IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v277y2019i1p42-51.html
   My bibliography  Save this article

Continuous patrolling and hiding games

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221719301547
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2019.02.026?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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, December.
    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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Bui, Thuy & Lidbetter, Thomas, 2023. "Optimal patrolling strategies for trees and complete networks," European Journal of Operational Research, Elsevier, vol. 311(2), pages 769-776.
    2. Wang, Jian & Cui, Lei, 2023. "Patrolling games with coordination between monitoring devices and patrols," Reliability Engineering and System Safety, Elsevier, vol. 233(C).
    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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Alpern, Steve & Lidbetter, Thomas & Papadaki, Katerina, 2019. "Optimizing periodic patrols against short attacks on the line and other networks," European Journal of Operational Research, Elsevier, vol. 273(3), pages 1065-1073.
    2. Alpern, Steve & Fokkink, Robbert & Simanjuntak, Martin, 2016. "Optimal search and ambush for a hider who can escape the search region," European Journal of Operational Research, Elsevier, vol. 251(3), pages 707-714.
    3. Katerina Papadaki & Steve Alpern & Thomas Lidbetter & Alec Morton, 2016. "Patrolling a Border," Operations Research, INFORMS, vol. 64(6), pages 1256-1269, December.
    4. 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.
    5. Patriksson, Michael, 2008. "A survey on the continuous nonlinear resource allocation problem," European Journal of Operational Research, Elsevier, vol. 185(1), pages 1-46, February.
    6. Hohzaki, Ryusuke & Iida, Koji, 2001. "Optimal ambushing search for a moving target," European Journal of Operational Research, Elsevier, vol. 133(1), pages 120-129, August.
    7. Darlington, Matthew & Glazebrook, Kevin D. & Leslie, David S. & Shone, Rob & Szechtman, Roberto, 2023. "A stochastic game framework for patrolling a border," European Journal of Operational Research, Elsevier, vol. 311(3), pages 1146-1158.
    8. Vic Baston & Kensaku Kikuta, 2009. "Technical Note---An Ambush Game with a Fat Infiltrator," Operations Research, INFORMS, vol. 57(2), pages 514-519, April.
    9. Vic Baston & Kensaku Kikuta, 2004. "An Ambush Game with an Unknown Number of Infiltrators," Operations Research, INFORMS, vol. 52(4), pages 597-605, August.
    10. I.D. Woodward, 2003. "Discretization of the continuous ambush game," Naval Research Logistics (NRL), John Wiley & Sons, vol. 50(5), pages 515-529, August.
    11. Steve Alpern & Thomas Lidbetter, 2019. "Approximate solutions for expanding search games on general networks," Annals of Operations Research, Springer, vol. 275(2), pages 259-279, April.
    12. K. T. Lee, 1990. "On ruckle's game of ambush," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(3), pages 355-363, June.
    13. Yolmeh, Abdolmajid & Baykal-Gürsoy, Melike, 2021. "Weighted network search games with multiple hidden objects and multiple search teams," European Journal of Operational Research, Elsevier, vol. 289(1), pages 338-349.
    14. Lidbetter, Thomas, 2020. "Search and rescue in the face of uncertain threats," European Journal of Operational Research, Elsevier, vol. 285(3), pages 1153-1160.
    15. V. J. Baston & F. A. Bostock, 1987. "A continuous game of ambush," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(5), pages 645-654, October.
    16. Zoroa, N. & Fernández-Sáez, M.J. & Zoroa, P., 2012. "Patrolling a perimeter," European Journal of Operational Research, Elsevier, vol. 222(3), pages 571-582.
    17. Alpern, Steve & Lidbetter, Thomas, 2020. "Search and Delivery Man Problems: When are depth-first paths optimal?," European Journal of Operational Research, Elsevier, vol. 285(3), pages 965-976.
    18. Peter Kolesar & Kellen Leister & Daniel Stimpson & Ronald Woodaman, 2013. "A simple model of optimal clearance of improvised explosive devices," Annals of Operations Research, Springer, vol. 208(1), pages 451-468, September.
    19. Bui, Thuy & Lidbetter, Thomas, 2023. "Optimal patrolling strategies for trees and complete networks," European Journal of Operational Research, Elsevier, vol. 311(2), pages 769-776.
    20. Lidbetter, Thomas, 2017. "On the approximation ratio of the Random Chinese Postman Tour for network search," European Journal of Operational Research, Elsevier, vol. 263(3), pages 782-788.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:ejores:v:277:y:2019:i:1:p:42-51. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/eor .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.