IDEAS home Printed from https://ideas.repec.org/a/wly/navres/v63y2016i1p23-31.html
   My bibliography  Save this article

The solution to an open problem for a caching game

Author

Listed:
  • Endre Csóka
  • Thomas Lidbetter

Abstract

In a caching game introduced by Alpern et al. (Alpern et al., Lecture notes in computer science (2010) 220–233) a Hider who can dig to a total fixed depth normalized to 1 buries a fixed number of objects among n discrete locations. A Searcher who can dig to a total depth of h searches the locations with the aim of finding all of the hidden objects. If he does so, he wins, otherwise the Hider wins. This zero‐sum game is complicated to analyze even for small values of its parameters, and for the case of 2 hidden objects has been completely solved only when the game is played in up to 3 locations. For some values of h the solution of the game with 2 objects hidden in 4 locations is known, but the solution in the remaining cases was an open question recently highlighted by Fokkink et al. (Fokkink et al., Search theory: A game theoretic perspective (2014) 85–104). Here we solve the remaining cases of the game with 2 objects hidden in 4 locations. We also give some more general results for the game, in particular using a geometrical argument to show that when there are 2 objects hidden in n locations and n→∞, the value of the game is asymptotically equal to h/n for h≥n/2. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 23–31, 2016

Suggested Citation

  • Endre Csóka & Thomas Lidbetter, 2016. "The solution to an open problem for a caching game," Naval Research Logistics (NRL), John Wiley & Sons, vol. 63(1), pages 23-31, February.
  • Handle: RePEc:wly:navres:v:63:y:2016:i:1:p:23-31
    DOI: 10.1002/nav.21674
    as

    Download full text from publisher

    File URL: https://doi.org/10.1002/nav.21674
    Download Restriction: no

    File URL: https://libkey.io/10.1002/nav.21674?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
    ---><---

    References listed on IDEAS

    as
    1. V. J. Baston & F. A. Bostock, 1991. "A generalized inspection game," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(2), pages 171-182, April.
    2. Avenhaus, Rudolf & Canty, Morton & Marc Kilgour, D. & von Stengel, Bernhard & Zamir, Shmuel, 1996. "Inspection games in arms control," European Journal of Operational Research, Elsevier, vol. 90(3), pages 383-394, May.
    3. Thomas S. Ferguson & Costis Melolidakis, 1998. "On the inspection game," Naval Research Logistics (NRL), John Wiley & Sons, vol. 45(3), pages 327-334, April.
    4. Noemí Zoroa & Procopio Zoroa & José Fernández‐Sáez, 2001. "New results on a Ruckle problem in discrete games of ambush," Naval Research Logistics (NRL), John Wiley & Sons, vol. 48(1), pages 98-106, February.
    5. Steve Alpern & Robbert Fokkink & Christos Pelekis, 2012. "A Proof of the Kikuta–Ruckle Conjecture on Cyclic Caching of Resources," Journal of Optimization Theory and Applications, Springer, vol. 153(3), pages 650-661, June.
    6. K. Kikuta & W. H. Ruckle, 1997. "Accumulation Games, Part 1: Noisy Search," Journal of Optimization Theory and Applications, Springer, vol. 94(2), pages 395-408, August.
    7. Stanley J. Benkoski & Michael G. Monticino & James R. Weisinger, 1991. "A survey of the search theory literature," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(4), pages 469-494, August.
    8. Kensaku Kikuta & William H. Ruckle, 2002. "Continuous accumulation games on discrete locations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 49(1), pages 60-77, February.
    9. 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.
    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. Bastián Bahamondes & Mathieu Dahan, 2024. "Hide-and-Seek Game with Capacitated Locations and Imperfect Detection," Decision Analysis, INFORMS, vol. 21(2), pages 110-124, June.
    2. Dömötör Pálvölgyi, 2018. "All or Nothing Caching Games with Bounded Queries," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 20(01), pages 1-9, March.
    3. Baston, Vic & Kikuta, Kensaku, 2019. "A search problem on a bipartite network," European Journal of Operational Research, Elsevier, vol. 277(1), pages 227-237.

