EVALUATING OPTIMAL STOPPING RULES IN THE MULTIPLE BEST CHOICE PROBLEM USING THE CROSS-ENTROPY METHOD

Best choice problems can be considered one of the most interesting problems of sequential decision analysis. Problems of this type can arise in a wide variety of fields, including psychological, economical, and ecological applications. In this study, we consider a generalization of the best choice problem when it is possible to make more than one choice. We use the Cross-Entropy method to determine the optimal stopping rules and the value of a game. We include results of numerical experiments illustrating the effectiveness of the approach. We obtain estimates of the thresholds in the optimal stopping rules and compare the accuracy of these estimates with those obtained via asymptotic approximation.

[1]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Optimal stopping rules , 1977 .

[2]  M. L. Nikolaev.  On a Generalization of the Best Choice Problem , 1977 .

[3]  Darryl A. Seale,et al.  Sequential Decision Making with Relative Ranks: An Experimental Investigation of the "Secretary Problem"> , 1997 .

[4]  M. L. Nikolaev,et al.  A multiple optimal stopping rule for sums of independent random variables , 2007 .

[5]  Israel David,et al.  Explicit results for a class of asset-selling problems , 1998, Eur. J. Oper. Res..

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

[8]  Jonathan M. Keith,et al.  An optimal sequential procedure for a buying-selling problem with independent observations , 2006, Journal of Applied Probability.

[9]  Ryan O. Murphy,et al.  Sequential Observation and Selection with Rank-Dependent Payoffs: An Experimental Study , 2006, Manag. Sci..

[10]  John M C Hutchinson,et al.  Mate choice when males are in patches: optimal strategies and good rules of thumb. , 2004, Journal of theoretical biology.

[11]  Amnon Rapoport,et al.  Analysis of heuristic solutions to the best choice problem , 2003, Eur. J. Oper. Res..

[12]  Tatiana V. Polushina.  Estimating optimal stopping rules in the multiple best choice problem with minimal summarized rank via the Cross-Entropy method , 2009, 2009 IEEE Congress on Evolutionary Computation.

[13]  S. Albright.  A Bayesian Approach to a Generalized House Selling Problem , 1977 .

[14]  George Y. Sofronov,et al.  An optimal sequential procedure for a multiple selling problem with independent observations , 2013, Eur. J. Oper. Res..

[15]  D. Rosenfield,et al.  Optimal Strategies for Selling an Asset , 1983 .

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

[17]  Israel David,et al.  Asset-selling problems with holding costs , 2001 .

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

[19]  A. Janetos.  Strategies of female mate choice: A theoretical analysis , 1980, Behavioral Ecology and Sociobiology.

[20]  Leslie A. Real,et al.  Search Theory and Mate Choice. II. Mutual Interaction, Assortative Mating, and Equilibrium Variation in Male and Female Fitness , 1991, The American Naturalist.

[21]  C. Gabor,et al.  Sequential mate choice by multiply mating smooth newts: females become more choosy , 1997 .

[22]  Dirk P. Kroese,et al.  The Cross-Entropy Method: A Unified Approach to Combinatorial Optimization, Monte-Carlo Simulation and Machine Learning , 2004 .

[23]  G. Simons.  Great Expectations: Theory of Optimal Stopping , 1973 .

[24]  F. Riedel,et al.  The best choice problem under ambiguity , 2013 .

[25]  Y. Dombrovsky,et al.  On Adaptive Search and Optimal Stopping in Sequential Mate Choice , 1994, The American Naturalist.

[26]  F. H. Rodd,et al.  Multiple mating and sequential mate choice in guppies: females trade up , 2003, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[27]  T. Birkhead.  The Evolution of Avian Breeding Systems , 1999, Heredity.

[28]  S. Lippman,et al.  THE ECONOMICS OF JOB SEARCH: A SURVEY* , 1976 .

[29]  L. Real.  Search Theory and Mate Choice. I. Models of Single-Sex Discrimination , 1990, The American Naturalist.

[30]  J. Neil Bearden,et al.  A new secretary problem with rank-based selection and cardinal payoffs , 2006 .

[31]  J. Ligon,et al.  The Evolution of Avian Breeding Systems , 1999 .

[32]  Helmut Schneider,et al.  Buying and selling an asset over the finite time horizon: A non-parametric approach , 2002, Eur. J. Oper. Res..