It's just a worst-case scenario for militia.Borsook said:
Unfortunately, what I remember my numbers showing is that Mil '36 and Mil '43 optimize for very different things (TC and MP [maybe IC], IIRC).Borsook said:
I'll add it to the list.
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I'll add it to the list.
More than that, it turns out.theokrat said:
First Inf has no chance to cause a double shot.
Second Inf has a 1/9 chance to cause a double shot, and an 8/9 chance to cause a regular shot.
If the second Inf causes a regular shot, the third Inf has a 2/9 chance to cause a double shot, and a 7/9 chance to cause a regular shot.
If the second Inf causes a double shot, the third Inf has a 1/9 chance to cause a triple shot (effectively double at this point) and an 8/9 chance to cause a regular shot.
So there are 729 possible targeting combinations (1-5-2, 1-2-3, 7-8-9, etc.), but they collapse into 81 if we arbitrarily assign number 1 to the militia that the first Inf shoots. 24 of these 81 combinations result in 1 double shot. 1 of these results in 2 double shots (or 1 triple shot, depending on your point of view).
So the odds of at least one doubleshot happening in the 9 mil attacked by 3 inf situation is 30.86%.
Each single shot does 14 + (14 - 8) adjusted shots per inf, or 20.
Each double shot does (28 + (28 - 8)) / 2 adjusted shots per inf, or 24.
Each triple shot does (42 + (42 - 8)) / 3 adjusted shots per inf, or 25.33.
The adjusted shots number in aggregate, counting MTE, works out to be:
3 * ((56 * 20 [single shots] + 24 * 24 [double shots] + 1 * 25.33 [triple shots]) / 81) = 63.75.
The number ignoring MTE is 60 shots, so the MTE effects here adjust upward 6%. You are correct that they favor infantry in this context.
ulmont said:
My statement of 25% was for 3INF vs 3INF.
Stay tuned i will post a general formula for what we did up to now (so without any MTE, but - contrary to the very first post- with exceeded DEF) in a minute.
A general expression
As we know BWA is the product of attack and ORG.
The attack is easy to calculate, its the sum of the number of units times their respective attack over all different unit types. So in this case its the number of Infantry N_i times their Softattack SA_i plus the number of Militia N_m times their Softattack SA_m.
At first glance one might think the ORG is just the number of all of our units. However the ORG of the Militias will wear of much faster than the ORG of the INF, because their Defensiveness is exceeded. If a INF41 is targeted by one INF41 it receives 14 shots, a militia receives 2*14-8=20 shots. Thus the Org of the militia has to be multiplied by the factor 14 / 20. We call this factor gamma.
So the second factor becomes N_i + gamma * N_m.
Now we have a formula for BWA, depending on a few variables. The Attack and Def stats for different years can be read of from the Wiki, that leaves us with the number of INFs and Militias as variables. Now obviously we are in need of some sort of constraint, be it IC, TC or Manpower. Let us say we could build alpha Militias for each INF. Hence the term alpha*N_i + N_m has to be kept constant, we call it c. The value of c is pretty irrelevant. If we manipulate this equation a bit we get a way to get rid of the variable N_m:
pluging this in our bwa formula:
Where we introduced the abbreviations beta and gamma for convenience:
So now we have a one dimensional problem, a mere quadratic equation. We do know how to handle these: we take the derivative and set it equal to zero. (Now you see why i did not bother to determine the N^0 term above):
So we arrive at the optimal numbers N_i and N_m (or their ratio):
An easy enough expression.
Lets say we choose the year 41 and the constraint of TC ->
SA_i = 14 ; SA_m = 2 ; (beta = 14/2 =7) ; gamma = 14 / 20 ; alpha = 1/0.2 = 5
So the ratio becomes negative, due to N_i becoming negative, which means we should not use any INF (as shown by ulmont). However if we assume 99% combat eff instead of 100% the parameter beta changes from 14/2=7 to 13/1=13. Then the ratio becomes 11, meaning we should use 11 militias for every INF. B big stack i admit.
Now if i knew how to create a chart here i could easily post all ratios for all constraints, on the attack and defence and all years...
As we know BWA is the product of attack and ORG.
