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