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Combined Games with Randomly Delayed Beginnings

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  • F. Thomas Bruss

    (Département de Mathématique, Faculté des Sciences, Université Libre de Bruxelles, Campus Plaine, CP 210, B-1050 Brussels, Belgium)

Abstract

This paper presents two-person games involving optimal stopping. As far as we are aware, the type of problems we study are new. We confine our interest to such games in discrete time. Two players are to chose, with randomised choice-priority, between two games G 1 and G 2 . Each game consists of two parts with well-defined targets. Each part consists of a sequence of random variables which determines when the decisive part of the game will begin. In each game, the horizon is bounded, and if the two parts are not finished within the horizon, the game is lost by definition. Otherwise the decisive part begins, on which each player is entitled to apply their or her strategy to reach the second target. If only one player achieves the two targets, this player is the winner. If both win or both lose, the outcome is seen as “deuce”. We motivate the interest of such problems in the context of real-world problems. A few representative problems are solved in detail. The main objective of this article is to serve as a preliminary manual to guide through possible approaches and to discuss under which circumstances we can obtain solutions, or approximate solutions.

Suggested Citation

  • F. Thomas Bruss, 2021. "Combined Games with Randomly Delayed Beginnings," Mathematics, MDPI, vol. 9(5), pages 1-16, March.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:5:p:534-:d:510067
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    References listed on IDEAS

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    1. L. Bayón & P. Fortuny Ayuso & J. M. Grau & A. M. Oller-Marcén & M. M. Ruiz, 2018. "The Best-or-Worst and the Postdoc problems," Journal of Combinatorial Optimization, Springer, vol. 35(3), pages 703-723, April.
    2. Tomomi Matsui & Katsunori Ano, 2016. "Lower Bounds for Bruss’ Odds Problem with Multiple Stoppings," Mathematics of Operations Research, INFORMS, vol. 41(2), pages 700-714, May.
    3. Szajowski, Krzysztof, 2007. "A game version of the Cowan-Zabczyk-Bruss' problem," Statistics & Probability Letters, Elsevier, vol. 77(17), pages 1683-1689, November.
    4. Nowak, Andrzej S. & Szajowski, Krzysztof, 1998. "Nonzero-sum Stochastic Games," MPRA Paper 19995, University Library of Munich, Germany, revised 1999.
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