@VictorIn_Pacific Impossibility... I do not think it means what you think it means. Mathematical impossibility simply means that the negation of a given statement is true under certain axioms.

You were looking for a function that gives back the strength of a unit. What I have shown is that you were asking the wrong question. You cannot evaluate the value of a unit, not even that of an army.

Linear formulas will also break down. If the value of a unit is *scalar1 * hp + scalar2 * power*, then the unit will, under its lifetime, kill *expected_combat_rounds_survived * power / dicesides* hitpoints, so you can write the equation *expected_combat_rounds_survived * power / dicesides * scalar1 = scalar2 * power*. But *expected_combat_rounds_survived* depends both on the average power and the deviation of power of ALL units on the board, and so does the ratio of *scalar1* and *scalar2*.

In short, if all units have 1 power, then 1 power is worth X utilons. If all units have 2 power, then 1 power is worth X/2 utilons.

As for the differential equations for which I admittedly do not have a great affinity, my attempts at giving any sort of useful general solution failed miserably, so did those of others I asked about it. The simplified case where every unit has the same power is clearly solvable, but uninteresting, so I decided not to bother.

Anyway... I really shouldn't be spending time writing comments here. Or if I focused my attention on TripleA, I could at least get back to working on my map. So... good luck with those nasty differential equations.