Dispersed combat as mass action with finite search

Abstract

Improvements to models of battle attrition are necessary because current models cannot explain battle attrition. Agent based simulations indicate that calculated attrition is substantially different when agents are not assumed to have unlimited detection capabilities. However, agent based models are limited to small force sizes and there is no evidence that the changes in calculated attrition occur for large force sizes. We develop a probabilistic model, based on Bernoulli trials, to check if limited detection capabilities result in significant changes to calculated attrition when force sizes are large, as in battle datasets. Our model is a search model and we convert it to an attrition model via the same processes used in current models, and include the same assumptions for factors other than detection range. We find two series solutions to the model, one for small force sizes, the other for large force sizes, and find numerically that the two solutions strongly overlap. The new model makes a difference to calculated attrition when force sizes are small, but not when they are large. However, the model makes a difference to calculated attrition for all force sizes if the battlefield area is increased to maintain a sparse force density. Our approach is mathematical, not requiring application knowledge, and several of the assumptions underlying mass action models are raised in our discussion. References J. V. Chase. Sea fights: A mathematical investigation of the effect of superiority of force in combats upon the sea . Naval War College Archives, RG 8, Box 109, XTAV (1902), 1902. N. R. Franks and L. W. Partridge. Lanchester battles and the evolution of combat in ants. Anim. Behav. , 45:197–199, 1993. doi:10.1006/anbe.1993.1021 G. S. Gradschtein and I. M. Ryzhik. Tables of Series, Products and Integrals . Deutcher Verlag der Wissenschaften, 1996. N. C. Grassly and C. Fraser. Mathematical models of infectious disease transmission. Nat. Rev. Microbiol. , 6:477–487, 2008. doi:10.1038/nrmicro1845 D. Kahneman. Thinking Fast and Slow . Penguin, London, 2013. L. R. Kosowski, A. Pincombe and B. Pincombe. Irrelevance of the fractal dimension term in the modified fractal attrition equation. ANZIAM J. , 52:C988–C1011, 2011. doi:10.21914/anziamj.v52i0.3963 F. W. Lanchester. Aircraft in warfare: The dawn of the fourth arm . Constable, London, 1916. http://edc448uri.wikispaces.com/file/view/Lanchester+-+Aircraft+in+Warfare.pdf T. W. Lucas and T. Turkes. Fitting Lanchester equations to the Battles of Kursk and Ardennes. Nav. Res. Log. , 51:95–116, 2004. doi:10.1002/nav.10101 T. W. Lucas and J. A. Dinges. The effect of battle circumstances on fitting Lanchester equations to the Battle of Kursk. Mil. Oper. Res. , 9:17–30, 2004. http://www.mors.org/Publications/MOR-Journal/Online-Issues P. M. Morse and G. E. Kimball. Methods of Operations Research . Wiley, 1951. M. Osipov. The influence of the numerical strength of engaged forces in their casualties. Translated by R. L. Helmbold and A. S. Rehm. Nav. Res. Log. , 42:435–490, 1995. 3.0.CO;2-2>doi:10.1002/1520-6750(199504)42:3 3.0.CO;2-2 R. Peterson. On the logarithmic law of combat and its application to tank combat. Oper. Res. , 15:557–558, 1967. doi:10.1287/opre.15.3.557 A. H. Pincombe, B. M. Pincombe and C. E. M. Pearce, Putting the art before the force. ANZIAM J. , 51:C482–C496, 2010. doi:10.0000/anziamj.v51i0.2584 . A. H. Pincombe, B. M. Pincombe and C. E. M. Pearce. A simple battle model with explanatory power. ANZIAM J. , 51:C497–C511, 2010. doi:10.21914/anziamj.v51i0.2585 . B. M. Pincombe and A. H. Pincombe. Mass action models of Falklands War battles. ANZIAM J. , 57:C235–C252, 2016. doi:10.21914/anziamj.v57i0.10450 J. G. Taylor. Lanchester models of warfare. Operations Research Society of America, Arlington, 1983.