Force-Ratio Considerations for Some Lanchester-Type Models of Warfare

This paper studies the Riccati equation satisfied by the force ratio of two homogeneous forces in deterministic Lanchester combat. This study provides qualitative insight as to the “direction” in which combat is moving (in the sense that the force ratio is changing to the advantage of one of the combatants). “Instantaneous” linear and square laws that are “local” conditions for winning are shown to apply for Lanchester-type formulations with time-varying system effectiveness. The value of considering the force-ratio equation lies in one's ability to determine the outcome of battle without explicitly solving the force-level equations, an important feature for variable-coefficient formulations. Moreover, in some cases observation of the initial time-behavior of the force ratio in a battle allows one to predict the outcome, sometimes even without explicit knowledge of attrition-rate coefficients. By additionally considering the instantaneous loss ratio, one gains insight into the consequences of concentration of forces in aimed-fire combat between two homogeneous forces with superimposed effects of supporting fires (which are not subject to attrition).