# Proving the Manual Wrong RE: Stacking Penalty

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#### dhelmet99

##### First Lieutenant
I had earlier come up with an approximation of the stacking penalty via in-game testing of the 1939 scenario as Germany with all leaders removed to avoid the coordination bonus. I recorded the number of brigades and the stacking penalty for a wide range of combats. The resulting correlation was pretty good, but not great.

I assumed the stacking penalty was based on the number of brigades as per the manual:

L2.0-
There is a Stacking Penalty which penalizes each side for
each Brigade they have in either the Front or the Reserves. It
becomes difficult to coordinate so many Units, but effective

Also,

L5.1 Stacking Penalty
The more Divisions involved in a combat, the harder it
is for them to effectively coordinate. A skilled Theatre
Commander may add to this ability, avoiding the penalty.
This is basically a stacking penalty. Certain Doctrine
Research and Tech Advances can also help. The Stacking
Penalty is approximately 1% per brigade, even if it is not
in combat (reserve), and there will be a 1% reduction for
every skill level of the Theatre Commander. As a result of
this Penalty, an army comprised mostly of Support-heavy
Divisions (i.e. only one Combat Brigade) may face problems
with coordination.

It was recently suggested that perhaps it is the number of divisions and not brigades that determines the stacking penalty. This would explain the variation in the data that I observed. I therefore re-ran some tests tonight and kept track of number of divisions, brigades and the stacking penalty for a number of combats.

The results are below:

Even a quick perusal verifies that it is not brigades that affect the stacking penalty. To further solidify this conclusion I created two graphs and fit some regression curves to them.

First is the number of brigades versus the stacking penalty with a linear equation. The equation is close but there is significant variation.

Next is the same with the number of divisions. Using a second order polynomial the fit is perfect.

The definitive equation for stacking penalty is therefore:

Penalty = -0.1069*(# Div)^2 + 5.9502*(# Div) - 22.066

(Note that this should only be used >4 divisions)

The only assertion in the manual, that the stacking penalty depends on all of the units in the province and not just those involved in combat, is also erroneous. I performed several tests by initiating combat with all the divisions in one province and then with only part of them. The stacking penalty only depended on the number of divisions actually involved in combat.

I believe this definitively proves that the manual is incorrect in this particular. It's possible they will update the calculation in a future version, but I would guess that it is a low priority.

What implications does this have on how to play? There is no penalty to packing in as many brigades as possible into each division. Additionally, there is a clear incentive to stuff more support brigades into each division such that more brigades can be involved in combat simultaneously.

#### unmerged(110676)

##### A Single Spark
You just made something way more complicated then it needs to be.

#### Mendeth

##### Colonel
You just made something way more complicated then it needs to be.

Though his argument is absolute, at least in terms of the statistical data. There's nothing wrong with that. Complexity is not always to be frowned upon.

Although I don't expect this to happen as there are far more important issues to deal with; it would be probably be helpful if the manual were to be updated to correct this.

#### Jorath13

##### Major
I for one say thank you for your time and dedication...too bad you weren't a beta tester (or were you?). This is interesting also in that I had read the relevant sections the other night that you've highlighted and was starting to solidify that information in my head. Now I'll simply go with a rough equation of "if >4 divisions in battle, 5% penalty per div" though I admit that's not accurate. It will induce me though to avoid putting in 20, 30, or more divisions in one battle.

#### unmerged(138184)

##### Second Lieutenant
You just made something way more complicated then it needs to be.

Completely disagree. To the original poster: from one statistician to another - good work!

#### Markusw7

##### Slave to the Magyar
you just proved a paradox manual to be incorrect, kudos

#### pnt

##### Colonel
What implications does this have on how to play? There is no penalty to packing in as many brigades as possible into each division. Additionally, there is a clear incentive to stuff more support brigades into each division such that more brigades can be involved in combat simultaneously.

Not having a penalty for packing a lot into each division kind of makes sense, though, doesn't it? Coordination within a division is usually much easier than between divisions. And as you pointed out in another thread, large divisions are penalized by the fact that smaller ones get off their shots quicker (a very small effect, but still a penalty).

The second part does not follow from the first, however. If you are limited by frontage, adding support brigades will increase the number of divisions, which is subject to a stacking penalty. If you are not limited by frontage, you could just as well add more combat brigades.