The attack is easy to calculate, its the sum of the number of units times their respective attack over all different unit types. So in this case its the number of Infantry N_i times their Softattack SA_i plus the number of Militia N_m times their Softattack SA_m.
At first glance one might think the ORG is just the number of all of our units. However the ORG of the Militias will wear of much faster than the ORG of the INF, because their Defensiveness is exceeded. If a INF41 is targeted by one INF41 it receives 14 shots, a militia receives 2*14-8=20 shots. Thus the Org of the militia has to be multiplied by the factor 14 / 20. We call this factor gamma.
So the second factor becomes N_i + gamma * N_m.
Now we have a formula for BWA, depending on a few variables. The Attack and Def stats for different years can be read of from the Wiki, that leaves us with the number of INFs and Militias as variables. Now obviously we are in need of some sort of constraint, be it IC, TC or Manpower. Let us say we could build alpha Militias for each INF. Hence the term alpha*N_i + N_m has to be kept constant, we call it c. The value of c is pretty irrelevant. If we manipulate this equation a bit we get a way to get rid of the variable N_m:
pluging this in our bwa formula:
Where we introduced the abbreviations beta and gamma for convenience:
So now we have a one dimensional problem, a mere quadratic equation. We do know how to handle these: we take the derivative and set it equal to zero. (Now you see why i did not bother to determine the N^0 term above):
So we arrive at the optimal numbers N_i and N_m (or their ratio):
An easy enough expression.
Lets say we choose the year 41 and the constraint of TC ->
SA_i = 14 ; SA_m = 2 ; (beta = 14/2 =7) ; gamma = 14 / 20 ; alpha = 1/0.2 = 5
So the ratio becomes negative, due to N_i becoming negative, which means we should not use any INF (as shown by ulmont). However if we assume 99% combat eff instead of 100% the parameter beta changes from 14/2=7 to 13/1=13. Then the ratio becomes 11, meaning we should use 11 militias for every INF. B big stack i admit.
Now if i knew how to create a chart here i could easily post all ratios for all constraints, on the attack and defence and all years...
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Ah. That's even farther off.theokrat said:
First Inf has no chance to cause a double shot.
Second Inf has a 1/3 chance to cause a double shot, and an 2/3 chance to cause a regular shot.
If the second Inf causes a regular shot, the third Inf has a 2/3 chance to cause a double shot, and a 1/3 chance to cause a regular shot.
If the second Inf causes a double shot, the third Inf has a 1/3 chance to cause a triple shot (effectively double at this point) and an 2/3 chance to cause a regular shot.
27 possible combinations of targets for the 3 inf (1-2-3, 2-2-3, etc.), but the first one is irrelevant (set to 1), leaving 9 useful combinations.
1-1 yields 2 double shots (all 3 inf fired at the same target).
1-2 yields 1 double shot (inf 1 and 2 fired at the same target).
1-3 yields 1 double shot (same).
2-1 yields 1 double shot (inf 1 and 3 fired at the same target).
2-2 yields 1 double shot (inf 2 and 3).
2-3 yields 0 double shots.
3-1 yields 1 double shot (inf 1 and 3 fired at the same target).
3-2 yields 0 double shots.
3-3 yields 1 double shot (inf 2 and 3).
2/9 chance of 0 double shots, 6/9 chance of 1 double shot, 1/9 chance of 2 double shots.
So actually there's a 78% chance that double shots will happen when 3 Inf attack 3 Inf.
Each single shot does 14 + (Max(0, 14 - 24) * 2) adjusted shots per inf, or 14.
Each double shot does (28 + (Max(0, 28 - 24)) / 2 adjusted shots per inf, or 16.
Each triple shot does (42 + (Max (0, 42 - 24)) / 3 adjusted shots per inf, or 20.
The adjusted shots number in aggregate, counting MTE, works out to be:
3 * ((2 * 14 [single shots] + 6 * 16 [double shots] + 1 * 20 [triple shots]) / 9) = 48.
The number ignoring MTE is 42 shots, so the MTE effects here adjust upward 14%. Militia suffer less from the MTE effects than the infantry do.
Edit: Which is expected. Militia have already had their defensiveness exceeded by 1/2 with the first shot, and have more units to diffuse incoming targets, reducing the odds of MTE.
