This paper shows that one can determine whether or not it is beneficial forthe victor to initially commit as many forces as possible to battle inLanchester-type combat between two homogeneous forces by considering theinstantaneous casualty-exchange ratio. It considers the initial-commitmentdecison as a one-sided static optimization problem and examines this nonlinearprogram for each of three decision criteria (victor's losses, lossratio, and loss difference) and for each of two different battle-termination conditions (given force-level breakpoint and given force-ratio breakpoint).The paper's main contribution is to show how to determine the sign of thepartial derivative of the decision criterion with respect to the victor'sinitial force level for general combat dynamics without explicitly solving theLanchester-type combat equations. Consequently, the victor's optimal initialcommitmentdecision many times may be determined from how the instantaneouscasualty-exchange ratio varies with changes in the victor's force level andtime. Convexity of the instantaneous casualty-exchange ratio is shown toimply convexity of the decision criterion so that conditions of decreasingmarginal returns may be identified also without solving the combat equations.The optimal initial-commitment decision is shown to be sensitive to the decisioncriterion for fixed force-ratio breakpoint battles.
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