IDEAS home Printed from https://ideas.repec.org/a/eee/ejores/v264y2018i1p46-54.html
   My bibliography  Save this article

Lanchester model for three-way combat

Author

Listed:
  • Kress, Moshe
  • Caulkins, Jonathan P.
  • Feichtinger, Gustav
  • Grass, Dieter
  • Seidl, Andrea

Abstract

Lanchester (1960) modeled combat situations between two opponents, where mutual attrition occurs continuously in time, by a pair of simple ordinary (linear) differential equations. The aim of the present paper is to extend the model to a conflict consisting of three parties. In particular, Lanchester’s main result, i.e. his Square Law, is adapted to a triple fight. However, here a central factor – besides the initial strengths of the forces – determining the long run outcome is the allocation of each opponent’s efforts between the other two parties. Depending on initial strengths, (the) solution paths are calculated and visualized in appropriate phase portraits. We are able identify regions in the state space where, independent of the force allocation of the opponents, always the same combatant wins, regions, where a combatant can win if its force allocation is wisely chosen, and regions where a combatant cannot win itself but determine the winner by its forces allocation. As such, the present model can be seen as a forerunner of a dynamic game between three opponents.

Suggested Citation

  • Kress, Moshe & Caulkins, Jonathan P. & Feichtinger, Gustav & Grass, Dieter & Seidl, Andrea, 2018. "Lanchester model for three-way combat," European Journal of Operational Research, Elsevier, vol. 264(1), pages 46-54.
    • Handle: RePEc:eee:ejores:v:264:y:2018:i:1:p:46-54
    DOI: 10.1016/j.ejor.2017.07.026
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0377221717306537
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.ejor.2017.07.026?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. Niall J MacKay, 2015. "When Lanchester met Richardson, the outcome was stalemate: A parable for mathematical models of insurgency," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 66(2), pages 191-201, February.
    2. Patrick S. Chen & Peter Chu, 2001. "Applying Lanchester's linear law to model the Ardennes campaign," Naval Research Logistics (NRL), John Wiley & Sons, vol. 48(8), pages 653-661, December.
    3. Peng, R. & Zhai, Q.Q. & Levitin, G., 2016. "Defending a single object against an attacker trying to detect a subset of false targets," Reliability Engineering and System Safety, Elsevier, vol. 149(C), pages 137-147.
    4. Kjell Hausken & John F. Moxnes, 2005. "Approximations and empirics for stochastic war equations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 52(7), pages 682-700, October.
    5. Ryusuke Hohzaki & Takehiro Higashio, 2016. "An attrition game on a network ruled by Lanchester’s square law," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 67(5), pages 691-707, May.
    6. Ye, Zhi-Sheng & Peng, Rui & Wang, Wenbin, 2017. "Defense and attack of performance-sharing common bus systemsAuthor-Name: Zhai, Qingqing," European Journal of Operational Research, Elsevier, vol. 256(3), pages 962-975.
    7. Jerome Bracken, 1995. "Lanchester models of the ardennes campaign," Naval Research Logistics (NRL), John Wiley & Sons, vol. 42(4), pages 559-577, June.
    8. Moshe Kress & Roberto Szechtman, 2009. "Why Defeating Insurgencies Is Hard: The Effect of Intelligence in Counterinsurgency Operations---A Best-Case Scenario," Operations Research, INFORMS, vol. 57(3), pages 578-585, June.
    9. Edward H. Kaplan & Moshe Kress & Roberto Szechtman, 2010. "Confronting Entrenched Insurgents," Operations Research, INFORMS, vol. 58(2), pages 329-341, April.
    10. S. J. Deitchman, 1962. "A Lanchester Model of Guerrilla Warfare," Operations Research, INFORMS, vol. 10(6), pages 818-827, December.
    11. Hausken, Kjell & Moxnes, John F., 2002. "Stochastic conditional and unconditional warfare," European Journal of Operational Research, Elsevier, vol. 140(1), pages 61-87, July.
    12. Brams, Steven J. & Kilgour, D. Mark, 1997. "The Truel," Working Papers 97-05, C.V. Starr Center for Applied Economics, New York University.
    13. M Kress & I Talmor, 1999. "A new look at the 3:1 rule of combat through Markov Stochastic Lanchester models," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 50(7), pages 733-744, July.
    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. Gerardo Minguela-Castro & Ruben Heradio & Carlos Cerrada, 2021. "Automated Support for Battle Operational–Strategic Decision-Making," Mathematics, MDPI, vol. 9(13), pages 1-15, June.
    2. Moshe Kress, 2020. "Lanchester Models for Irregular Warfare," Mathematics, MDPI, vol. 8(5), pages 1-14, May.
    3. Anelí Bongers & José L. Torres, 2021. "A bottleneck combat model: an application to the Battle of Thermopylae," Operational Research, Springer, vol. 21(4), pages 2859-2877, December.
    4. Hausken, Kjell, 2019. "Governmental combat of the dynamics of multiple competing terrorist organizations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 166(C), pages 33-55.

