You can tell that, with the right defense, an increase of 1.5 AR, can actually lead to such things as three times the damage. Considering how I don't really feel like breaking down and analyzing it any more then they have already down. (At least at this moment) I'll just copy and paste what they said in a spoiler for those who can't get the website to load. Ok?
- How damage calculation works.:
- As I wrote in my previous review blog (here: http://www.teamliquid.net/blogs/viewblog.php?id=395908), Dark Souls is an amazing game. But there is one thing I think is lacking in many of the Dark Souls resources, including the wiki pages, which is a more in-depth statistical analysis of the game itself. I intend to discuss a number of combat stats, suggestions for optimization, and give a few pieces of gear to look for based on this discussion. Originally, I was planning to do a lot more in this blog, but I wrote so much about this topic that I thought it would feel rushed to tack everything else on. Consider this blog one of a small handful (provided there is any interest of course).
Damage, Attack and Defense
This is likely the best place to start, as it’s difficult to make judgments about weapons and armor having different attack and defense values if we don’t know what these stats actually do for damage taken and damage dealt, and therefore, we don’t know what these stats do to keep us alive and to make our enemies dead faster.
The way damage works in Dark Souls is confusing. I’ve often found myself confused when switching between a weapon like the Silver Knight Sword (AP 262 at +5) and Great Club (AP 337 at +15). Why is it that one appears to do 400-500 damage and the other only about 150?
The first thing to consider is the difference between personal AP and weapon AP. All weapons come with a certain amount of AP initially, and then (with the exception of certain enchants such as lightning) additional AP which scales with a certain stat. In our example above, the Great Club gets a ton of AP from strength (about 8 per point under 40, about 1 thereafter), whereas the Silver Knight Sword has a much smaller coefficient with dex.
The second thing to consider is that Attack power differences don’t translate to damage on a set ratio. This is probably a good time to state the best approximation I’ve found for the damage calculation in dark souls:
When Atk < Def, Damage = 0.4*(Atk^3/ Def^2) - 0.09*(Atk^2/ Def)+0.1*Atk
When Atk >= Def, Damage = Atk - 0.79* Def*e^(-0.27* Def/Atk)
(Note: when attack = defense, Damage ~ 0.4* Atk in both equations, so this damage function is continuous.)
You can see from the outset that if you suddenly get 100 more AP, that’s certainly going to translate to more damage, but the amount more damage it translates into varies greatly based on the amount of attack and defense values both combatants have.
Let’s take these equations a little further (feel free to skip this section, as this gets into pretty in depth math). First, let’s take a look at partial derivatives.
dDamage/dAttack = 1.2*(Atk^2/ Def^2) - 0.18 * (Atk/ Def) + 0.1 when Attack < Defense,
dDamage/dAttack = 1-0.21*e^(-0.27* Def/Atk)*( Def^2/Atk^2) when Attack >= Defense.
It’s worthwhile to note that these functions are not continuous, meaning that at the point Attack = Defense, you go from having an increasing function with 1 AP scaling to more than 1 damage, and continuing to increase to a function which (while still being increasing) is increasing to a function with a horizontal asymptote at 1, instead of continuing to grow higher and higher. It’s also worthwhile to note what happens at very low values and very high values of AP. At very low values, we get a very small value of damage per AP – while at very high values, we get (as we mentioned a minute ago, due to the fact that the derivative has a horizontal asymptote at 1) approximately 1 damage per point of additional AP.
dDamage/dDefense = 0.09 (Atk^2/Def^2)-0.8*Attack^3/Def^3 when Attack < Defense,
dDamage/dDefense= -0.79*e^(-0.27* Defense/AttacK)+0.79*0.27*(Def/Atk)*e^(-0.27* Defense/Attack) when Attack >= Defense.
First of all, purely for comedic value, let’s take a moment to note that for extremely high values of defense, or extremely low values of attack, increasing defense actually increases the amount of damage we take (at 400 attack and 4,000 defense, for instance, we take 38 damage, while at 400,000 defense, we take 40 damage)! This is because our partial derivative is positive when Attack < Defense *0.09/0.8. Also note, slightly more meaningfully, note that at very low values of Defense, our partial derivative goes to -0.79 damage per point of defense, while at higher values, it tends towards 0. Also, notice that as we pass the point where Attack = Defense (due to the point of discontinuity), we jump back up to -0.71 damage per point of defense (from 0.44). Also notice that the defense side of the equation has a lesser impact on damage than the attack side of the equation (this is evident in the damage formula as well).
But we’re not quite done yet. Consider that when a person deals 20 damage per swing, 20 additional damage is a lot – but when a person deals 1000 damage per swing, not so much. Therefore, let’s consider what happens when we consider the change in damage not as a value – but as a percent (i.e. (dDamage/dAttack)/Damage). This is where we start to see diminishing returns from both sides of the equation for high values of AP/Defense, and larger returns for smaller values (which explains why smaller returns early in the game which allow you to damage black knights – although it’s less damage per point of AP – has a larger impact on the fight than it does after you’re already dealing 100-200 damage per swing or more). Note that when we say “high values” of AP/Defense - in dark souls, we mean to say higher values than will generally be attainable.
Conclusions (Note: if you skimmed over math, this is where to pick it up again.)
We can determine that higher attack power is even better than we would think it would be – because not only does the higher value translate to higher damage, it translates better than linearly, meaning that (for instance) there are occasions (at 400 defense, for instance) where 750 AP will literally do three times more damage than 400 AP. That doesn’t seem very intuitive, does it?
