The problem we consider here is a full-information best-choice problem in which n applicants appear sequentially, but each applicant refuses an offer independently of other applicants with known fixed probability 0≤q<1. The objective is to maximize the probability of choosing the best available applicant. Two models are distinguished according to when the availability can be ascertained; the availability is ascertained just after the arrival of the applicant (Model 1), whereas the availability can be ascertained only when an offer is made (Model 2). For Model 1, we can obtain the explicit expressions for the optimal stopping rule and the optimal probability for a given n. A remarkable feature of this model is that, asymptotically (i.e. n→∞), the optimal probability becomes insensitive to q and approaches 0.580 164. The planar Poisson process (PPP) model provides more insight into this phenomenon. For Model 2, the optimal stopping rule depends on the past history in a complicated way and seems to be intractable. We have not solved this model for a finite n but derive, via the PPP approach, a lower bound on the asymptotically optimal probability.

Keywords

Best-choice problemsecretary problemoptimal stoppingplanar Poisson processinsensitivityfull-history dependenceRobbins' problem

MSC classification

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory 62L15: Optimal stopping
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 4 , December 2009 , pp. 1086 - 1099
Copyright
Copyright © Applied Probability Trust 2009 

References

Berezovskĭ, B. A. and Gnedin, A. V. (1984). The Problem of Optimal Choice. Nauka, Moscow (in Russian).Google Scholar
Bruss, F. T. (2005). What is known about Robbins' problem? J. Appl. Prob. 42, 108–120.Google Scholar
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
Bruss, F. T. and Swan, Y. C. (2009). A continuous-time approach to Robbins' problem of minimizing the expected rank. J. Appl. Prob. 46, 1–18.Google Scholar
Das, S. and Tsitsiklis, J. N. (2009). When is it important to know you've been rejected? A search problem with probabilistic appearance of offers. To appear in J. Econom. Behavior Organization.Google Scholar
Gilbert, J. P. and Mosteller, F. (1966). Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.CrossRefGoogle Scholar
Gnedin, A. V. (1996). On the full information best-choice problem. J. Appl. Prob. 33, 678–687.Google Scholar
Gnedin, A. V. (2004). Best choice from the planar Poisson process. Stoch. Process. Appl. 111, 317–354.Google Scholar
Petruccelli, J. D. (1982). Full-information best-choice problems with recall of observations and uncertainty of selection depending on the observation. Adv. Appl. Prob. 14, 340–358.Google Scholar
Porosinski, Z. (1987). The full-information best choice problem with a random number of observations. Stoch. Process. Appl. 24, 293–307.Google Scholar
Sakaguchi, M. (1973). A note on the dowry problem. Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs. 20, 11–17.Google Scholar
Samuels, S. M. (1982). Exact solutions for the full information best choice problem. Mimeo Ser. 82–17, Department of Statistics, Purdue University.Google Scholar
Samuels, S. M. (1991). Secretary problems. In Handbook of Sequential Analysis, eds Gosh, B. K. and Sen, P. K., Marcel Dekker, New York, pp. 381–405.Google Scholar
Samuels, S. M. (2004). Why do these quite different best-choice problems have the same solutions? Adv. Appl. Prob. 36, 398–416.Google Scholar
Smith, M. H. (1975). A secretary problem with uncertain employment. J. Appl. Prob. 12, 620–624.Google Scholar
Tamaki, M. (1991). A secretary problem with uncertain employment and best choice of available candidates. Operat. Res. 39, 274–284.Google Scholar