Lets imagine we have two armies, represented by lists of pairs of positive numbers, like this: [($attack1$,$defence1$),($a2$,$d2$)...($an$,$dn$)] face each other in combat. The rules of combat are the following: each unit (pair of attack $a$ and defence $d$) may at any time choose a target; it then does damage continously to its target at a rate of $a$ per second until it either chooses another target or "dies" (i.e. its defence becomes 0). Units may switch targets at any time. Each army plays optimally to either maximise its strength at the time all enemy units are dead (if it will win) or to minimise the strength of the enemy army (if it will be routed). An army wins by killing every unit on the opposing side. An army $A$ is of greater strength than the army $B$ if it can defeat $B$ and any army $B$ can defeat or ties with.
I conjecture that the function $strength(A) = d1*(a1+a2...an) + d2*(a2+a3...an)...dn*an$ , where $(a1,d1)...(an,dn)$ are the units of $A$ ordered by $(attack /defence)$ in descending order is a measure of an army's strength such that:
If $A1$ faces $A2$ and $strength(A1) > strength(A2)$ then A1 will emerge victorious with a remaining strength $strength(A1) - strength(A2)$
If $strength(A1) = strength(A2)$ then they draw, either through both being routed or the battle never ending. I can't prove it. Can you?