This has a couple immediate ramifications. Before growing accustomed to Dark Souls damage mechanics, one might think that faster weapons are way, way better than slower ones – because the lower recovery time gives you more time to avoid damage coming your way (even in circumstances where the difference is only fairly small – e.g. the leaping attack of a greataxe vs. the normal attack of a 1-handed sword), but this analysis above tells us that to compensate for the lessened ability to avoid damage (and to compensate for the small windows in which slow weapons don’t get to attack) – you deal a ton more of it.
Perhaps the most applicable thing we learn here is what it tells us about weapons like “Fire” and “Lightning”. A fire +10 Great Club, for instance, lists itself as 756 AP. This might be tempting for those who are currently wielding a +15 Great Club that only has <700 AP. But then something weird happens… you take it out and it deals less damage. A lot less – wtf? What’s going on here? With the math above, we can make sense of this, because fire/lightning weapons actually make 2 attacks, one physical and one elemental – each at half the listed AP. So remember what I said about a 750 AP attack dealing three times the damage of a 400 AP attack? It stands to reason, then that there are situations (350 fire/phys defense) where a Fire Great Club + 10 (attacking twice at 378 AP each time, dealing 326 damage in total) would actually deal less damage than a Great Club +15 with only 600 AP (which would hit 350 defense for 364 damage).
Also, spells which add AP to attacks fall under a similar category. If you’ve ever read up on Greater Magic Weapon, you might be really excited (wow, nearly 200 AP!) until you realize that you only actually get a fraction of that in damage (since a 200 AP attack doesn’t deal more than 80 damage against anything with 200 defense or more).
Also, this tells us that scaling stats may be more important than we thought they were. Many people are under the assumption that stats like Strength and Dex are stats you only raise to unlock the ability to use more gear, and to an extent that’s true (in fact, for much of the endgame gear, the discussion is moot because you need a ton of stats to use the gear anyway), but for some gear (like the great club/large club – yes, I like clubs, ok?), it’s worth noting that going from 28-40 strength increases your damage output by 25-30%. So if it’s worth 28 strength to use a weapon which does x damage, shouldn’t it be worth 40 strength to use a weapon that does 1.3x damage (sure, 30% is only against armored enemies, but even against lightly armored ones, it’s 20% or more)? At least, when 1 point of vitality is only worth about 2-3% health, I’d think 1 point of Str being worth 2-3% damage would be pretty good. In theory, this wouldn’t mean that we should stack AP up too high (because as we noted, the increase of damage goes up as our AP is higher, but the increase of damage as a percentage of the damage we’re already dealing goes down) – except that for all but those with the lowest defenses, even stacking STR all the way to 40(/amount required to wield weapon) leaves us competitive with increases in vitality (and increases in vitality tend to be of lumpier significance – meaning that in many fights vitality ends up being insignificant because you die in 3 hits with or without the increased health, for instance, but once it is siginificant it’s the difference between life and death, hence “lumpy” – because there is usually overkill damage during deaths).
Defense, on the other hand, is a stat which seems to be reasonably worthwhile at all but the highest levels (incoming attack *1.5ish or more). For example, against an enemy with 400 AP, our defense is worth 0.2% damage reduction per point or more (multiplicatively, not additively, so 2 points would be worth 1-.98*.98 = .396% or more) up until 640 defense, and never goes higher than about 0.4% - so its value is pretty much the same going from 100 to 200 as it is going from 200 to 300 – which is to say (in a highly damaging game like dark souls) quite valuable.
Obviously, this would be situational, but based on our analysis it seems like the weapon of choice would be the slowest, best-scaling one you can get away with. For Dex builds, this means something like Server, Scythe, or Polearms. For str builds – you’d want Greatclubs, Greataxes, or Greatswords (Demon’s Greataxe or Large Club are probably the best candidates). And Int and Faith builds using melee weapons might consider adding more STR to pick up one of the slower weapons, too. Remember - going from 300-450 Magic/Divine/Fire AP practically doubles your damage, even against the least armored opponent. Though, of course, the section about the Lightning and Fire weapons should probably have warned you –it’s going to be difficult for a Divine/Fire/Magic Weapon to compete with a Normal +15 weapon for damage output.
Now, of course, if you don’t actually have a window in which to use your slow attack – it’s a bit hard for the slow attack to be better. But in many fights where you can have a few moments to collect yourself and attack, the slower attack is going to deal a lot more damage – and if you get good enough and knowing and using a slow weapon’s quicker attacks (the leaping attack, generally), it really isn’t that much slower. Speaking as someone who uses a great club a lot, it’s definitely harder to play. But you definitely also do a lot more damage when you can get away with it.
But at the very least, avoid looking at the number of AP it gives you on your stat screen as if that is a relevant number. Higher AP on the stats screen doesn’t always mean higher damage. If you take nothing else away from this blog, that should be the minimum.
Personally the most surprising part of this is
"First of all, purely for comedic value, let’s take a moment to note that for extremely high values of defense, or extremely low values of attack, increasing defense actually increases the amount of damage we take (at 400 attack and 4,000 defense, for instance, we take 38 damage, while at 400,000 defense, we take 40 damage)!"
Who would ever expect that?
Ohhh, btw, for any member who feels like taking this and editing the information to make it work in the wiki, gets a free cookie... and a scooby snack.
Ohh, I just found a second part... here it is. http://www.teamliquid.net/blogs/viewblog.php?topic_id=398745