Shelf life of candidates in the generalized secretary problem

The study presents a version of the secretary problem called the duration problem in which the objective is to maximize the time of possession of the relatively best or the second best objects. It is shown that in this duration problem there are threshold numbers such that the optimal strategy is determined by them. When the number of objects tends to infinity the thresholds values are ź 0.120381 N ź and ź 0.417188 N ź , respectively, and the asymptotic mean time of shelf life is 0.403827 N .

[1]  Thomas S. Ferguson,et al.  Who Solved the Secretary Problem , 1989 .

[2]  L. P. Kok,et al.  Table errata: Handbook of mathematical functions, with formulas, graphs, and mathematical tables [Dover, New York, 1966; MR 34 #8606] , 1983 .

[3]  Geoffrey F. Yeo.  DURATION OF A SECRETARY PROBLEM , 1997 .

[4]  Z. Porosiński.  The full-information best choice problem with a random number of observations , 1987 .

[5]  V. Mazalov,et al.  An explicit formula for the optimal gain in the full-information problem of owning a relatively best object , 2006 .

[6]  T. Ferguson,et al.  Maximizing the duration of owning a relatively best object 1 , 2001 .

[7]  Isaac M. Sonin,et al.  The Best Choice Problem for a Random Number of Objects , 1973 .

[8]  Z. Porosiński.  On best choice problems having similar solutions , 2002 .

[9]  F. Mosteller,et al.  Recognizing the Maximum of a Sequence , 1966 .

[10]  S. M. Gusein-Zade,et al.  The Problem of Choice and the Sptimal Stopping Rule for a Sequence of Independent Trials , 1966 .

[11]  W. E. H. B.,et al.  Aufgaben und Lehrsätze aus der Analysis. , 1925, Nature.

[12]  Geoffrey F. Yeo,et al.  SELECTING SATISFACTORY SECRETARIES , 1994 .

[13]  P. Freeman.  The Secretary Problem and its Extensions: A Review , 1983 .

[14]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[15]  Best Choice from the Planar Poisson Process , 2002, math/0209050.

[16]  E. Platen,et al.  About secretary problems , 1980 .

[17]  G. Pólya,et al.  Problems and Theorems in Analysis I: Series. Integral Calculus. Theory of Functions , 1976 .

[18]  S. M. Samuels.  Why do these quite different best-choice problems have the same solutions? , 2004, Advances in Applied Probability.

[19]  Optimal Stopping with Rank-Dependent Loss , 2007, Journal of Applied Probability.

[20]  J. Petruccelli.  On a Best Choice Problem with Partial Information , 1980 .

[21]  Mitsushi Tamaki.  Recognizing both the maximum and the second maximum of a sequence , 1979 .

[22]  Luc Pronzato,et al.  Optimal and asymptotically optimal decision rules for sequential screening and resource allocation , 2001, IEEE Trans. Autom. Control..

[23]  M. Quine,et al.  Exact results for a secretary problem , 1996, Journal of Applied Probability.

[24]  F Thomas Bruss,et al.  Strategies for Sequential Search and Selection in Real Time , 1992 .

[25]  Jorge Nuno Silva,et al.  Mathematical Games , 1959, Nature.

[26]  C. Pearce,et al.  Multiple Choice Problems Related to the Duration of the Secretary Problem , 1998 .

[27]  C. Pearce,et al.  The duration problem with multiple exchanges , 2008, 0812.3765.

[28]  M. Kawai,et al.  Choosing either the best or the second best when the number of applicants is random , 2003 .

[29]  Helly.  Aufgaben und Lehrsätze aus der Analysis , 1928 .