Regarding artillery retreating from the back row and the AI.
I want to compare how large the difference in army strength is between the optimal army composition and the average AI army composition in the following two scenarios:
(i) As implemented right now in the 1.33 beta,
(ii) As implemented right now in the 1.33 beta but with artillery not retreating from the back row.
In order to do this we need to determine the optimal army compositions, the average AI compositions and their associated strength depending on the number of troops involved.
To do so, define the following:
T:= the total number of regiments in a given army,
A:= the number of artillery regiments in the army,
F:= T-A the number of front line (inf +cav) regiments in the army,
a:= the relative strength advantage of a front line regiment supported by artillery vs a front line regiment not supported by artillery, (this depends on a lot of things like tech and modifiers but can be explicitly calculated for any given situation)
b:= the relative strength of an artillery regiment in the front line vs a front line regiment.
CW:= the combat width of the province where the battle takes place.
For ease of strength calculation, we normalize in such a way that the strength of a front line regiment is 1.
We will focus on situation (ii) first:
Optimal compositions:
Clearly we need to make a case distinction here:
Case I:
If T=<2CW then the optimal army composition is A=T/2,
We then calculate the strength of the army as:
Str_O(T)=0.5Ta
Case II:
If T>=2CW then the optimal army composition is A=CW,
We then calculate the strength of the army as:
Str_O(T)=(T-CW)a
AI compositions:
Note that for most of the game the AI runs a fixed ratio of approx. A=0.4T. This means we need to distinguish between 3 cases:
Case I:
If T=<2CW then the AI army composition is A=0.4T, where A<CW
We then calculate the strength of the army as:
Str_AI(T)=0.4Ta+0.2T
Case II:
If 2,5CW=<T=<2.5CW then the AI army composition is A=0.4T, where A=<CW
We then calculate the strength of the army as:
Str_AI(T)=(2CW-0.8T)+(1,4T-2CW)a
Case III:
If 2,5CW=<T then the AI army composition is A=0.4T, where A>=CW
We then calculate the strength of the army as:
Str_AI(T)=0.6Ta+(0.4T-CW)b.
Strength Quotient:
We can now calculate the quotient of army strength between the optimal composition and the AI:
Clearly we have to look at three cases again:
Case I:
If T=<2CW then
(Str_O/Str_AI)(T)=0.5Ta/(0.4Ta-0.2T)=0.5a/(0.4a-0.2)
Case II:
If 2,5CW=<T=<2.5CW
then
(Str_O/Str_AI)(T)=(T-CW)a/(2CWa-1,4Ta-2CW+0,8T)
Case III:
If 2,5CW=<T
then
(Str_O/Str_AI)(T)=(T-CW)a/(0,6Ta-0.4Tb+CWb)
We will look at situation (i) next:
With artillery retreating everything is a bit easier to analyze, as combat stays consistent when exceeding CW.
We get:
Optimal compositions:
Optimal composition is A=0.5T and we get:
Str_O(T)=0.5Ta
AI compositions:
Note that for most of the game the AI runs a fixed ratio of approx. A=0.4T.
Therefore:
Str_AI(T)=0.4Ta+0.2T
Strength Quotient:
We can now calculate the quotient of army strength between the optimal composition and the AI:
(Str_O/Str_AI)(T)=0.5Ta/(0.4Ta-0.2T).
So what do we learn from this. As long as T=<2CW the AI performance is not affected by the change at all.
When T>=2.5CW the combat without artillery retreating degenerates in such a way that A/T is dependent on T, clearly an AI which operates on the assumption A/T constant is not well adapted to the situation.
The case 2CW=<T=<2.5CW can be seen as an interpolation of the other two cases where combat only partially degenerates.
Therefore, the crucial case to analyze is T>=2.5CW. Here the 1.33 system acts as a continuation of the system with T=<2CW, while the other system starts to degenerate more and more.
So we compare: 0.5Ta/(0.4Ta-0.2T) and (T-CW)a/(0,6Ta-0.4Tb+CWb)
Now under the assumption that one does not want to have artillery in the front row one has b=<0 (So b=0, because one can retreat in EU4).
In this case the quotient of situation (ii) divided by situation (i) is:
(T-CW)a/(0,6Ta-0.4Tb+CWb)/0.5Ta/(0.4Ta-0.2T)=(T-CW)(0.4a-0.2)/0.5a/0.6T=(0.4Ta+0.2T-0.4CWa-0.2CW)/(0.3aT)=1+(0.1Ta+0.2T-0.4CWa-0.2CW)/(0.3aT).
Therefore, the AI is better in situation (i) if and only if 0.1Ta+0.2T-0.4CWa-0.2CW>=0.
Solving for T yields T>=4CW-(0.6CW)/(0.1a+0.2). With 4 as an upper bound for a being a safe bet, at least with the current techs and modifiers, we get that, if T>=3, we have 0.1Ta+0.2T-0.4CWa-0.2CW>=0 and therefore the AI is better with the current changes in 1.33, i.e. situation (i).
Therefore, the AI is better in almost every case, if the change to artillery is not reverted. There is only a very small gap near T=2.5CW, where the AI would be better, if one where to revert the change.
With the complicated part out of the way, two remarks to other concerns:
1. How much additional army reorganization is needed?
In 1.32 every stack which contains infantry (so in the MP meta every stack) takes casualties and therefore needs to be constantly reorganized. One way around this is to include more infantry than CW. In 1.33 the number of stacks which contain infantry is smaller or equal than in 1.32 (I think this should be clear).
Now what could lead to reorganization being more complicated? Wanting to avoid more infantry than artillery in any given stack.
But is this necessary? Lets compare:
One stack with A=F and one with A<F. The fear is now that, if all artillery retreat and only infantry remains, the infantry fights inefficiently. But what happens to the army with A=F when all artillery retreats.? It loses the battle. Therefore infantry entering the front row without artillery support needs to be worse than losing the battle.
Is this the case?
If you can retreat, certainly not, as you can just retreat your infantry. Which gives you losing the battle as an option and is therefore strictly better.
If 12 day have not passed yet, then it probably prevents a stack wipe which is also better.
So in conclusion, adding more infantry to stacks is maybe not needed but also not harmful for the battle.
2. The MP meta will change. Yes it will, but it is not clear that it will consist only of large mixed stacks. At least, if you do not ignore attrition. Probably just watch the video by
@Distinct for this. It is very insightful.
And maybe some other potential upside of cannons not retreating, which should be pretty clear when looking at the strength calculations above:
1) Battles are shorter (up to 50%) and having more troops gives you less of an advantage (also up to 50%). In 1.32 the effect of troop quantity on strength is superlinear and 1.33 it is linear.
2) This also leads to gathering all troops for one battle being less efficient for large countries, which could lead to multiple fronts being more advantageous. This could also increase skill in MP.
Sorry for all the math. But discussing something like this requires at least some mathematics.
Edit: Clarified the AI strength calculation a bit, using quotients instead of differences.
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