Some thoughts on unit sizes

[Aesculapius]

Perhaps I haven't made myself clear.

I certainly agree that stronger units will take fewer casualties and do faster damage than weak ones. But total numbers still have a bearing: for example, a Warband unit (n=61, attack=7, defence=10) will probably beat a unit of Arcani (n=16, attack=12, defence=16), assuming neither routs.

But consider, in the case of the Gauls, a Warband unit (n=61, attack=7, defence=10) and a unit of Chosen Swordsmen (n=41, attack=13, defence=17).

A Warband with no chevrons has:
: an attack strength of 61 * 7 = 427
: a defence strength of 61 * 10 = 610

Chosen Swordsmen with no chevrons have:
: an attack strength of 41 * 13 = 533
: a defence strength of 41 * 17 = 697

So with no chevrons, the Chosen Swordsmen look like the stronger unit (disregarding other factors like charge bonus, cost, morale etc.). Now consider the case with three chevrons (e.g. from Large Temple of Toutatis) and an armourer:

An armoured Warband with three chevrons has:
: an attack strength of 61 * 12 = 732
: a defence strength of 61 * 15 = 915

and an armoured unit of Chosen Swordsmen with three chevrons has:
: an attack strength of 41 * 18 = 738
: a defence strength of 41* 22 = 902

So with these bonuses, multiplied by the larger number of men in the unit, the Warband has become about as strong as the Chosen Swordsmen. And the effect will be even more marked compared to smaller units such as cavalry.

In a nutshell: bonuses are bigger overall when applied to larger units. This tends to cancel out their intrinsic weaknesses, and make them more competitive with smaller, 'stronger' units that have the same bonuses.

[Deus ret.]

Of course: As units grow in strength due to bonuses (chevrons, upgrades), the relative difference between weaker and stronger units is diminished. Thus, 3-gold chevron units are overall far more equipollent than newly built ones.

There is a restriction to your calculation, though. If you take whole units and determine their overall attack strength, even if taking into account equal morale/general etc you were right if the whole unit would fight at once. Since they very seldomly do so, overall "weaker" units with better individual stats stand a far better chance of deciding a combat in their favour. An extreme example would be a battle on the wall where the match-up arcani vs. warband might well result in a victory of the former.

[Puzz3D]

In a nutshell: bonuses are bigger overall when applied to larger units. This tends to cancel out their intrinsic weaknesses, and make them more competitive with smaller, 'stronger' units that have the same bonuses.

That isn't how combat works. The relative difference in combat power between two units is related to the difference in combat points, i.e. (attack - defense). If you add the same number of combat points to both sides, the difference doesn't change. However, increasing chevrons does raise morale which will help a larger, lower morale unit more than it does a small high morale unit.

[Productivity]

You're assuming that it's an absolute difference that matters, and not a relative difference, or do you have evidence that it isn't?

ie.

Absolute difference = (defence(unitB) + (exp)) - (attack(unitA) + (exp)).

In which case, the experience does fall out. ie. Compare defence(unitB) = 10, attack(unitA) = 15, at exp 0,4 and 8.

(10+0)-(15+0) = -5
(10+4)-(15+4) = -5
(10+4)-(15+4) = -5

However if there is a scaling factor, then experience might matter.

Relative difference = ((defence(unitB) + exp) - (attack(unitA) + exp))/(defence(unitB) + exp)

Same numbers

((10+0)-(15+0))/10 = -0.5
((10+4)-(15+4))/14 = -0.3574
((10+4)-(15+4))/18 = -0.287778

Now this could be shown theoretically - higher stat units, with the same absolute difference should take longer to kill if the final number is relative, rather than absolute.

ie. same equation as above, try pairs of 10/5,15/10,20/15,25/20, gives differences of -1,-0.5,-.333,-0.25.


I might do a few trial runs to see if this is true - certainly it would make more sense to me, although they would have to do somethign to get around the divide by zero issue - maybe that is why the minium defenseive value is zero, with them getting around it by making the denominator (defence(unitB) + 1) or something.

