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Double stopping by two decision-makers

Part of: Stochastic processes

Published online by Cambridge University Press:  01 July 2016

K. Szajowski
K. Szajowski*
Affiliation:
Delft University of Technology
*
∗Permanent address: Institute of Mathematics, Technical University of Wrocław, Wybrzeże Wyspiańskiego 27, PL-50-370 Wroclaw, Poland.
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Abstract

A problem of optimal stopping of the discrete-time Markov process by two decision-makers (Player 1 and Player 2) in a competitive situation is considered. The zero-sum game structure is adopted. The gain function depends on states chosen by both decision-makers. When both players want to accept the realization of the Markov process at the same moment, the priority is given to Player 1. The construction of the value function and the optimal strategies for the players are given. The Markov chain case is considered in detail. An example related to the generalized secretary problem is solved.

Keywords

OPTIMAL STOPPINGGAME VARIANTMARKOV PROCESSSECRETARY PROBLEMZERO-SUM TWO-PERSON GAME: PRIORITY OF THE PLAYERS

MSC classification

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory
Type
Research Article
Information
Advances in Applied Probability , Volume 25 , Issue 2 , June 1993 , pp. 438 - 452
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

Research partially supported by the Research Fellowship Committee of Delft University of Technology, while the author was visiting the Faculty of Technical Mathematics and Informatics there.

References

[1] Ano, K. (1990) Bilateral secretary problem recognizing the maximum or the second maximum of a sequence. J. Informat. Optim. Sci. 11, 177–188.Google Scholar
[2] Dynkin, E. (1969) Game variant of a problem on optimal stopping. Soviet Math. Dokl. 10, 270–274.Google Scholar
[3] Dynkin, E. and Yushkevich, A. (1969) Theorems and Problems on Markov Processes. Plenum, New York.Google Scholar
[4] Eidukjavicjus, R. (1979) Optimalna ostanovka markovskoj cepi dvumia momentami ostanovki. Lit. Mat. Sb. 19, 181–183.Google Scholar
[5] Enns, E. and Ferenstein, E. (1985) The horse game. J. Operat. Res. Soc. Japan 28, 51–62.Google Scholar
[6] Enns, E. and Ferenstein, E. (1987) On a multi-person time-sequential game with priorities. Sequential Anal. 6, 239–256.Google Scholar
[7] Enns, E. and Ferenstein, E. (1990) A competitive best-choice problem with Poisson arrivals. J. Appl. Prob. 27, 333–342.CrossRefGoogle Scholar
[8] Ferenstein, E. (1990) On a two-person time-sequential game with priorities on Markov chains. Sequential Anal. Submitted.Google Scholar
[9] Ferenstein, E. (1992) Two-person non-zero-sum games with priorities. In Strategies for Sequential Search and Selection in Real Time, Proc. AMS-IMS-SIAM Joint Summer Research Conf. 1990, ed. Ferguson, T. S. and Samuels, S. M., Contemporary Mathematics , 125, pp. 119–123, University of Massachusetts at Amherst.Google Scholar
[10] Freeman, P. (1983) The secretary problem and its extensions: a review. Internat. Statist. Rev. 51, 189–206.Google Scholar
[11] Fushimi, M. (1981) The secretary problem in a competitive situation. J. Operat. Res. Soc. Japan 24, 350–358.Google Scholar
[12] Gilbert, J. and Mosteller, F. (1966) Recognizing the maximum of a sequence. J. Amer. Statist. Assoc. 61, 35–73.Google Scholar
[13] Haggstrom, G. (1967) Optimal sequential procedures when more than one stop is required. Ann. Math. Statist. 38, 1618–1626.Google Scholar
[14] Luce, R. and Raiffa, H. (1957) Games and Decisions. Wiley, New York.Google Scholar
[15] Majumdar, A. (1985) Optimal stopping for a two-person sequential game in the continuous case. Pure Appl. Math. Sci. 22, 79–89.Google Scholar
[16] Majumdar, A. (1986) Optimal stopping for a two-person sequential game in the discrete case. Pure Appl. Math. Sci. 23, 67–75.Google Scholar
[17] Moler, C., Little, J. and Bangert, S. (1987) PC-MATLAB User's Guide. The Math. Works, Sherborn.Google Scholar
[18] Mucci, A. (1973) Differential equations and optimal choice problem. Ann. Statist. 1, 104–113.CrossRefGoogle Scholar
[19] Neveu, J. (1975) Discrete-Parameter Martingales. North-Holland, Amsterdam.Google Scholar
[20] Preater, J. (1988) A survey of secretary problems with multiple appointments. Research Report 88.12, Dept, of Math., Univ. of Keele, Keele, ST5 5BG, UK.Google Scholar
[21] Radzik, T. and Szajowski, K. (1988) On some sequential game. Pure Appl. Math. Sci. 28, 51–63.Google Scholar
[22] Radzik, T. and Szajowski, K. (1990) Sequential games with random priority. Sequential Anal. 9, 361–377.CrossRefGoogle Scholar
[23] Ravindran, G. and Szajowski, K. (1992) Non-zero sum game with priority as Dynkin's game. Math. Japonica 37, 401–413.Google Scholar
[24] Rose, J. (1982) Twenty years of secretary problems: a survey of developments in the theory of optimal choice. Management Studies 1, 53–64.Google Scholar
[25] Sakaguchi, M. (1988) Some infinite problems in classical secretary problems. Math. Japonica 34, 307–318.Google Scholar
[26] Sakaguchi, M. (1988) Some two-person bilateral games in the generalized secretary problem. Math. Japonica 34, 637–654.Google Scholar
[27] Shiryaev, A. (1978) Optimal Stopping Rules. Springer-Verlag, New York.Google Scholar
[28] Stadje, W. (1985) On multiple stopping rules. Optimization 16, 401–418.CrossRefGoogle Scholar
[29] Stadje, W. (1987) An optimal k-stopping problem for the Poisson process. Proc. 6th Pannonian Symp. Math. Stat, Bad Tazmannsdorf, Austria , Vol B, Reidel, Dordrecht.Google Scholar
[30] Szajowski, K. (1982) Optimal choice problem of ath object. Matem. Stos. 19, 51–65 (in Polish).Google Scholar
[31] Szajowski, K. (1993) Markov stopping games with random priority. Z. Operat. Res. Submitted.Google Scholar
[32] Yasuda, M. (1985) On a randomized strategy in Neveu's stopping problem. Stoch. Proc. Appl. 21, 159–166.Google Scholar