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theokrat

sexy shoeless god of dice
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Feb 6, 2007
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Motivation

Every division has the stats "Defensiveness" and "Toughness" . Since these stats are higher than the respective Attack values they are commonly toughed to be more or less irrelevant. This is because only one Defensive point can be used for each shot that is fired at a unit. The wide-spread argument is that if additionally Defensiveness points (eg from brigades) can only make sense if the enemy has more attack points than you have Defensive points thus he must have more attack points than you (remember defensiveness>attack) thus you fight a loosing battle which you should not do first place.
But as far as we know during combat units target each other on a divisional basis. Meaning Division A randomly picks an enemy division and fires its shots on it. Now obviously by chance it might happen that two units pick the same target. Now the combined shots of these two units will surely exceed the targets defence. So let us have a look at this effect to determine whether or not it can create some significant difference in battle.

Method

I assume each Division picks the enemy it targets randomly and fires all its shots at it. Now lets have a look how many enemy divisions (called Attackers from now) will target one of our divisions (called Defenders from now). We will call this number k. Each Attacker will target this division with the probability of 1/N_D. Where N_D is the number of Defenders. Obviously enough this will lead to a binomial distribution. The probability p(k) of k Attackers targeting our division then reads:

binompd2.jpg

Where N_A is the number of attackers (of whom k target our division), N_D the number of defenders.

E.g. for 36 enemies attacking 12 divisions the probability for each of these divisions looks like this:

binoci0.jpg

You can see the distribution is peeked around 3, which is no big surprise, as in average each unit will be targeted by 3 enemies.

Now if only one enemy targets a division of us nothing surprising happens, the Defensiveness (or Toughness) will surely not be exceed, so each enemy shot counts only once.
But if two enemies target us their combined shots will exceed our Defensiveness. For example two INF39 attack one INF39: 2*12=24 attack points vs only 18 defence points. That means 24+(24-18)=30 shots. Or in general 2*attack-defence. (as long as defence is smaller than attack(which is the combined attack of all attackers)). Since i dont want to compare high numbers i will just chancel out a bit and introduce a new variable "Def". Def is the ratio of Defence (or Toughness, i admit it might be misleading) to the Soft attack of one enemy division. In the above example of INF39 vs INF 39 without modifiers this would be Def = 1.5 (18/12=1.5). The formula for the number of (effective) shots the k enemy divisions fire then reads: (2*k-Def), which is valid as long as k>1 and Def<2.
Now we multiply this factor on the probability of k enemy targeting this division (as derived above) and sum over all k:

weightedbp4.jpg

obviously k=1 was considered seperatly as one enemy will never exceed the Defence (or toughness) at all. Last thing we have to do is normalize this with the ratio of N_D and N_A.

Results

When we compare the effective shots fired at a unit this way and take the ratio to the number where we assume infinite Defencivness (meaning every attack only counts once) we get the following.

First scenario is a pinched fight. N Divisions vs N divisions. I think if a player finds himself in such a battle he will usually be the defender. Thus i choose teh variable Def in a way to represent INF39(45) Soft attack vs INF39(45) Defencivness so Def = 1.5 (2):

pitchedhitswb6.jpg

The x axis is the number of units on each site ranging from 2vs2 to 12vs12. The red graph is for the 1939 case, the blue one for 1945. the y-axis shows how much more shots are fired when compared to ignoring the double (tripple etc) targeting. so 1.2 measn 20% more shots. If we translate this into the % of our man dying because of exceeded Defence we find:

pitchedloosfb7.jpg

As you can see as many as 17.5% of our men die (and our ORG is lost) just because of more units than one targeting. It might be surprising to see the value rising for more units involved, as one miht assume that statistics even out with high numbers. However this does not seem to be the case because the high weighting of high k propabilities (that indeed drop). We also see that late in the war (or good modifiers) the effect is essentially halved.