[Puzz3D]

You're assuming that it's an absolute difference that matters, and not a relative difference, or do you have evidence that it isn't?

Yes. The designer gave us the combat formula for STW and MTW, and it's similar for RTW eventhough they won't reveal the exact formula.

For STW and MTW:
chance to kill = 1.9% * 1.2 ** (df)
df = (attack - defend + bonus)

attack, defend and bonus are all integers to simplify computation.
attack is the striker's attack value.
defend is the target's defend value.
bonus is any situational combat modifier that applies.
df is limited to a range -x < df < x
x is 20 for STW and MTW, but most likely a larger number for RTW since the 1.2 constant is closer to 1.1 in RTW according to my tests.
lethality in RTW is a new parameter that most likely comes in as an additional multiplier.

they would have to do somethign to get around the divide by zero issue - maybe that is why the minium defenseive value is zero

Is the minimum defensive value zero? Both attack and defend values could be negative in STW and MTW, but I haven't tested that in RTW.

[Productivity]

Hmm - that seems strange, so high quality units can hack each other to bits just as easily as low quality units can hack each other to bits - strange.

Edit, sorry is that chance to kill actually chance to take off a hit point? Or is it an absolute kill value?

Is the minimum defensive value zero? Both attack and defend values could be negative in STW and MTW, but I haven't tested that in RTW.

Well I'm pulling it out of the hard coded limits thread. It shouldn't be too difficult to check - that's actually what triggered me to think that it might be scaled somewhat, the change from allowing negative values to not allowing them - somewhere you're going to get a divide by zero, and even if you can avoid that, you're going to get strange results due to the negative sign swapping things around.

[Puzz3D]

Hmm - that seems strange, so high quality units can hack each other to bits just as easily as low quality units can hack each other to bits.

Yes because a high quality blow against a high quality defense would have the same effectiveness as a low quality blow against a low quality defense.

Edit, sorry is that chance to kill actually chance to take off a hit point? Or is it an absolute kill value?

It's the chance to take off hitpoints. A different parameter determines how many hitpoints are removed by a successful strike. In the previous games, only some projectile weapons could remove more than one hitpoint at a time, and it may still be that way in RTW. I'm not sure.

Well I'm pulling it out of the hard coded limits thread. It shouldn't be too difficult to check - that's actually what triggered me to think that it might be scaled somewhat, the change from allowing negative values to not allowing them - somewhere you're going to get a divide by zero, and even if you can avoid that, you're going to get strange results due to the negative sign swapping things around.

No one checked negative values of attack and defend in that thread, but it doesn't matter. The difference (df) can still be negative, and there is no discontinuity in that formula I gave at df = zero. The exponential 1.2**(0) = 1. A -20 < df < +20 range maps onto a chance to kill range of 0.05% to 73%. That's right out of the MTW Official Strategy Guide, and that chapter of the guide was written by LongJohn who designed the battle engine. I think the range of df in RTW is more like -40 < df < +40 with a base constant of 1.1 so each combat point gives a 10% change in the chance to kill.

[Aesculapius]

That isn't how combat works. The relative difference in combat power between two units is related to the difference in combat points, i.e. (attack - defense). If you add the same number of combat points to both sides, the difference doesn't change.

That's true, but it misses the point. I'm NOT adding the same number of combat points to both sides - I'm pitting whatever unit I train plus bonuses against whatever the AI throws at me. My point still stands that my bonuses count for more when applied to larger units.

Thus, 3-gold chevron units are overall far more equipollent than newly built ones.

There is a restriction to your calculation, though. If you take whole units and determine their overall attack strength, even if taking into account equal morale/general etc you were right if the whole unit would fight at once. Since they very seldomly do so, overall "weaker" units with better individual stats stand a far better chance of deciding a combat in their favour. An extreme example would be a battle on the wall where the match-up arcani vs. warband might well result in a victory of the former.

