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2012, 2(2): 333-355. doi: 10.3934/naco.2012.2.333

Duration problem with multiple exchanges

1. 

The University of Adelaide, Department of Applied Mathathematics, Adelaide, South Australia 5005, Australia

2. 

Institute of Mathematics and Computer Science, Wrocław University of Techechnology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland

3. 

Department of Business Administration, Aichi University, Nagoya Campus, Hiraike 4-60-6, Nakamura, Nagoya, Aichi 453-8777, Japan

Received  October 2011 Revised  May 2012 Published  May 2012

We treat a version of the multiple-choice secretary problem called the multiple-choice duration problem, in which the objective is to maximize the time of possession of relatively best objects. It is shown that, for the $m$--choice duration problem, there exists a sequence $(s_1,s_2,\ldots,s_m)$ of critical numbers such that, whenever there remain $k$ choices yet to be made, then the optimal strategy immediately selects a relatively best object if it appears at or after time $s_k$ ($1\leq k\leq m$). We also exhibit an equivalence between the duration problem and the classical best-choice secretary problem. A simple recursive formula is given for calculating the critical numbers when the number of objects tends to infinity. Extensions are made to models involving an acquisition or replacement cost.
Citation: Charles E. M. Pearce, Krzysztof Szajowski, Mitsushi Tamaki. Duration problem with multiple exchanges. Numerical Algebra, Control & Optimization, 2012, 2 (2) : 333-355. doi: 10.3934/naco.2012.2.333
References:
[1]

M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,", Dover, (2000).   Google Scholar

[2]

K. Ano, Optimal selection problem with three stops,, J. Oper. Res. Soc., 32 (1989), 491.   Google Scholar

[3]

D. Assaf, L. M. Goldstein and E. Samuel-Cahn, Ratio prophet inequalities when the mortal has several choices,, Ann. Appl. Probab., 12 (2002), 972.  doi: 10.1214/aoap/1031863177.  Google Scholar

[4]

D. Assaf, L. M. Goldstein and E. Samuel-Cahn, Two-choice optimal stopping,, Adv. Appl. Probab., 36 (2004), 1116.  doi: 10.1239/aap/1103662960.  Google Scholar

[5]

D. Assaf and E. Samuel-Cahn, Simple ratio prophet inequalities for a mortal with multiple choices,, J. Appl. Probab., 37 (2000), 1084.  doi: 10.1239/jap/1014843085.  Google Scholar

[6]

Y. Chow, H. Robbins and D. Siegmund, "Great Expectations: The Theory of Optimal Stopping,", Houghton Mifflin Co., (1971).   Google Scholar

[7]

E. Dynkin and A. Yushkevich, "Markov Process, Theorems and Problems,", Plenum Press, (1969).   Google Scholar

[8]

R. Eidukjavicjus, Optimalna ostanovka markovskoj cepi dvumia momentami ostanovki,, Lit. Mat. Sb., 13 (1979), 181.   Google Scholar

[9]

T. Ferguson, Who solved the secretary problem?, Statist. Sci. 4 (1989), 4 (1989), 282.  doi: 10.1214/ss/1177012493.  Google Scholar

[10]

T. Ferguson, J. Hardwick and M. Tamaki, Maximizing the duration of owning a relatively best object, Strategies for sequential search and selection in real time (Amherst, MA, 1990)., Contemp. Math., 125 (1992), 37.  doi: 10.1090/conm/125/1160608.  Google Scholar

[11]

A. Frank and S. Samuels, On an optimal problem of Gusein-Zade,, Stoch. Proc. Appl., 10 (1980), 299.  doi: 10.1016/0304-4149(80)90013-7.  Google Scholar

[12]

J. P. Gilbert and F. Mosteller, Recognizing the maximum of a sequence,, J. Am. Stat. Assoc., 61 (1966), 35.   Google Scholar

[13]

A. V. Gnedin, Best choice from the planar poisson process,, Stoch. Proc. Appl., 111 (2004), 317.  doi: 10.1016/j.spa.2003.12.005.  Google Scholar