    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. Anelí Bongers & José L. Torres, 2021. "A bottleneck combat model: an application to the Battle of Thermopylae," Operational Research, Springer, vol. 21(4), pages 2859-2877, December.
    2. C-Y Hung & G K Yang & P S Deng & T Tang & S-P Lan & P Chu, 2005. "Fitting Lanchester's square law to the Ardennes Campaign," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 56(8), pages 942-946, August.
    3. Kjell Hausken & John F. Moxnes, 2005. "Approximations and empirics for stochastic war equations," Naval Research Logistics (NRL), John Wiley & Sons, vol. 52(7), pages 682-700, October.
    4. Moshe Kress, 2020. "Lanchester Models for Irregular Warfare," Mathematics, MDPI, vol. 8(5), pages 1-14, May.
    5. Sakai, Kazuki & Hohzaki, Ryusuke & Fukuda, Emiko & Sakuma, Yutaka, 2018. "Risk evaluation and games in mine warfare considering shipcounter effects," European Journal of Operational Research, Elsevier, vol. 268(1), pages 300-313.
    6. Chad W. Seagren & Donald P. Gaver & Patricia A. Jacobs, 2019. "A stochastic air combat logistics decision model for Blue versus Red opposition," Naval Research Logistics (NRL), John Wiley & Sons, vol. 66(8), pages 663-674, December.
    7. Ian R. Johnson & Niall J. MacKay, 2011. "Lanchester models and the battle of Britain," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(3), pages 210-222, April.
    8. Michael P. Atkinson & Moshe Kress & Niall J. MacKay, 2021. "Targeting, Deployment, and Loss-Tolerance in Lanchester Engagements," Operations Research, INFORMS, vol. 69(1), pages 71-81, January.
    9. Kjell Hausken & Jun Zhuang, 2011. "Governments' and Terrorists' Defense and Attack in a T -Period Game," Decision Analysis, INFORMS, vol. 8(1), pages 46-70, March.
    10. Shan, Xiaojun & Zhuang, Jun, 2013. "Hybrid defensive resource allocations in the face of partially strategic attackers in a sequential defender–attacker game," European Journal of Operational Research, Elsevier, vol. 228(1), pages 262-272.
    11. Michael J. Armstrong, 2013. "The salvo combat model with area fire," Naval Research Logistics (NRL), John Wiley & Sons, vol. 60(8), pages 652-660, December.
    12. Qingqing Zhai & Rui Peng & Jun Zhuang, 2020. "Defender–Attacker Games with Asymmetric Player Utilities," Risk Analysis, John Wiley & Sons, vol. 40(2), pages 408-420, February.
    13. Gerardo Minguela-Castro & Ruben Heradio & Carlos Cerrada, 2021. "Automated Support for Battle Operational–Strategic Decision-Making," Mathematics, MDPI, vol. 9(13), pages 1-15, June.
    14. Kolebaje, Olusola & Popoola, Oyebola & Khan, Muhammad Altaf & Oyewande, Oluwole, 2020. "An epidemiological approach to insurgent population modeling with the Atangana–Baleanu fractional derivative," Chaos, Solitons & Fractals, Elsevier, vol. 139(C).
    15. Pettit, L. I. & Wiper, M. P. & Young, K. D. S., 2003. "Bayesian inference for some Lanchester combat laws," European Journal of Operational Research, Elsevier, vol. 148(1), pages 152-165, July.
    16. Patrick S. Chen & Peter Chu, 2001. "Applying Lanchester's linear law to model the Ardennes campaign," Naval Research Logistics (NRL), John Wiley & Sons, vol. 48(8), pages 653-661, December.
    17. Manh D Hy & My A Vu & Nam H Nguyen & Anh N Ta & Dinh V Bui, 2020. "Optimization in an asymmetric Lanchester (n, 1) model," The Journal of Defense Modeling and Simulation, , vol. 17(1), pages 117-122, January.
    18. Edward H. Kaplan & Moshe Kress & Roberto Szechtman, 2010. "Confronting Entrenched Insurgents," Operations Research, INFORMS, vol. 58(2), pages 329-341, April.
    19. Michael J Armstrong, 2014. "The salvo combat model with a sequential exchange of fire," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 65(10), pages 1593-1601, October.
    20. Gao, Kaiye & Yan, Xiangbin & Liu, Xiang-dong & Peng, Rui, 2019. "Object defence of a single object with preventive strike of random effect," Reliability Engineering and System Safety, Elsevier, vol. 186(C), pages 209-219.

    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:264:y:2018:i:1:p:46-54. 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.