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ulmont said:
the other way round.
I calculated the chance for each individual unit to be target by more than one unit. I should of course have m,ulitplied this by three.
Its easy: the first untit picks a target (does not matter which), the secound has a 2/3 chance to hit undoubled as well, the third has only 1/3 to not double target, leaves us with 2/9=22.2%. Or a 78% chance of at least some double shots. As i said i failed to multiply by three above.
theokrat said:
I think there are two bugs in this equation.
First, if a negative ratio indicates that we should not use any INF, how do you determine that you should not use any MIL?
Second, taking your formula, and using '41 numbers (beta 7 and gamma 14 / 20):
Nm / Ni = (alpha + alpha * beta * gamma - 2 * beta) / (1 + beta * gamma - 2 * alpha * gamma)
Nm / Ni = (alpha + alpha * 7 * (14 / 20) - 2 * 7) / (1 + 7 * (14 / 20) - 2 * alpha * 14 / 20)
Nm / Ni = (alpha + 4.9 * alpha - 14) / (5.9 - 1.4 * alpha)
Nm / Ni = (5.9 * alpha - 14) / (5.9 - 1.4 * alpha)
For IC, alpha = 665 / 200 = 3.325
Nm / Ni = (5.9 * 3.325 - 14) / (5.9 - 1.4 * 3.325)
Nm / Ni = 5.6175 / 1.245 = 4.51:1
So, based on IC, you end up with 9 mil for every 2 infantry.
For MP, alpha = 10 / 5 = 2.
Nm / Ni = (5.9 * 2 - 14) / (5.9 - 1.4 * 2)
Nm / Ni = -2.2 / 3.1 = -0.70:1.
For MP, we have a negative ratio again. If this suggests that no infantry should be used, then I suggest the calculation is broken. If that suggests that no militia should be used, how do you tell?
Edit: As noted below, if Nm is negative, use no militia. If Ni is negative, use no infantry. For MP constraints, the formula suggests no militia, in line with previous observations.
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Right. I eyeballed the 7/9 conversion, and picked 8/9 for some reason. Corrected above. I believe the ultimate adjusted shot numbers, and percentage increase, are still correct.theokrat said:
ulmont said:
The formulars for both N_i and N_m are given, one has to look which is negative. If its not obvious (which by now it is
)I dont see what you mean with the second bug?
Ok. That makes sense (on both counts).theokrat said:The formulars for both N_i and N_m are given, one has to look which is negative. If its not obvious (which by now it is
I had not realized how to determine which component should be omitted, so I was afraid your formula was suggesting a pure Mil army for the MP constraint (obviously flawed).theokrat said:
Now it makes sense.
So the followup questions are:
1) What combat efficiency should one plan for, on attack and on defense? I suggest that attack efficiency is usually < 100%, and that defense is typically > 100%, but that's just my observed experience.
2) What are the TC / IC / MP optimal ratios for Mil '36 v. Inf '36, and for Mil '36 v. Inf '39?
Ok, so I got 3 questions:
1.)
Why did you substitute c by c tilde? Don´t you like the pure letter? Did it confuse you somehow? I´m just curious....
2.)
What happened to SA_M here? I don´t see why it can be neglected? Should it not be 2? This would reduce the value of N_I to half its current number.
3.)
What did you replace c tilde with to get from the first line of N_M to the second line?
1.)
2.)
3.)
SAm can be neglected because it is a constant multiplier of the entire dBWA / dNi term. Since that term is being set to 0, you can just divide both sides by SAm (always non-zero), yielding 0 / SAm on the left or 0.General Failure said:2.)
What happened to SA_M here? I don´t see why it can be neglected? Should it not be 2? This would reduce the value of N_I to half its current number.
Edit: I have no idea what that second line means on Nm, but I don't think it matters, since the c~ terms cancel out in Nm / Ni.General Failure said:3.)
What did you replace c tilde with to get from the first line of N_M to the second line?
ulmont said:
The non-trivial sollutions are only 36 (beta =5, gamma = 10/12) on manpower account (alpha=2) , which leaves us with a ratio of 1 milita per 5 INFs.
