We're not talking what works here, we're talking about what's best. Of course a 10/0/10 can obliterate an under-tech 20/0/0. But really, a 10/0/0 can beat a 20/0/0 with a disc/tactics lead, you don't need either. What you need to show is that adding 10 arty to the back row will create a better outcome than adding 10 inf to the front row. I don't see how this works out.
Yes, exactly. Is 10/0/10 better or worse than 20/0/0? Or to put it differently, are the benefits of the artillery in the 10/0/10 army are greater than the benefit of the extra width of a 20/0/0 army (assuming your opponent is 20 wide).
I don't think it's possible to completely run the number on this, since some parts of the combat math aren't fully documented (e.g. how exactly ART defends the unit in front of it, how units in the battle but not actually participating in combat get morale damage). But as a rough example:
At tech 20 INF is Fire[3/2], Shock[3/2], Morale[3/2], and ART is F[3/2], S[1/1], M[3/2]. So for fire and morale, compared to a single INF regiment, an INF+ART combo has an extra 50% defense and 50%+ more offence (50% when the units are both at full strength, more if the INF has taken casulties); for shock, the bonus is ~ 25% on defense and 15% on offence. So, very back of the envelop, you can say a 10/0/10 army is 50% stronger on both attack and defense vs. a 10/0/0 army. (At the beginning of a battle when both are at full strength it's a little less; as both front lines take casualties the advantage become bigger because the ART keeps attacking at full strength and by limiting damage to the INF).
Another rough way to think about it is that the ART is effectively adding 50% of its pips to the INF unit in front of it (ART attacks with full pips but 50% strength, and adds 1/2 of its pips on defense). At tech level 20, that's 6 pips. So you go from 15 pip INF to 21 pips. Actual 21 pip INF is only available at tech level 28. So a 10/0/10 tech 20 army is roughly comparable to a 10/0/0 tech 28 army. (The 10/0/10 army should actually be better, as the art won't take casulties in combat, but it's in the same general ballpark).
So, a tech 20 "Narrow" 10/0/10 army is about 50% stronger across its width compared to a same tech "Wide" 20/0/0 army. At lower techs it's a little weaker, at higher techs a little stronger (since ART pips are on a steeper slope than INF). But at the same time, a narrow army is open to flanking attacks on the outside of its line and will have to be able to defeat enemy regiments in sequence as new regiments move in from the flanks to fill holes in the center of the line. The question then is, is that extra strength in the center from a full back line in the "Narrow" army more valuable than the extra width in the "Wide" amry?
Let's think about how each would perform in a battle vs. a 20/0/0 army. In the case of a Wide vs. Wide battle, it's clear the two sides are equally matched, assuming same tech, modifiers, etc.. In the case of a Narrow vs. Wide batttle it's more complex. Think of the battle having 2 phases:
In the beginning of a battle between the two, the wide army has 2 units able to attack from a flanking position. These do 100% damage while taking 0 casulties. Again very back of the envelop, let's say that's roughly the same as a 1 on 1 matchup with +100% offence and +50% defence. So we can think of this phase of the battle as a 10/0/0 vs. 10/0/10 battle, where the narrow army regiments all have +50% on offence and defence and the outer 2 wide regiments have +100% offence and +50% defence. In this phase, the narrow army clearly has a big advatnage over the wide one--not only does it have an advantage in the central 8 of the unit-vs-unit matchups, but the advantage it has there is bigger than the advantage the wide army gets in the outer 2 matchups.
In the second phase of the battle, after the central units of the wide army are destroyed, replacements come in from the flanks, and you again have a 10/0/0 vs. 10/0/10 battle. Now the key is the relative strength of the individual regiments. All the previously flanking regiments in the wide army will be at full strength, although they will have taken some morale damage. The INF in the narrow army, on the other hand, will have taken casualties and morale damage. If these regiments are down to ~ 50% strength, then the battle is roughly equal--the narrow INF will be attacking with 500 men, but the full strength ART behind them will be attacking at 50% (i.e. as if it had 500 men), which combines to the same as a full strength INF regiment for fire/morale damage; shock damage will be lower than a full strength INF, since ART is worse than INF in this phase, but they will still have +50% defense). Wherever the narrow army's INF regiments are below half strength, then the wide army now is at an advantage; wherever they are above half strength, the narrow army has an advantage.
So, in theory, a narrow army could be better than a wide one, provided the pips/modifier/etc. are such that a INF/ART combo kills/breaks the morale of an enemy INF regiments before your own troops take enough casualties to bring them down to half strength and without losing, say, 50% of your morale. In practice, I'm not sure you what happens, but I suspect that at some tech levels this is the case.
Multipliers like disc/combat ability would push the balance further in favor of narrow, since they stack with the effective bonus from the ART+INF combo. E.g. going back to tech 20 example, where an INF+ART combo is roughly the same as a 1500 man INF regiment on offense, +20% discipline/CA is effectively a +80% bonus to attack (1.5*1.2) for a narrow army, vs. +20% for the wide one.
Anyway, I haven't tested this at all, but it seems to me that there should be some situations where it is better to build a 10/0/10 army than a 20/0/0 one.
