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D. Grass, M. Kress, J. P. Caulkins, G. Feichtinger, A. Seidl:
"Lanchester Model for Three-Way Combat.";
Research Reports (Vienna University of Technology, Institute of Statistics and Mathematical Methods in Economics, Operations Research and Control Systems), 2017-02 (2017), 2; 19 pages.

Lanchester (1916) 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.

system dynamics, Lanchester model, Square Law, three combatants

http://publik.tuwien.ac.at/files/publik_261856.pdf