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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: tamaki@vega.aichi-u.ac.jp
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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 

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