The secound Scenario is an "overkill" one:
3 N units attack N. This seems to be more common for player attacks. Thus i now choose the Toughness of INF 39 and INF45 to their Attack values. This means we are considering the attacker now:

overhitsdn1.jpg

Or again in % of our men dying (and ORG dropping) because of this effect:

overloosxd4.jpg

Surprisingly the value is still quite high - 10% altough we have only 12 enemies targeting 36 of ours. One might have thought that double targeting is pretty negeltcable in this case.

Conclusion:

The effect seems to be quite sizeable at some points. However given the moderate increase in the stats brigades provide they still seem to be hardly worht it.

Outlook:
The parameter "Def" is the obvios change one can tune. Unmodified sats seem to be hardly realistic, as well as equal tech units fighting each other might be a stretch. Feel free to state what modifiers you think to be a good average number.
Same goes for attacker vs defender ratios. Do you think the 1:1 ratio for the defence and 3:1 for the offense seem to be realistic.
Last of all i am retty unsure on how to include brigades. I might model the additional Toughness/Defenceivness of a Engeener and reduce the number of our units by 25%. Good way?


Thanks for your attention :) Feel free to coment anything.
 
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Yessss..... :D

Finally after all these years a 'poster' after my own heart!
:cool:
 
And here I thought that math had nothing to do with my history major in college.
 
Black Lotus said:
And here I thought that math had nothing to do with my history major in college.

...because actual battles were fought by rolling dice and calculating binomial distribution.

"Damn, we missed our defense roll, our company took 48.346 casualties. McNeill is going to have to be amputated to make the fractions work out."

"Our artillary is landing infront of them and behind them, so on average they are taking a direct hit."
 
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Black Lotus said:
And here I thought that math had nothing to do with my history major in college.

Some of the most important developments in ASW were a product of statistical analysis. Code-breaking was all about number theory. Engineering underpins all big military projects. There is a lot of math in history... :p
 
At this stage, I have mostly questions, not observations...

How about a graph that assumes that 40% (30%? 25%?) of the fighting takes place at night? Most players do the majority of their fighting during the Summer half of the year, when nights are shorter... and some of the battles (especially those fought at high odds) wouldn't last until dark.

How do Engineers / Armored Cars / Artillery change the odds, on a Manpower-equivalent basis? On an IC-equivalent basis? Is the Softness reduction of a hard brigade (AC / SP-Art / SP-AT) signifigant? I suspect not. Is Offensive Doctrine more effective in preserving your Manpower than Defensive Doctrine? I expect it is, slightly, given the common "attacking" playing style.
 
RobbieAB1981 said:
Some of the most important developments in ASW were a product of statistical analysis. Code-breaking was all about number theory. Engineering underpins all big military projects. There is a lot of math in history... :p
Exactly... the blending of mathematical theory with warfare was called "Operational Analysis", and produced some very important results, especially in the ASW field.

A very simple (very, very simple) application of the technique indicated that the convoys being used were inefficiently small, and that increasing the convoy size to sixty or more ships would more-or-less halve the losses of ships in convoy. When this step was actually taken by the Admiralty (commendably quickly), losses of ships in convoy dropped by more than 40%.

Freeman Dyson (Freeman Dyson!) in his memoirs tells a story about the officer who was working at the desk next to his in the Operational Analysis section... who had fought the system literally for years in order to get the floor escape-hatch on a particular model of Bomber widened by two inches. His calculations had indicated that this measure would save the lives of hundreds of Air-crew. When this step was finally taken in mid-war (1942 or 43), the survival rate of shot-down Air-crew (for that particular type of Bomber) improved dramatically. By that time, the career and promotion-prospects of the officer concerned had been seriously damaged... he was considered to be "a trouble-maker" by his superiors.
 
You guys are killing me with a thread title like this. I thought it was only my advanced theological and medical papers that had titles like, "On statistical fluctuations and ... " :p
 
RobbieAB1981 said:
Some of the most important developments in ASW were a product of statistical analysis. Code-breaking was all about number theory. Engineering underpins all big military projects. There is a lot of math in history... :p

There is a lot of math IN history, but that doesn't mean mathematics are a part of most historical studies. Personally, I'm usually content with knowing something like "Some of the most important developments in ASW were a product of statistical analysis." Not how statistical analysis works. Too "numberey".
 