[14]

A. V. Gnedin, Objectives in the best-choice problems,, Sequential Analysis, 24 (2005), 1.  doi: 10.1081/SQA-200056196.  Google Scholar

[15]

I. Gradshteyn and I. Ryzhik, "Tables of Integrals, Series, and Products,", 6th Edition, (2000).   Google Scholar

[16]

S. M. Gusein-Zade, The problem of choice and the optimal stopping rule for a sequence of independent trials,, Theory Probab. Appl., 11 (1966), 472.  doi: 10.1137/1111050.  Google Scholar

[17]

G. W. Haggstrom, Optimal sequential procedures when more then one stop is required,, Ann. Math. Stat., 38 (1967), 1618.  doi: 10.1214/aoms/1177698595.  Google Scholar

[18]

M. Hu, M. Tamaki and K. Ohno, A continuous time duration problem with an unknown number of options,, Math. Japonica, 48 (1998), 233.   Google Scholar

[19]

A. Kurushima and K. Ano, On optimal stopping problems for possession-period maximization associated with Poisson arrival,, Mathematics of decision-making under uncertainty (Japanese) (Kyoto, 1306 (2003), 11.   Google Scholar

[20]

A. Kurushima and K. Ano, Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations,, J. Appl. Probab., 46 (2009), 402.  doi: 10.1239/jap/1245676096.  Google Scholar

[21]

A. Kurushima and K. Ano, Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations,, 13-th International Symposium on Dyn. Games and Appl., (2008).   Google Scholar

[22]

R. Kühne and L. Rüschendorf, On optimal two-stopping problems,, In, II (2002), 261.   Google Scholar

[23]

A. Lehtinen, The best-choice problem with an unknown number of objects,, Z. Oper.Res., 37 (1993), 97.   Google Scholar

[24]

V. V. Mazalov and M. Tamaki, Explicit solutions to the duration problem,, Aichi Keiei Ronsyu, 147 (2003), 69.   Google Scholar

[25]

V. V. Mazalov and M. Tamaki, An explicit formula for the limiting optimal gain in the full information duration problem,, Mathematics of decision-making under uncertainty (Japanese) (Kyoto, 1306 (2003), 217.   Google Scholar

[26]

V. V. Mazalov and M. Tamaki, An explicit formula for the optimal gain in the full-information problem of owning a relatively best object,, J. Appl. Probab., 43 (2006), 87.  doi: 10.1239/jap/1143936245.  Google Scholar

[27]

V. V. Mazalov and M. Tamaki, Duration problem on trajectories,, Stochastics, 79 (2007), 211.  doi: 10.1080/17442500601072704.  Google Scholar

[28]

T. F. Móri, The random secretary problem with multiple choice,, Ann. Univ. Sci. Budapest. de Rolando Eötvös Nominatae. Sect. Comput., 5 (1984), 91.   Google Scholar

[29]

A. G. Mucci, Differential equations and optimal choice problem,, Ann. Stat., 1 (1973), 104.  doi: 10.1214/aos/1193342386.  Google Scholar

[30]

A. G. Mucci, On a class of secretary problems,, Ann. Probab., 1 (1973), 417.  doi: 10.1214/aop/1176996936.  Google Scholar

[31]

M. L. Nikolaev, Zadacha vybora dvokh ob"ektov z minimal'nom sumarnym rangom,, Izv. Vyss. Uchebn. Zaved. Mat., 166 (1976), 33.   Google Scholar

[32]

M. L. Nikolaev, Ob odnom obobshchenii zadachi nailuchshego vybora,, Teor. Veroyatn. Primen., 22 (1977), 191.  doi: 10.1137/1122023.  Google Scholar

[33]

M. L. Nikolaev, Obobshchennyje posledovatelnyje procedury,, Litov. Mat. Sb., 191 (1979), 35.   Google Scholar

[34]

