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Continuous patrolling and hiding games

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  • Garrec, Tristan

Abstract

We present two zero-sum games modeling situations where one player attacks (or hides in) a finite dimensional nonempty compact set, and the other tries to prevent the attack (or find him). The first game, called patrolling game, corresponds to a dynamic formulation of this situation in the sense that the attacker chooses a time and a point to attack and the patroller chooses a continuous trajectory to maximize the probability of finding the attack point in a given time. Whereas the second game, called hiding game, corresponds to a static formulation in which both the searcher and the hider choose simultaneously a point and the searcher maximizes the probability of being at distance less than a given threshold of the hider.

Suggested Citation

  • Garrec, Tristan, 2019. "Continuous patrolling and hiding games," European Journal of Operational Research, Elsevier, vol. 277(1), pages 42-51.
    • Handle: RePEc:eee:ejores:v:277:y:2019:i:1:p:42-51
    DOI: 10.1016/j.ejor.2019.02.026
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    1. Vic Baston & Kensaku Kikuta, 2009. "Technical Note---An Ambush Game with a Fat Infiltrator," Operations Research, INFORMS, vol. 57(2), pages 514-519, April.
    2. B. O. Koopman, 1956. "The Theory of Search. I. Kinematic Bases," Operations Research, INFORMS, vol. 4(3), pages 324-346, June.
    3. Gaëtan Fournier, 2019. "General distribution of consumers in pure Hotelling games," International Journal of Game Theory, Springer;Game Theory Society, vol. 48(1), pages 33-59, March.
    4. John M. Danskin, 1990. "On the Cookie-Cutter Game: Search and Evasion on a Disc," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 573-596, November.
    5. Kyle Y. Lin & Michael P. Atkinson & Timothy H. Chung & Kevin D. Glazebrook, 2013. "A Graph Patrol Problem with Random Attack Times," Operations Research, INFORMS, vol. 61(3), pages 694-710, June.
    6. Alan Washburn, 2014. "Two-Person Zero-Sum Games," International Series in Operations Research and Management Science, Springer, edition 4, number 978-1-4614-9050-0, April.
    7. Kyle Y. Lin & Michael P. Atkinson & Kevin D. Glazebrook, 2014. "Optimal patrol to uncover threats in time when detection is imperfect," Naval Research Logistics (NRL), John Wiley & Sons, vol. 61(8), pages 557-576, December.
    8. B. O. Koopman, 1956. "The Theory of Search. II. Target Detection," Operations Research, INFORMS, vol. 4(5), pages 503-531, October.
    9. Oléron Evans, Thomas P. & Bishop, Steven R., 2013. "Static search games played over graphs and general metric spaces," European Journal of Operational Research, Elsevier, vol. 231(3), pages 667-689.
    10. Zoroa, Noemi & Zoroa, Procopio & Jose Fernandez-Saez, M., 1999. "A generalization of Ruckle's results for an ambush game," European Journal of Operational Research, Elsevier, vol. 119(2), pages 353-364, December.
    11. William H. Ruckle & John R. Reay, 1981. "Ambushing Random Walks III: More Continuous Models," Operations Research, INFORMS, vol. 29(1), pages 121-129, February.
    12. Steve Alpern & Vic Baston & Shmuel Gal, 2008. "Network search games with immobile hider, without a designated searcher starting point," International Journal of Game Theory, Springer;Game Theory Society, vol. 37(2), pages 281-302, June.
    13. Lidbetter, Thomas, 2013. "Search games with multiple hidden objects," LSE Research Online Documents on Economics 55103, London School of Economics and Political Science, LSE Library.
    14. William H. Ruckle, 1981. "Ambushing Random Walks II: Continuous Models," Operations Research, INFORMS, vol. 29(1), pages 108-120, February.
    15. Vic Baston & Kensaku Kikuta, 2004. "An Ambush Game with an Unknown Number of Infiltrators," Operations Research, INFORMS, vol. 52(4), pages 597-605, August.
    16. W. H. Ruckle & K. Kikuta, 2000. "Continuous Accumulation Games in Continuous Regions," Journal of Optimization Theory and Applications, Springer, vol. 106(3), pages 581-601, September.
    17. Katerina Papadaki & Steve Alpern & Thomas Lidbetter & Alec Morton, 2016. "Patrolling a Border," Operations Research, INFORMS, vol. 64(6), pages 1256-1269, December.
    18. Bernard O. Koopman, 1957. "The Theory of Search," Operations Research, INFORMS, vol. 5(5), pages 613-626, October.
    19. Zoroa, N. & Fernández-Sáez, M.J. & Zoroa, P., 2012. "Patrolling a perimeter," European Journal of Operational Research, Elsevier, vol. 222(3), pages 571-582.
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    Cited by:

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    3. Deutsch, Yael, 2021. "A polynomial-time method to compute all Nash equilibria solutions of a general two-person inspection game," European Journal of Operational Research, Elsevier, vol. 288(3), pages 1036-1052.

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