Lovely word, that - equipollent........ Your point about the amount of 'battlefront' a unit presents is well taken. I guess one should try to get as many men in the unit engaged as fast as possible, by setting a 'loose' formation or a long thin line to envelop the enemy (as long as there's no incoming cavalry charge.........). However, even in a restricted engagement, a larger unit also translates into a greater duration of withstanding attrition. So I'm not sure whether your example of an Arcani vs Warband battle on a war holds true - anyone feel like doing some testing?

[Watchman]

A larger unit has more in the way of "attrition resistance" owing to the sheer numbers of buggers having to be killed, true, but the thing is, unless you've turned morale off or something those mounting casualties will tend to have a nasty tendency to cause routs... all the more so as the low-tech "mass" unit types also tend to have rather low base morale.

Besides, as long as the stronger-but-smaller unit maintains formation for most intents and purposes only the members in contact with their opposite numbers will actually fight; and the members of the elite unit will tend to win those match-ups. Sure, there's more guys coming up, but it doesn't really matter if they too just keep losing those match-ups and getting slaughtered until the whole bunch routs. It's not quite as bad in RTW as it was with the "Jedi Generals" of MTW, but the principle is there.

Trying to wear out strong cohesive units by swamping them with hordes of disposable weenies does work to a degree, but especially given that you can only deploy a limited number of units to the field that's probably not such an optimal technique if you have alternatives. Didn't work all that well historically either most of the time - hand-to-hand combat just plain is an endeavour where the quality of arms and their bearers tends to have ludicrously disproportionate effects. The TW series does a fairly decent job representing that, too.

[CBR]

In a nutshell: bonuses are bigger overall when applied to larger units. This tends to cancel out their intrinsic weaknesses, and make them more competitive with smaller, 'stronger' units that have the same bonuses.

As the RTW formula is most likely based on the same system as STW/MTW it is an exponential system and you therefore cant just multiply the stat with number of men to get an overall "combat value"

In your example you gave the two units the same type of upgrade which means the overall increase in combat power is the same for both units: in this case of 10 points increase that could mean around 160% stronger than without upgrades.

Now lots of experience upgrades could mean a low quality units doesnt run as easy and therefore it will actually be even better than what the actual increase in attack/defense suggest but that would be it really.

If you still want to try and get an overall combat power based on both stats and number of men then you would have to take into account the exponential increase from the attack/defense values. We just dont know the precise numbers but if we use what some of us are guessing then it would be the following.

(Im just combining attack and defense values)

Warband: 1.1^17 x 61 = 308
Swordsmen 1.1^30 x 41 = 715

With upgrades:

Warband: 1.1^27 x 61 = 799
Swordsmen: 1.1^40 x 41 = 1855

These numbers are much closer to what a units combat power is. But that is not everything.

Now if we just look at attack/defense to find the difference in kill rate (still assuming our formula is correct) the 13 point difference would be 1.1^13 = 3.45 so overall two chosen swordsmen will die for every seven spearmen.

I just did a few quick tests with each unit having 9 valor so they wouldnt run (the warband still ran a few times but it was after heavy losses) and I got anywhere between 3 and 4 times higher losses for the warband so at least it looks pretty much what we can expect with the limited knowledge of how combat works.

So what about that total combat power for units I just did? Well as I wanted a units overall combat power I multiplied effective individual combat power by number of of men but as you rarely get all men to fight at once and larger units normally just stand in deeper ranks thats still not an accurate way.

The ratio between them is (715/308 and 1855/799) 2.32/1. And I cant really use that number to much as combat results shows the killratio actually is higher.

Of course I would have to run a lot more battles to get a more accurate average of kills. But as I do know the deeper ranks will mean the larger unit wont get its full potiential and that a losing unit will end up having lots of men being doubled, it does seem that going by the attack/defense stats alone is close enough to get a picture of a units overall combat power (assuming the size difference is not too big but large units having 50% more men doesnt seem to do that much)


CBR

[Aesculapius]

CBR - much thanks for that very convincing work, both arithmetical and playtesting. Tollo galeam!