In defense of artillery attack splitting

[fibol]

I have several times read the notion that as overall stats increase artillery (and similars) splitting attack value into multiple attacks reduces their effective damage output (because the thing they are fireing at grows its hp).
As I was thinking about combat armor I decided to finally write up why this is not the case.

First of all, if we do the math on a 1v1, then the notion is correct. If one artillery fights against one something else, splitting attack value into multiple attacks reduces the overall chance of achieving at least one hit. However in Shadow Empire 1v1s don't happen.
So, for now let's assume that there are infinite numbers of enemies and calculate how many an artillery hits.
For this we will calculate the expected value of hits. I will denote as X the chance of artillery with unsplit attack value to score a hit, E1 will be the expected amount of hits with unsplit attack value, E2 will be the expected amount of hits with split attack value*.
E1 = X (quite simply)
E2 = 2 * ((X/2)^2) + (X/2) * (1-(X/2)) = (X^2)/2 + X - (X^2)/2 = X
We have therefore shown that the number of hits we can expect from an artillery subunit remains exactly equal, irrespective of having split attack value or not, as long as there is something to shoot at. And well, if you run out of things to shoot at you already won the battle, so that's irrelevant.
Therefore we can conclude that on average attack splitting does in no circumstance reduce the damage output of artillery.

*For the purpose of this post I will only observe cases of 1 or 2 attacks. For attack numbers above 2 the formula gets rather long.

[Voker57]

Have somebody said it reduces the total damage? It increases it if anything, because of possibility of PINNED hits and improved chances at targets with lowered readiness.

[fibol]

Not recently, but I've seen it a handful of times and it's always bugged me and as I was doing math on artillery vs mass infantry to delay combat armor if I don't have the resources for tanks I got the motivation to finally write this up.

[Thrake]

Having more attack than the enemy has hit points leads to diminishing returns where more attack has less effect than if attack was lower than hit points. So in my book the multiple attacks are a pro as it means that it always uses the more favourable to hit formula.

I would be curious to know how you came up with your calculation. I could never get the maths down to how the hit formula exactly works.

[fibol]

That is an extremely good question that I entirely messed up. The scaling of the to hit chance is not linear with a linear in/decrease of hp/attack values. I didn't think to hard about that and the few numbers I plugged coincidentally fit.
So for now everything I wrote in my initial post is worthless and useless, I'm gonna have to see if I can figure something out for the actual formulas.

[fibol]

Alright, I can't find a solid closed formula, but I think I got enough to still validate my initial findings.

Regarding the formulas to achieve a hit:
While Attack(A) <= HP(H):
((H-A)/H * 0) + ((A/H) * 0.5) = 0.5 * A / H
While Attack(A) >= HP(H):
((A-H)/A * 1) + ((H/A) * 0.5)

What we can observe is that while attack value is equal or lower than hp the linear relationship holds, so my original calculation is an accurate representation.
For cases where attack outstrips hp this is not the case, but we can observe that the the growth in hit probability is always equal or lower than the linear growth. Therefore if in this area a halving of attack value adding together the hit probabilites of the split attack values will actually be greater than that of the unsplit attack value. Therefore in these cases a split attack will actually outperform an unsplit attack in number of expected hits even on the simplified model I used here, not considering entrenchment degradation or other modifiers that could profit from previous hits etc.

Conclusion: Although my initial calculation was very sloppy and error prone, it turns out the conclusions were still correct or understated.