#### dhelmet99

##### First Lieutenant
So a little further testing revealed that my equation is extremely close but not exact. It starts to underestimate the penalty past about 16-17 divisions. I doubt that very many people will perform combats past that point. I suspect Paradox is using a direct table look-up and not an equation to represent this penalty.

If anyone is curious I also explored the effect of attacking from multiple provinces on the stacking penalty. The equation itself is still a good approximation with one minor exception: the number of divisions in the equation is equal to the number of divisions in the province with the most divisions plus half of the divisions in every other province. This worked for both two and three province attacks. I didn't have the patience for more.

#### TheLoneGunman

##### NO STEP ON SNEK
Moderator
I always figured it was for the number of divisions and not brigades... But people told me I was wrong...

Good work!

This means it can be useful to pack in a lot of support into your divisions, but you might also want to try and cut down on the total amount of divisions you send into combat and maximize the combat width each division can cover.

My opinion, this system gives you way more options than a system that simply counts the number of brigades.

#### Rensslaer

##### Strategy GuidAAR
Dhelmet,

This is a good discovery. If I update the manual, I will take note of it, though if it's an error (and it seems to be -- I don't doubt your calculations) I'm sure it's something they did intentionally and the word didn't get to me. They will surely leave things as they are, and we'll correct the manual. As one poster mentioned, it does make more sense that coordination between divisions would get crosswise, rather than communication within a division.

On the second point, I think you misread. I didn't actually say stacking is dependent on the number of units in the province. I said it depends on the number of brigades (divisions) in combat, even if they're in the reserve (i.e. not in direct combat). The number of units in the province is irrelevant.

If anyone is curious I also explored the effect of attacking from multiple provinces on the stacking penalty. The equation itself is still a good approximation with one minor exception: the number of divisions in the equation is equal to the number of divisions in the province with the most divisions plus half of the divisions in every other province. This worked for both two and three province attacks. I didn't have the patience for more.
This actually makes sense, though I was unaware of this calculation. Since a regular combat (the first province) is across 10 squares, and an additional front (from a second province) only adds 5 squares, the calculation apparently matches the number of squares, regardless of where the units in combat actually come from.

I'll add to the praise for this work -- good job! Analyses like this are not easy, and certainly do require knowledge and attention to detail.

Rensslaer

#### pnt

##### Colonel
Good work!

Agree!

This means it can be useful to pack in a lot of support into your divisions, but you might also want to try and cut down on the total amount of divisions you send into combat and maximize the combat width each division can cover.

Not really. You only benefit from support if you have maxed out your frontage. Optimal division size is, however, four combat brigades, as I tried to point out earlier.

You can add one support brigade to each size 4 division, or two to size three divisions. Anything else will penalize your frontage.

#### unmerged(90249)

##### Colonel
Good work, and it's confirmed by earlier Johann Post :

a province frontage is 10, so the best way to attack (most efficient) is 3 divs having 3 brigades + support and a fourth div having 4 combat brigades.

That is consistent with your search, frontage is related to brigades, but stacking is related to divisions, starting at a base of 4 divs.

#### pnt

##### Colonel
the best way to attack (most efficient) is 3 divs having 3 brigades + support and a fourth div having 4 combat brigades.

Well, 4 + 4 + 5 = 13 as well.
But regardless of how you sum things up, making sure that you disperse this odd brigade correctly at all times is a huge administrative task.

#### Kuciwalker

##### Major
The reason for the inaccuracy in the equation is because the penalties as reported are almost certainly rounded to one decimal.

#### dhelmet99

##### First Lieutenant
Okay, I extended my data out to 24 divisions. It's actually impossible to have >20 divisions involved in combat in a single province (and even 20 requires the militia tech that reduces width and single militia + support divisions). It's at least possible for this to be exceeded by attacking from multiple provinces, but that should be an incredibly rare circumstance.

My final data is below.

In table form with my final approximation:

Penalty = -0.0813*(# Div)^2 + 5.4661*(# Div) - 20.094

That's not quite perfect, but it is certainly close enough to fit the data. I think I have beaten this horse hard enough. But wait, now that I think of it I could quantify the effect of the theater's field marshal's skill level on stacking penalty. I think that can wait for tomorrow.

Good work, and it's confirmed by earlier Johann Post :

a province frontage is 10, so the best way to attack (most efficient) is 3 divs having 3 brigades + support and a fourth div having 4 combat brigades.

