A lot of the independent research I've done in the last year has been the application of combat models to analyze battlefield outcomes. There's been a lot written about these types of models, and most seem to point to the Lanchester equations as the first mathematically rigorous attempt to understand battlefield dynamics, even if that means making a lot of assumptions. Other models are even more complicated, treating groups of fighting soldiers as if they were two merging crowds to get a sense of where the instabilities in the lines are, kind of like the motion of fluids around objects.
Okay, that's cool math and all. But my main problem with these types of models is how deterministic they are. The model I've been working with, derived by Stephen Biddle in his book Military Power, gives a set of equations with a bunch of parameters and asks you to plug and chug to get a result on whether or not an attacking offensive force will break through a defense. But that's just it - take the initial conditions and variables, throw them into the model, and get a yes/no answer about the success of an attack. Not the likelihood of success, but a single number that says to an analyst, "under these conditions, an attacker will or will not achieve a breakthrough of the defender's lines."
Wait wait wait. What about Clausewitz's friction? ("Everything in war is simple, but the simplest ting is difficult.") What about variables that are difficult to quantify, like morale or military readiness? In my mind, for a combat model to be analytically useful, it should give results in terms of probabilities, not deterministic yes/no answers. Something like, "under these conditions, an attacker will achieve a breakthrough around 80% of the time." There are remarkably few, if any, situations where an analyst can predict an outcome in war with absolute certainty.
I guess predicting the results of a battle should be less like an engineering problem with a definitive answer, and more like predicting the weather.