The RC said:
There is a lot of math IN history, but that doesn't mean mathematics are a part of most historical studies. Personally, I'm usually content with knowing something like "Some of the most important developments in ASW were a product of statistical analysis." Not how statistical analysis works. Too "numberey".
The point that RobbieAB1981 was making... and which I supported in my above post... is that the mathematics was not just used retrospectively. It was used at the time, in order to make decisions that affected the lives (or deaths) of many thousands of people. See my examples above.
 
Speaking of night battles...

Has anyone ever tried actually breaking off attacks at nighttime and renewing them again at dawn? In theory that should be the most efficient way to fight battles, although it would add considerably to micromanagement.
 
screwtype said:
Speaking of night battles...

Has anyone ever tried actually breaking off attacks at nighttime and renewing them again at dawn? In theory that should be the most efficient way to fight battles, although it would add considerably to micromanagement.
You can't renew the battle again at dawn... when you break of a battle, you must wait at least 24 hours before the game engine will allow you to issue new orders to those troops.

You could only attack every second dawn... which allows the opponent some time to re-ORG.
 
The RC said:
There is a lot of math IN history, but that doesn't mean mathematics are a part of most historical studies. Personally, I'm usually content with knowing something like "Some of the most important developments in ASW were a product of statistical analysis." Not how statistical analysis works. Too "numberey".

To appreciate some of the points you need to understand statistics. Otherwise you can end up misunderstanding the data. (Which result in idiotic conclusions like banana consumption causes divorce, for which there are real statistical correlations.) Everbody should do basic statistics and mathematical methods, as the skills they require are highly useful in any field.
 
RobbieAB1981 said:
To appreciate some of the points you need to understand statistics. Otherwise you can end up misunderstanding the data. (Which result in idiotic conclusions like banana consumption causes divorce, for which there are real statistical correlations.) Everbody should do basic statistics and mathematical methods, as the skills they require are highly useful in any field.

Ah, now I understand why I had to pass 4 exams on Statistics while studying Economics :)

/me takes note: never buy bananas when married.

The work is nice, but I think it would be nice to compare different brigade types to see how they affect manpower losses and a comparison between a fixed ICdays amount spent in various combinations to see what would be the optimal build setup, considering someone who wants to keep a good ability to fight and limited manpower losses (almost all countries beside USSR and China, I'd say).
 
gorion83 said:
... a comparison between a fixed ICdays amount spent in various combinations to see what would be the optimal build setup, considering someone who wants to keep a good ability to fight and limited manpower losses (almost all countries beside USSR and China, I'd say).
Comparisons based on fixed Manpower and fixed TC-load are both more relevent than fixed IC-cost, I would say... pretty well every country I've played has the ultimate size of their Army limited either by Manpower or by TC limits... and not by their ability to actually build the forces.
 
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screwtype said:
Speaking of night battles...

Has anyone ever tried actually breaking off attacks at nighttime and renewing them again at dawn? In theory that should be the most efficient way to fight battles, although it would add considerably to micromanagement.
Setting the night attack and -defence modifiers for all ground units to -90% makes most action come to a halt at night.
 
Wobbler said:
Setting the night attack and -defence modifiers for all ground units to -90% makes most action come to a halt at night.
That would remove entirely the current advantage that Defenders enjoy at night... which would be quite ahistorical. Why else did attackers tend to wait until dawn in real life?
 
blue emu said:
Comparisons based on fixed Manpower and fixed TC-load are both more relevent than fixed IC-cost, I would say... pretty well every country I've play has the ultimate size of their Army limited either by Manpower or by TC limits... and not by their ability to actually build the forces.
That's true.
I think that we could compare some basic corps being used often by most players.

I'd start with:
1 ARM-2 MOT
3 INF

And then I'd attach several brigades and perform the test to see the effect of those brigades against a similar vanilla corps using the same TC or manpower or ICdays.