That is consistent with your search, frontage is related to brigades, but stacking is related to divisions, starting at a base of 4 divs.

Actually the best way to attack will almost certainly turn out to be 10 different Inf + 3 Art brigade divisions. That combo only has a 26.7% penalty on combat. The result is that the 10 division stack has 218% more soft attack after correcting for the stacking penalty. Note that the highlighted cells are corrected for stacking penalty. This is not necessarily the most cost effective way of building divisions, but it is unquestionably powerful. I'll be doing more analyses like this one in my quantitative division design thread tomorrow (http://forum.paradoxplaza.com/forum/showthread.php?t=420352).

#### TheLoneGunman

##### NO STEP ON SNEK
Moderator
Out of curiousity, are individual brigades counted as divisions in regards to the stacking penalty?

I'd assume that they would.

#### pnt

##### Colonel
Actually the best way to attack will almost certainly turn out to be 10 different Inf + 3 Art brigade divisions. That combo only has a 26.7% penalty on combat. The result is that the 10 division stack has 218% more soft attack after correcting for the stacking penalty. Note that the highlighted cells are corrected for stacking penalty. This is not necessarily the most cost effective way of building divisions, but it is unquestionably powerful.

Since we are perhaps not seeking to reinvent the megastack, let me suggest a slightly different focus. It seems that a reasonable stacking penalty would be about 10% per division past the first four (this does not work very well for aircraft, but would for ground units), which is a little higher than the current value. Why not make a more detailed suggestion that could go into the patch? The mechanism for the stacking penalty is not broken, but it would be nice if you can help fine tuning the parameters.

#### unmerged(152756)

##### Corporal
<snip>

What implications does this have on how to play? There is no penalty to packing in as many brigades as possible into each division. Additionally, there is a clear incentive to stuff more support brigades into each division such that more brigades can be involved in combat simultaneously.

Thankyou for the clarification.

#### Peekee

##### Field Marshal
Penalty = -0.0813*(# Div)^2 + 5.4661*(# Div) - 20.094

Some playing with a spreadsheet and your values (Spreadsheet did some rounding on values displayed here ):

Code:
``````D         0.95^(D-4)  (1-X)*100
4         1           0
5         0.95        5
6         0.9         9.75
7         0.86        14.26
8         0.81        18.55
9         0.77        22.62
10        0.74        26.49
11        0.7         30.17
12        0.66        33.66
13        0.63        36.98
14        0.6         40.13
15        0.57        43.12
16        0.54        45.96
17        0.51        48.67
18        0.49        51.23
19        0.46        53.67
20        0.44        55.99
21        0.42        58.19
22        0.4         60.28
23        0.38        62.26
24        0.36        64.15
25        0.34        65.94
26        0.32        67.65
27        0.31        69.26
28        0.29        70.8
29        0.28        72.26
30        0.26        73.65
31        0.25        74.97
32        0.24        76.22
33        0.23        77.41
34        0.21        78.54
35        0.2         79.61
36        0.19        80.63
37        0.18        81.6
38        0.17        82.52
39        0.17        83.39
40        0.16        84.22
41        0.15        85.01
42        0.14        85.76
43        0.14        86.47
44        0.13        87.15
45        0.12        87.79
46        0.12        88.4``````

Exponential not quadratic (although some rounding issues??). What is especially interesting is if you look at D*the second column. This gives you a value of the fighting strength in terms of original division strength (assuming all divisions have equal fighting strength).

Code:
``````D       Power
4       4
5       4.75
6       5.42
7       6
8       6.52
9       6.96
10      7.35
11      7.68
12      7.96
13      8.19
14      8.38
15      8.53
16      8.65
17      8.73
18      8.78
19      8.8
20      8.8
21      8.78
22      8.74
23      8.68
24      8.6``````

So if you ever attack with > 20 divisions you will actually decrease your overall attacking strength rather than increase. If you have lots of width 1 divisions (Inf +3*Art) this is not impossible if attacking from 4 sides (width 25). Arguably in that case you would win regardless, however you would actually be better off with slightly less artillery and slightly more infantry. As the infantry would soak up incoming attacks better. (better toughness/defensiveness and higher strength).

-----

Edit: Does the number of provinces attacking from / effect the stacking penalty???

