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Theory of Combat: The Probability of Winning

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  • Richard H. Brown

    (Washington College, Chestertown, Maryland)

Abstract

As two forces engage in combat without replacements, the numbers of survivors on the two forces will diminish. The force first reaching zero loses, the other wins Let P ( x , y ) be the probability that the first force wins if there are x members for it and y members for the second force, and let A ( x , y ) be the probability that the next casualty is on the second force. It is shown that P ( x , y ) = A ( x , y ) P ( x , y - 1) + [1 - A ( x , y )] P ( x - 1, y ). From this equation and the conditions that P ( x , 0) = 1 and P (0, y ) = 0, P ( x , y ) can be given in terms of A . Lanchester's Linear Law governs the case that A ( x , y ) is a constant, p , and his Square Law governs the case that A ( x , y ) = rx ( rx + y ), where r is the ratio of the relative effectiveness of a member of the first force to that for a member of the second force. The exact solution of the problem of finding P ( x , y ) is given for both the linear law and the square law. The exact solution is cumbersome, however, and useful approximations to the exact solutions are also given in terms of the normal probability integral.

Suggested Citation

  • Richard H. Brown, 1963. "Theory of Combat: The Probability of Winning," Operations Research, INFORMS, vol. 11(3), pages 418-425, June.
    • Handle: RePEc:inm:oropre:v:11:y:1963:i:3:p:418-425
    DOI: 10.1287/opre.11.3.418
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    Cited by:

    1. Kyle Y. Lin, 2014. "New results on a stochastic duel game with each force consisting of heterogeneous units," Naval Research Logistics (NRL), John Wiley & Sons, vol. 61(1), pages 56-65, February.
    2. Hwang, Sung-Ha, 2012. "Technology of military conflict, military spending, and war," Journal of Public Economics, Elsevier, vol. 96(1), pages 226-236.
    3. C. J. Ancker & A. V. Gafarian, 1987. "The validity of assumptions underlying current uses of Lanchester attrition rates," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(4), pages 505-533, August.
    4. Liu, Liwei & Yu, Jun & Guo, Zhi, 2006. "A kind of stochastic duel model for guerrilla war," European Journal of Operational Research, Elsevier, vol. 171(2), pages 430-438, June.

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