M. Nikolaev, Optimal multi-stopping rules,, Obozr. Prikl. Prom. Mat., 5 (1998), 309.   Google Scholar

[35]

M. L. Nikolaev, On optimal multiple stopping of Markov sequences,, Teor. Veroyatnost. i Primenen., 43 (1998), 374.   Google Scholar

[36]

J. Petruccelli, On the best-choice problem when the number of observation is random,, J. Appl. Probab., 20 (1983), 165.  doi: 10.2307/3213731.  Google Scholar

[37]

Z. Porosiński, The full-information best choice problem with a random number of observations,, Stoch. Proc. Appl., 24 (1987), 293.  doi: 10.1016/0304-4149(87)90020-2.  Google Scholar

[38]

Z. Porosinski, On best choice problems having similar solutions,, Stat. Probab. Lett., 56 (2002), 321.  doi: 10.1016/S0167-7152(01)00201-2.  Google Scholar

[39]

J. Preater, On multiple choice secretary problem,, Math. Oper. Res., 19 (): 597.  doi: 10.1287/moor.19.3.597.  Google Scholar

[40]

E. Presman and I. Sonin, The best choice problem for a random number of objects,, Theory Prob. Appl., 17 (1972), 657.  doi: 10.1137/1117078.  Google Scholar

[41]

S. M. Ross, "Applied Probability Models with Optimization Applications,", Holden-Day, (1970).   Google Scholar

[42]

M. P. Quine and J. S. Law, Exact results for a secretary problem,, J. Appl. Probab., 33 (1996), 630.  doi: 10.2307/3215345.  Google Scholar

[43]

M. Sakaguchi, Dowry problem and OLA policies,, Rep. Stat. Appl. Res. Union Jap. Sci. Eng. JUSE, 25 (1978), 124.   Google Scholar

[44]

M. Sakaguchi, Generalized secretary problems with three stops,, Math. Japonica, 32 (1987), 105.   Google Scholar

[45]

S. M. Samuels, Secretary problems,, In, (1991), 381.   Google Scholar

[46]

S. M. Samuels, Why do these quite different best-choice problems have the same solutions?, Adv. Appl. Probab., 36 (2004), 398.  doi: 10.1239/aap/1086957578.  Google Scholar

[47]

W. Stadje, On multiple stopping rules,, Optimization, 16 (1985), 401.  doi: 10.1080/02331938508843030.  Google Scholar

[48]

A. Suchwałko and K. Szajowski, Non standard, no information secretary problems,, Sci. Math. Japonicae, 56 (2002), 443.   Google Scholar

[49]

K. Szajowski, Optimal choice problem of $a$-th object,, Matem. Stos., 19 (1982), 51.   Google Scholar

[50]

K. Szajowski, A two-disorder detection problem,, Applicationes Mathematicae, 24 (1996), 231.   Google Scholar

[51]

K. Szajowski and M. Tamaki, Shelf life of candidates in the generalized secretary problem,, preprint, (2006).   Google Scholar

[52]

K. Szajowski, On a random number of disorders,, Probab. Math. Statist., 31 (2011), 17.   Google Scholar

[53]

K. Szajowski, On stopping games when more than one stop is possible,, In, (2000), 57.   Google Scholar

[54]

M. Tamaki, A secretary problem with double choice,, J. Oper. Res. Soc. Jap., 22 (1979), 257.   Google Scholar

[55]

M. Tamaki, OLA policy and the best choice problem with random number of objects,, Math. Japonica, 24 (1979), 451.   Google Scholar

[56]

M. Tamaki, C. E. Pearce and K. Szajowski, Multiple choice problems related to the duration of the secretary problem,, Sūrikaisekikenkyūsho Kōkyūroku 1068 (1998), 1068 (1998), 75.   Google Scholar

[57]

J. Wilson, Optimal choice and assignment of the best $m$ of $n$ randomly arriving items,, Stoch. Proc. Appl., 39 (1991), 325.  doi: 10.1016/0304-4149(91)90086-R.  Google Scholar