And 39 (beta = 6, gamma = 12/16) on IC (alpha = 3.325). That leaves a 12 Milita per INF.
The 39INF vs 36Mil on manpower favours total INF, all other cases (36 on TC and IC and 39 on TC) favour total Militia stacks. IF ce is 100%.
No big surprise here. TC and IC constraint and early years favour Militia, Manpower and later years INF.
Dammit! I was confused by the braces..... *demolishes his table with his head*ulmont said:
I thought the angled braces ended after the N_I ...
General Failure said:
good question. In a previous calculation i used the inverse of alpha instead of alpha, so i always had to carry the alpha times c around, so i redefined it (drops out anyway). Its not nes. anymore now, right.
General Failure said:
As ulmont said, i just devided by it, so it drops out. In the secound line of bwa i assumed its unequal zero without stating it.
General Failure said:
Right again, i forgot it as it chancels out right in the next line. SHould be ther tough, just tought it might be a nice representation.
Edit: changed it, thx.
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You guys are officially a bunch of freaks! Woah, are you all wannabe-mathematicians? I bet the paradox guys didnt even do such enormous calculations.
I just wanna express my deep appreciation of your works.
But can you come up with a "killer-stack", or a stack which is most versatile and useful? Like 1 Inf+4 Mil? The usual "killer stack", or rather the most versatile is the 3 Inf stack. What would that equate to, in Militia terms? What is really the best bang for a buck?
I just wanna express my deep appreciation of your works.
But can you come up with a "killer-stack", or a stack which is most versatile and useful? Like 1 Inf+4 Mil? The usual "killer stack", or rather the most versatile is the 3 Inf stack. What would that equate to, in Militia terms? What is really the best bang for a buck?
The problem with this whole thread is it takes a very basic calculation with some basic assumptions.
The basic assumptions therefore render the whole point of the thread useless.
Here are the things to consider.
1. Attack efficiency is almost always less then 100% so therefore militia are hurt more by the reduction making them even more useless for attacking.
2. Secondly defense efficiency is almost always above 100% so therefore other units gain more bonuses then milita do because of their extremely low values.
3. These calculations again do not take into effect leader bonuses which in turn help infantry more then militia.
4. Also they do not factor in unit experience. Unit experience benefits infantry more then militia. In addition militia lose more men and therefore lose more experience through replenishment.
SO yes militia and inf combined armies make interesting mathematical examples but are hardly feasible in the game as an effective strategy.
I would like to see how a russian player in a mp game who makes mixed armies of militia fare versus an experienced german player. Even a mix of 3 militia for every one infantry would make him very impotent on the counter attack. Which is how russia is supposed to survive.
So therefore logically militia have a minor situational role when you apply logic instead of pure mathematics with limits imposed.
The basic assumptions therefore render the whole point of the thread useless.
Here are the things to consider.
1. Attack efficiency is almost always less then 100% so therefore militia are hurt more by the reduction making them even more useless for attacking.
2. Secondly defense efficiency is almost always above 100% so therefore other units gain more bonuses then milita do because of their extremely low values.
3. These calculations again do not take into effect leader bonuses which in turn help infantry more then militia.
4. Also they do not factor in unit experience. Unit experience benefits infantry more then militia. In addition militia lose more men and therefore lose more experience through replenishment.
SO yes militia and inf combined armies make interesting mathematical examples but are hardly feasible in the game as an effective strategy.
I would like to see how a russian player in a mp game who makes mixed armies of militia fare versus an experienced german player. Even a mix of 3 militia for every one infantry would make him very impotent on the counter attack. Which is how russia is supposed to survive.
So therefore logically militia have a minor situational role when you apply logic instead of pure mathematics with limits imposed.
OttomanSipahi said:
In the first post i adressed this by ploting several different graphs for different CEs. I am not sure if defence is always above 100%, a 60 infra province already means a -20% modifier. However if you have any feeling of a typical battle: please post it! I do have a 1:1 maping of combat, but its useless if there is no input, the more realistic input there is the better i can determine the use of milita, brigades etc...
In the last math heavy post there was no requirement for the modifiers, other than the milita attack may not be 0. You can easily take every modifers (leader, terrain, exp, ESE...) into account by just changing the variables beta and gamma accordingly.