Ok, I haven't had time to look through the lua files, so that's yet another case of the manual being wrong: L4.1 Firing Phase: "So if a defending Division’s modified Defensiveness is 6.3, it may avoid being hit by the first six shots aimed at it, while any additional shots would hit." However, it's worth noting that it still gives double the chance to hit and therefore any divisions that receive shots in excess of their def/tough will receive (on average) double damage from those shots.If you would take a look at defines.lua, you will find:
BASE_CHANCE_TO_AVOID_HIT = 0.8,
-- Base chance to avoid hit if defences left.
CHANCE_TO_AVOID_HIT_AT_NO_DEF = 0.6,
-- chance to avoid hit if no defences left.
In other words, anyone could still get hit with defense points, and that anyone could still avoid hits without defense points.
Concluding from this, the only benefit for having more divisions is to allow the inclusion of more support brigades to end a fight quicker before random factors average out.
There are plenty of instances where there's combat with big stacks of divisions, e.g. naval invasions (especially from the USA). The bulk of an army is generally plain infantry divisions, so if there's any advantages or disadvantages based on division size it's worth knowing about. That is the general point of this thread anyhow. So if you're playing GER, for example, and you've got stacks of small INF or GAR divisions in ports to defend against USA naval invasions then it's worth knowing if there's an advantage to be gained from merging those divisions!
Stats derived from dhelmet's, but for 2xinf & 5xinf (rounded down):
2xINF: SA:4, DEF:10, TOUGH:6
5xINF: SA:10, DEF:26, TOUGH:15
2 5xINF vs 5 2xINF:
Probability of 5xINF attacking separate 2xINF: 80%
SA 10 vs DEF 10: 10 shots at normal chance to hit of 20%, average 2 hits.
On average, total of 4 hits from both 5xINF.
Probability of both 5xINF attacking just one 2xINF: 20%
SA 20 vs DEF 10: 10 shots at normal chance 20%, 10 at double chance 40%.
On average, total of 6 hits from both 5xINF.
So 20% chance of doing 150% damage.
On average the 5xINF divisions get 0.8 * 4 + 0.2 * 6 = 4.4 hits.
5 2xINF vs 2 5xINF:
Even if all 5 2xINF divisions attack just one 5xINF division (0.2^4 = 0.16% chance!), their combined SA of 20 doesn't exceed the 5xINF's DEF of 26, so there's no possibility of increased damage, all shots get 20% chance to hit. On average the 2xINF divisons get 20 * 0.2 = 4 hits.
So the stack of 5xINF divisions get 4.4 / 4.0 = 110% the number of hits that the stack of 2xINF divisions do. I'd say that 10% extra damage just from a bit of division reorganisation is worth it, especially for your static port garrisons. Obviously for situations where you haven't got a high density of divisions, this isn't applicable. But there are cases where this could be useful to know. Note that this 10% extra damage is only for the case of attacking, for defending the bonus will be greater, but I'll leave you to work through the maths to prove that to yourself
So, point proven? Or do you see some nasty flaw in my calculations?For example, the hypothetical 2 X 5 Inf combat versus 5 X 2 Inf. While it is true that on each round the 5 brigade division will overwhelm whatever division it targets, it is equally true that the 5 divisions will overwhelm one of the 5 brigade divisions. The luck of the draw may favor one of these over the other for any given combat round, but it will eventually even out (or at least come close). I will remain unconvinced until someone sets up a rough numerical simulation to definitively prove otherwise.