Optimal allocation in combat models with multiple battlefields
We study the allocation of a force of a fixed size across independent battlefields against a known distribution of opposing forces. Formulating battlefield outcomes through loss and survival functions allows us to establish allocation results without assuming a specific system of combat equations. For convex enemy loss functions, an optimal allocation exists with at most one partially saturated battlefield. We prove NP-hardness for every prescribed family of strictly convex, strictly increasing loss functions satisfying an endpoint normalization, and give an exact pseudopolynomial algorithm for the special case when the total force size and saturation thresholds are integers. We also derive allocation rules for maximizing successful engagements, maximizing own survivors, and maximizing the ratio of enemy losses to own losses. Applications include a power-law family containing the classical Lanchester, guerrilla, and mixed combat models, as well as models with implicitly defined outcomes.