[58]

M. Yasuda, On a stopping problem involving refusal and forced stopping,, J. Appl. Probab., 20 (1983), 71.  doi: 10.2307/3213721.  Google Scholar

[59]

M. Yasuda and K. Szajowski, Dynkin games and its extension to a multiple stopping model,, Bulletin of the Japan Society for Industrial Mathematics 12 (2002), 12 (2002), 17.   Google Scholar

[60]

A. J. Yeo and G. F. Yeo, Selecting satisfactory secretaries,, Austral. J. Statist., 36 (1994), 185.   Google Scholar

[61]

G. F. Yeo, Duration of a secretary problem,, J. Appl. Probab., 34 (1997), 556.   Google Scholar

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables,", Dover, (2000).   Google Scholar

[2]

K. Ano, Optimal selection problem with three stops,, J. Oper. Res. Soc., 32 (1989), 491.   Google Scholar

[3]

D. Assaf, L. M. Goldstein and E. Samuel-Cahn, Ratio prophet inequalities when the mortal has several choices,, Ann. Appl. Probab., 12 (2002), 972.  doi: 10.1214/aoap/1031863177.  Google Scholar

[4]

D. Assaf, L. M. Goldstein and E. Samuel-Cahn, Two-choice optimal stopping,, Adv. Appl. Probab., 36 (2004), 1116.  doi: 10.1239/aap/1103662960.  Google Scholar

[5]

D. Assaf and E. Samuel-Cahn, Simple ratio prophet inequalities for a mortal with multiple choices,, J. Appl. Probab., 37 (2000), 1084.  doi: 10.1239/jap/1014843085.  Google Scholar

[6]

Y. Chow, H. Robbins and D. Siegmund, "Great Expectations: The Theory of Optimal Stopping,", Houghton Mifflin Co., (1971).   Google Scholar

[7]

E. Dynkin and A. Yushkevich, "Markov Process, Theorems and Problems,", Plenum Press, (1969).   Google Scholar

[8]

R. Eidukjavicjus, Optimalna ostanovka markovskoj cepi dvumia momentami ostanovki,, Lit. Mat. Sb., 13 (1979), 181.   Google Scholar

[9]

T. Ferguson, Who solved the secretary problem?, Statist. Sci. 4 (1989), 4 (1989), 282.  doi: 10.1214/ss/1177012493.  Google Scholar

[10]

T. Ferguson, J. Hardwick and M. Tamaki, Maximizing the duration of owning a relatively best object, Strategies for sequential search and selection in real time (Amherst, MA, 1990)., Contemp. Math., 125 (1992), 37.  doi: 10.1090/conm/125/1160608.  Google Scholar

[11]

A. Frank and S. Samuels, On an optimal problem of Gusein-Zade,, Stoch. Proc. Appl., 10 (1980), 299.  doi: 10.1016/0304-4149(80)90013-7.  Google Scholar

[12]

J. P. Gilbert and F. Mosteller, Recognizing the maximum of a sequence,, J. Am. Stat. Assoc., 61 (1966), 35.   Google Scholar

[13]

A. V. Gnedin, Best choice from the planar poisson process,, Stoch. Proc. Appl., 111 (2004), 317.  doi: 10.1016/j.spa.2003.12.005.  Google Scholar

[14]

A. V. Gnedin, Objectives in the best-choice problems,, Sequential Analysis, 24 (2005), 1.  doi: 10.1081/SQA-200056196.  Google Scholar

[15]

I. Gradshteyn and I. Ryzhik, "Tables of Integrals, Series, and Products,", 6th Edition, (2000).   Google Scholar

[16]

S. M. Gusein-Zade, The problem of choice and the optimal stopping rule for a sequence of independent trials,, Theory Probab. Appl., 11 (1966), 472.  doi: 10.1137/1111050.  Google Scholar

[17]

G. W. Haggstrom, Optimal sequential procedures when more then one stop is required,, Ann. Math. Stat., 38 (1967), 1618.  doi: 10.1214/aoms/1177698595.  Google Scholar

