Stacking penalty

[LimaTango]

I searched on this and got too many references without data.

I think I read once Pang confirming that optimal fleet size is 26 vessels. Is there a similar optimal amount of divisions in a province, where one more actually makes it worse? Does it make a difference if any or all divisions are brigaded? Does it differ if attacking or defending? Does it differ if some or all the divisions are half of full strength?

 

[Pang Bingxun]

The applied stacking penalty does only scale up by the number n of ones fighting divising n by roughly sqrt(n). So using fully brigaded divisions with strength is a good choice.

PS: The exact formula has been found by Mor_Rioghain on 26-01-2012:

Short version:
stacking penalty = 1 - (sqrt(8*X-1)-1) / (2*X)
where X is the number of units

Long version:
Before we start analysing the numbers it would be good to take a look at what kind of formulas could be used and how we could transform the data into something that makes sense.

Formulas are usualy composed of whole numbers and simple functions (addition, multiplication, division). Since the other explanations mention squareroots we'll include also this as an option. Since it doesn't make much sense for a penalty to go over 100% it will either be hard capped (with a maximum function) or the function itself is converging. This is best done by putting the number of units in the denominator. Also it will need to be a continuously rising function.

We can transform the data into either total lost power (= #units * penalty), efficiency (= 1- penalty) or total power (= #units * efficiency ). As it's mentioned that the total power of an army is proportional to the squareroot of the number of units (power ~ sqrt(#units) )this seems like the best option. We can use these formulas to express the penalty in function of the number of units:

penalty = 1 - power/#units ~ 1 - sqrt(#untis)/#units

This formula fits our earlier criteria of converging (towards 1 or 100%) and being a continuously rising function. So now that we have the general form of our function all we need to is determine the exact correlation between the power and the number of units.

our transformed data:
units power
1 __ 1
2 __ 1.56
3 __ 2
4 __ 2.37
5 __ 2.7
6 __ 3
7 __ 3.27
8 __ 3.53
9 __ 3.77
10 __ 4
we'll use the red numbers because whole numbers are easier to work with. The easiest way to get a quadratic correlation is to differentiate and then intigrate.

We get the following data:
dunits power
2 ___ 1.5
3 ___ 2.5
4 ___ 3.5
Or: dunits = power + 1/2
After integrating we get: units = power²/2 + power/2
After reworking this we get: power = (sqrt(8*units + 1) - 1) / 2

combined with the earlier formula we get:
penalty = 1 - (sqrt(8*units + 1) - 1) / (2*units)

 

[mcganyol]

well i understand the math behind it but probably a better explanation to many would be that there is not really "optimal army size". as a rule of thumb you are far better using more effective units than more units. quality over quantity is very important in aod. so yes brigades are very important and fighting with seriously under strengthen units is a very bad idea in general.
Bear in mind that the defensive abilities will not be lowered because of the more units. This is a very important difference from fleets!!

also giving very low-quality units to a fight is usually not a good idea. for example militias, hqs or early cavalry and so on as they will seriously reduce the effectiveness of your main forces. a unit being low quality is always a relative term compared to your other units, even tanks may be considered low quality in swamps or jungles if you have advanced marines. and all units with the over-command penalty (-75%) is nearly always could be referred as low quality.

As already told only the attack values are lowered through stacking (that's why aod battles took usually much longer than in hoi2). So, even militia can be useful as a canon fodder and thus keeping your valuable units longer in the fight.

Giving more units can't hurt but bear in mind that the more you already have the lesser you gain from giving more and more. Avoid having units with over command penalty, that's one of the worst mistakes you can make in AoD warfare. But as long as you not reach that limit your forces will be stronger the more unit involved (their effectiveness will be reduced but if you can afford the increased mp/ic cost that should be no matter) So there are no death-stacks in aod as they used to be in hoi2.

 

[Pang Bingxun]

[...]quality over quantity is very important in aod.[...]

For my taste you give too little for quantity. Of course all your units should be good enough, meaning no hard mali like overstacking or arm in marsh or being understrenght or being unbrigaded should be the case. But as long as quality is sufficient increasing quantity is a very good choice in general. If you attack with 24 Inf-Art instead of 18 Inf-Art the battle duration will be divided by at least sqrt(24/18) and the losses you will suffer will be divided by an even bigger number. Simply spamming Inf-Art is quite a good strategy as long as manpower and ESE suffice. Imo it is these 3 steps one needs to follow:

1. Get sufficient numbers. 6 fighting divisions is the reasonable minimum, below that fighting efficiency is unreasonably low.
2. Get sufficient quality. No fighting division should be unreasonable weaker than Inf-Art as a "reference fighting division".
3. Increase quantity till ICD, ESE or manpower will create a bottleneck that would kill your fighting efficiency.