Last edited:

#### dhelmet99

##### First Lieutenant
Some playing with a spreadsheet and your values (Spreadsheet did some rounding on values displayed here ):

Code:
``````D         0.95^(D-4)  (1-X)*100
4         1           0
5         0.95        5
6         0.9         9.75
7         0.86        14.26
8         0.81        18.55
9         0.77        22.62
10        0.74        26.49
11        0.7         30.17
12        0.66        33.66
13        0.63        36.98
14        0.6         40.13
15        0.57        43.12
16        0.54        45.96
17        0.51        48.67
18        0.49        51.23
19        0.46        53.67
20        0.44        55.99
21        0.42        58.19
22        0.4         60.28
23        0.38        62.26
24        0.36        64.15
25        0.34        65.94
26        0.32        67.65
27        0.31        69.26
28        0.29        70.8
29        0.28        72.26
30        0.26        73.65
31        0.25        74.97
32        0.24        76.22
33        0.23        77.41
34        0.21        78.54
35        0.2         79.61
36        0.19        80.63
37        0.18        81.6
38        0.17        82.52
39        0.17        83.39
40        0.16        84.22
41        0.15        85.01
42        0.14        85.76
43        0.14        86.47
44        0.13        87.15
45        0.12        87.79
46        0.12        88.4``````

Exponential not quadratic (although some rounding issues??). What is especially interesting is if you look at D*the second column. This gives you a value of the fighting strength in terms of original division strength (assuming all divisions have equal fighting strength).

Code:
``````D       Power
4       4
5       4.75
6       5.42
7       6
8       6.52
9       6.96
10      7.35
11      7.68
12      7.96
13      8.19
14      8.38
15      8.53
16      8.65
17      8.73
18      8.78
19      8.8
20      8.8
21      8.78
22      8.74
23      8.68
24      8.6``````

So if you ever attack with > 20 divisions you will actually decrease your overall attacking strength rather than increase. If you have lots of width 1 divisions (Inf +3*Art) this is not impossible if attacking from 4 sides (width 25). Arguably in that case you would win regardless, however you would actually be better off with slightly less artillery and slightly more infantry. As the infantry would soak up incoming attacks better. (better toughness/defensiveness and higher strength).

-----

Edit: Does the number of provinces attacking from / effect the stacking penalty???

First, the number of province does affect the stacking penalty. As I stated earlier in the thread the number of divisions in the calculation becomes the highest number of divisions in any of the provinces that is attacking plus HALF the number of divisions in every other province that is attacking. This makes it proportional to the amount of frontage added, since each additional province only adds 5 to the front. This also means that if there is an odd number of attacking divisions we can get values that are between points in my observed table above, which is something I directly observed.

Second the exponential equation you propose fits the data almost as well and is much simpler. There is still some weird rounding thing going on. The numbers in the combat screen always have two digits past the decimal, but are rounded to the nearest tenth.

Since the straight formula slightly underestimates the values at each point I changed it slightly as below and it was closer, but even still not exact.

(1-0.9495^(#_Div-4))*100

Out of curiousity, are individual brigades counted as divisions in regards to the stacking penalty?

I'd assume that they would.

I have not tested divisions composed of a single combat/support brigade, but HQ divisions with only the HQ brigade DO count as divisions with regards to the limit. As stated earlier this can easily be avoided by not directly attacking with your HQ units. There is no penalty for having more divisions in the province only the divisions directly involved in combat count toward the stacking penalty.

Since we are perhaps not seeking to reinvent the megastack, let me suggest a slightly different focus. It seems that a reasonable stacking penalty would be about 10% per division past the first four (this does not work very well for aircraft, but would for ground units), which is a little higher than the current value. Why not make a more detailed suggestion that could go into the patch? The mechanism for the stacking penalty is not broken, but it would be nice if you can help fine tuning the parameters.

I'm disappointed in this as I also did not want to see giant stacks of divisions being superior to more balanced approaches. I think 10% per division is a bit excessive, but I do have two proposals that I think would be better than the current values.

First, is to modify the exponential to increase the penalty. In particular I believe the following equation would be better.

100*(1-0.925^(#_Div-4))

Second, just a straight linear equation with 5% per division past the first four. This would start to massively penalize large stacks.

The shape of the curves are different and it would depend on how much the devs like having a progressive line instead of just a straight line. The proposed values are displayed in table and graphical form compared to the current penalty.

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