[18]

M. Hu, M. Tamaki and K. Ohno, A continuous time duration problem with an unknown number of options,, Math. Japonica, 48 (1998), 233.   Google Scholar

[19]

A. Kurushima and K. Ano, On optimal stopping problems for possession-period maximization associated with Poisson arrival,, Mathematics of decision-making under uncertainty (Japanese) (Kyoto, 1306 (2003), 11.   Google Scholar

[20]

A. Kurushima and K. Ano, Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations,, J. Appl. Probab., 46 (2009), 402.  doi: 10.1239/jap/1245676096.  Google Scholar

[21]

A. Kurushima and K. Ano, Maximizing the expected duration of owning a relatively best object in a Poisson process with rankable observations,, 13-th International Symposium on Dyn. Games and Appl., (2008).   Google Scholar

[22]

R. Kühne and L. Rüschendorf, On optimal two-stopping problems,, In, II (2002), 261.   Google Scholar

[23]

A. Lehtinen, The best-choice problem with an unknown number of objects,, Z. Oper.Res., 37 (1993), 97.   Google Scholar

[24]

V. V. Mazalov and M. Tamaki, Explicit solutions to the duration problem,, Aichi Keiei Ronsyu, 147 (2003), 69.   Google Scholar

[25]

V. V. Mazalov and M. Tamaki, An explicit formula for the limiting optimal gain in the full information duration problem,, Mathematics of decision-making under uncertainty (Japanese) (Kyoto, 1306 (2003), 217.   Google Scholar

[26]

V. V. Mazalov and M. Tamaki, An explicit formula for the optimal gain in the full-information problem of owning a relatively best object,, J. Appl. Probab., 43 (2006), 87.  doi: 10.1239/jap/1143936245.  Google Scholar

[27]

V. V. Mazalov and M. Tamaki, Duration problem on trajectories,, Stochastics, 79 (2007), 211.  doi: 10.1080/17442500601072704.  Google Scholar

[28]

T. F. Móri, The random secretary problem with multiple choice,, Ann. Univ. Sci. Budapest. de Rolando Eötvös Nominatae. Sect. Comput., 5 (1984), 91.   Google Scholar

[29]

A. G. Mucci, Differential equations and optimal choice problem,, Ann. Stat., 1 (1973), 104.  doi: 10.1214/aos/1193342386.  Google Scholar

[30]

A. G. Mucci, On a class of secretary problems,, Ann. Probab., 1 (1973), 417.  doi: 10.1214/aop/1176996936.  Google Scholar

[31]

M. L. Nikolaev, Zadacha vybora dvokh ob"ektov z minimal'nom sumarnym rangom,, Izv. Vyss. Uchebn. Zaved. Mat., 166 (1976), 33.   Google Scholar

[32]

M. L. Nikolaev, Ob odnom obobshchenii zadachi nailuchshego vybora,, Teor. Veroyatn. Primen., 22 (1977), 191.  doi: 10.1137/1122023.  Google Scholar

[33]

M. L. Nikolaev, Obobshchennyje posledovatelnyje procedury,, Litov. Mat. Sb., 191 (1979), 35.   Google Scholar

[34]

M. Nikolaev, Optimal multi-stopping rules,, Obozr. Prikl. Prom. Mat., 5 (1998), 309.   Google Scholar

[35]

M. L. Nikolaev, On optimal multiple stopping of Markov sequences,, Teor. Veroyatnost. i Primenen., 43 (1998), 374.   Google Scholar

[36]

J. Petruccelli, On the best-choice problem when the number of observation is random,, J. Appl. Probab., 20 (1983), 165.  doi: 10.2307/3213731.  Google Scholar

[37]

Z. Porosiński, The full-information best choice problem with a random number of observations,, Stoch. Proc. Appl., 24 (1987), 293.  doi: 10.1016/0304-4149(87)90020-2.  Google Scholar

[38]