    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. Guzmán, Cristóbal & Riffo, Javiera & Telha, Claudio & Van Vyve, Mathieu, 2022. "A sequential Stackelberg game for dynamic inspection problems," European Journal of Operational Research, Elsevier, vol. 302(2), pages 727-739.
    2. Bastián Bahamondes & Mathieu Dahan, 2024. "Hide-and-Seek Game with Capacitated Locations and Imperfect Detection," Decision Analysis, INFORMS, vol. 21(2), pages 110-124, June.
    3. Steve Alpern & Robbert Fokkink & Christos Pelekis, 2012. "A Proof of the Kikuta–Ruckle Conjecture on Cyclic Caching of Resources," Journal of Optimization Theory and Applications, Springer, vol. 153(3), pages 650-661, June.
    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. Ryusuke Hohzaki & Diasuke Kudoh & Toru Komiya, 2006. "An inspection game: Taking account of fulfillment probabilities of players' aims," Naval Research Logistics (NRL), John Wiley & Sons, vol. 53(8), pages 761-771, December.
    6. Yang, Zhisen & Yang, Zaili & Yin, Jingbo & Qu, Zhuohua, 2018. "A risk-based game model for rational inspections in port state control," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 118(C), pages 477-495.
    7. 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.
    8. Bernhard von Stengel, 2016. "Recursive Inspection Games," Mathematics of Operations Research, INFORMS, vol. 41(3), pages 935-952, August.
    9. Nourinejad, Mehdi & Gandomi, Amir & Roorda, Matthew J., 2020. "Illegal parking and optimal enforcement policies with search friction," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 141(C).
    10. Steven M. Shechter & Farhad Ghassemi & Yasin Gocgun & Martin L. Puterman, 2015. "Technical Note—Trading Off Quick versus Slow Actions in Optimal Search," Operations Research, INFORMS, vol. 63(2), pages 353-362, April.
    11. Ederlina Ganatuin‐Nocon & Tyrone Ang, 2020. "Revisiting inspection game and inspector leadership through reaction networks," Naval Research Logistics (NRL), John Wiley & Sons, vol. 67(6), pages 438-452, September.
    12. T. C. E. Cheng & B. Kriheli & E. Levner & C. T. Ng, 2021. "Scheduling an autonomous robot searching for hidden targets," Annals of Operations Research, Springer, vol. 298(1), pages 95-109, March.
    13. Kensaku Kikuta & William H. Ruckle, 2002. "Continuous accumulation games on discrete locations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 49(1), pages 60-77, February.
    14. Reiter, Johannes & Mauch, Franz & Jäckle, Josef, 1992. "Blocking transitions in lattice spin models with directed kinetic constraints," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 184(3), pages 493-498.
    15. Hoam Chung & Elijah Polak & Johannes O. Royset & Shankar Sastry, 2011. "On the optimal detection of an underwater intruder in a channel using unmanned underwater vehicles," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(8), pages 804-820, December.
    16. Rudolf Avenhaus & D. Marc Kilgour, 2004. "Efficient distributions of arms‐control inspection effort," Naval Research Logistics (NRL), John Wiley & Sons, vol. 51(1), pages 1-27, February.
    17. Morton Canty & Daniel Rothenstein & Rudolf Avenhaus, 2001. "A sequential attribute sampling inspection game for item facilities," Naval Research Logistics (NRL), John Wiley & Sons, vol. 48(6), pages 496-505, September.
    18. Fandel, G. & Trockel, J., 2013. "Avoiding non-optimal management decisions by applying a three-person inspection game," European Journal of Operational Research, Elsevier, vol. 226(1), pages 85-93.
    19. Dömötör Pálvölgyi, 2018. "All or Nothing Caching Games with Bounded Queries," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 20(01), pages 1-9, March.
    20. Guzman, Cristobal & Riffo, Javiera & Telha, Claudio & Van Vyve, Mathieu, 2021. "A Sequential Stackelberg Game for Dynamic Inspection Problems," LIDAM Discussion Papers CORE 2021036, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).

    More about this item

    Statistics

    Access and download statistics

    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:wly:navres:v:63:y:2016:i:1:p:23-31. 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: Wiley Content Delivery (email available below). General contact details of provider: https://doi.org/10.1002/(ISSN)1520-6750 .

    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.