Source: J. Appl. Probab.
Volume 46, Number 4
The problem we consider here is a full-information best-choice problem in
applicants appear sequentially, but each applicant refuses an
offer independently of other applicants with known fixed probability
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.
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.
Full-text: Access denied (no subscription
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Berezovski\u i, B. A. and Gnedin, A. V. (1984). The Problem of Optimal Choice. Nauka, Moscow (in Russian).
Mathematical Reviews (MathSciNet): MR768372
Bruss, F. T. (2005). What is known about Robbins' problem? J. Appl. Prob. 42, 108--120.
Bruss, F. T. and Rogers, L. C. G. (1991). Embedding optimal selection problems in a Poisson process. Stoch. Process. Appl. 38, 267--278.
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.
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. %Tech. Rep. 2699, MIT Laboratory for Information and Decision Systems.
Gilbert, J. P. and Mosteller, F. (1966). Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35--73.
Mathematical Reviews (MathSciNet): MR198637
Gnedin, A. V. (1996). On the full information best-choice problem. J. Appl. Prob. 33, 678--687.
Gnedin, A. V. (2004). Best choice from the planar Poisson process. Stoch. Process. Appl. 111, 317--354.
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.
Mathematical Reviews (MathSciNet): MR650127
Porosinski, Z. (1987). The full-information best choice problem with a random number of observations. Stoch. Process. Appl. 24, 293--307.
Mathematical Reviews (MathSciNet): MR893177
Sakaguchi, M. (1973). A note on the dowry problem. Rep. Statist. Appl. Res. Un. Japan. Sci. Engrs. 20, 11--17.
Mathematical Reviews (MathSciNet): MR329160
Samuels, S. M. (1982). Exact solutions for the full information best choice problem. Mimeo Ser. 82--17, Department of Statistics, Purdue University.
Samuels, S. M. (1991). Secretary problems. In Handbook of Sequential Analysis, eds B. K. Gosh and P. K. Sen, Marcel Dekker, New York, pp. 381--405.
Samuels, S. M. (2004). Why do these quite different best-choice problems have the same solutions? Adv. Appl. Prob. 36, 398--416.
Smith, M. H. (1975). A secretary problem with uncertain employment. J. Appl. Prob. 12, 620--624.
Mathematical Reviews (MathSciNet): MR378292
Tamaki, M. (1991). A secretary problem with uncertain employment and best choice of available candidates. Operat. Res. 39, 274--284.