Z. Porosinski, On best choice problems having similar solutions,, Stat. Probab. Lett., 56 (2002), 321.  doi: 10.1016/S0167-7152(01)00201-2.  Google Scholar

[39]

J. Preater, On multiple choice secretary problem,, Math. Oper. Res., 19 (): 597.  doi: 10.1287/moor.19.3.597.  Google Scholar

[40]

E. Presman and I. Sonin, The best choice problem for a random number of objects,, Theory Prob. Appl., 17 (1972), 657.  doi: 10.1137/1117078.  Google Scholar

[41]

S. M. Ross, "Applied Probability Models with Optimization Applications,", Holden-Day, (1970).   Google Scholar

[42]

M. P. Quine and J. S. Law, Exact results for a secretary problem,, J. Appl. Probab., 33 (1996), 630.  doi: 10.2307/3215345.  Google Scholar

[43]

M. Sakaguchi, Dowry problem and OLA policies,, Rep. Stat. Appl. Res. Union Jap. Sci. Eng. JUSE, 25 (1978), 124.   Google Scholar

[44]

M. Sakaguchi, Generalized secretary problems with three stops,, Math. Japonica, 32 (1987), 105.   Google Scholar

[45]

S. M. Samuels, Secretary problems,, In, (1991), 381.   Google Scholar

[46]

S. M. Samuels, Why do these quite different best-choice problems have the same solutions?, Adv. Appl. Probab., 36 (2004), 398.  doi: 10.1239/aap/1086957578.  Google Scholar

[47]

W. Stadje, On multiple stopping rules,, Optimization, 16 (1985), 401.  doi: 10.1080/02331938508843030.  Google Scholar

[48]

A. Suchwałko and K. Szajowski, Non standard, no information secretary problems,, Sci. Math. Japonicae, 56 (2002), 443.   Google Scholar

[49]

K. Szajowski, Optimal choice problem of $a$-th object,, Matem. Stos., 19 (1982), 51.   Google Scholar

[50]

K. Szajowski, A two-disorder detection problem,, Applicationes Mathematicae, 24 (1996), 231.   Google Scholar

[51]

K. Szajowski and M. Tamaki, Shelf life of candidates in the generalized secretary problem,, preprint, (2006).   Google Scholar

[52]

K. Szajowski, On a random number of disorders,, Probab. Math. Statist., 31 (2011), 17.   Google Scholar

[53]

K. Szajowski, On stopping games when more than one stop is possible,, In, (2000), 57.   Google Scholar

[54]

M. Tamaki, A secretary problem with double choice,, J. Oper. Res. Soc. Jap., 22 (1979), 257.   Google Scholar

[55]

M. Tamaki, OLA policy and the best choice problem with random number of objects,, Math. Japonica, 24 (1979), 451.   Google Scholar

[56]

M. Tamaki, C. E. Pearce and K. Szajowski, Multiple choice problems related to the duration of the secretary problem,, Sūrikaisekikenkyūsho Kōkyūroku 1068 (1998), 1068 (1998), 75.   Google Scholar

[57]

J. Wilson, Optimal choice and assignment of the best $m$ of $n$ randomly arriving items,, Stoch. Proc. Appl., 39 (1991), 325.  doi: 10.1016/0304-4149(91)90086-R.  Google Scholar

[58]

M. Yasuda, On a stopping problem involving refusal and forced stopping,, J. Appl. Probab., 20 (1983), 71.  doi: 10.2307/3213721.  Google Scholar

[59]

M. Yasuda and K. Szajowski, Dynkin games and its extension to a multiple stopping model,, Bulletin of the Japan Society for Industrial Mathematics 12 (2002), 12 (2002), 17.   Google Scholar

[60]

A. J. Yeo and G. F. Yeo, Selecting satisfactory secretaries,, Austral. J. Statist., 36 (1994), 185.   Google Scholar

[61]

G. F. Yeo, Duration of a secretary problem,, J. Appl. Probab., 34 (1997), 556.   Google Scholar

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