A Treatise on Combat Mathematics
By Charles Ensign

     So, you've finally found the perfect sword, scribed the final rune on your Platemail, slotted that one killer elite onto your skill bar from which it will never leave again. While you know that the numbers on your skills and equipment are large, do you actually know what they mean? Have you ever been pounding away at a rival Warrior and wondered why even your critical hits were barely scratching him, or Fireballed an Aloe and been confused when it registered for 300 damage? The reasons for these effects are not arcane secrets, but well understood mathematics - formulae that will be elucidated herein.


General Remarks

     Damage in Guild Wars is governed by two distinct sets of effects - those that affect the damage of the attack, and those that affect the defense of the target. These two sets are separable - if you just want to analyze weapon damage, for instance, you can completely ignore the effects of armor in your calculations and analysis.

     In mathematical form, this means that:

[Actualized Damage] = [Effective Damage] × [Defensive Adjustment]

     Sometimes a given attack will deal damage multiple times - such as with Conjure Flame or Stone Daggers. In these cases, treat each instance as a separate attack and add it all up in the end, exactly as you would if the attacks were done in sequence. For example, if you attacked with a bow while you had Conjure Flame up, the general equation would be:

[Actualized Damage] = ([Effective Bow Damage] × [Physical Defensive Adjustment]) + ([Conjure Flame Damage] × [Fire Defensive Adjustment])

     Let's approach the two terms in turn, first the offensive part, then the defensive.


Attacking and Damage

     The base damage of an attack depends on both your character's statistics, and on situational modifiers, such as attacking a target over a cliff. The latter are maddeningly difficult to evaluate quantitatively, and thus we'll limit ourselves to discussions of the former. As a basis for our analysis, we'll use characters ready to set foot into the Tombs as a basis - level 20 characters with top armor and maxed attributes. As we'll see later, the whole system works on a sliding scale.


Weapon Damage

     If you are attacking the target with a sword, axe, hammer, or bow, the amount of damage that you'll naturally deal depends on both the inherent damage of the weapon, and your level in the corresponding attribute. The relationship between your attribute level and damage is a simple exponential over a 0-12 attribute level range:

[Effective Damage] = [Weapon Damage] × 2((5 × [Attribute Level] - 60) / 40)

     Or, in tabular form:

Effect of Attributes on Weapon Damage

Attribute Level	Percent of Weapon Damage
0	35.6%
1	38.6%
2	42.0%
3	45.9%
4	50.0%
5	54.5%
6	59.5%
7	64.8%
8	70.7%
9	77.1%
10	84.1%
11	91.7%
12	100%
13	104%
14	107%

     As an example, if you had a normal, customized (20% more damage) 15-22 damage Dragon Sword, and a level 9 Swordsmanship attribute, each swing with the weapon would deal:

15 × 1.20 × 2(( 5 × 9 - 60 ) / 40) = 13.88 minimum damage
22 × 1.20 × 2(( 5 × 9 - 60 ) / 40) = 20.36 maximum damage

     While swinging the same sword with a level 12 attribute would deal:

15 × 1.20 × 2(( 5 × 12 - 60 ) / 40) = 18.0 minimum damage
22 × 1.20 × 2(( 5 × 12 - 60 ) / 40) = 26.4 maximum damage

     While pumping up your weapon attribute continues to give outstanding, exponential returns on damage all the way to level 12, it drops off quickly after that. Attribute level 13 is less than half as effective as the previous level at raising your damage, and attribute level 14 is even worse. So while having a level 13 or 14 attribute might look good on your stat screen, it isn't doing a whole lot in game. You'd be better served lowering that attribute and investing elsewhere.


Skill Damage

     The vast majority of the time, you aren't just swinging away mindlessly with your weapon of choice, you're using skills to augment, replace, or just outright deal damage to your opponent. As a baseline, the amount of damage that you'll deal with a skill is listed in that skill's description - rounded to the nearest integer. But as we're beginning to see, attacks rarely deal their baseline damage - indeed, in the case of skills your character's level plays a role in how much damage you'll actually deal:

[Effective Damage] = [Skill Damage] × 2((3 × [Character Level] - 60) / 40)

     Or, in tabular form:

Effect of Character Level on Skill Damage

Character Level Percent of Skill Damage 1 37.2% 2 39.2% 3 41.3% 4 43.5% 5 45.9% 6 48.3% 7 50.9% 8 53.6% 9 56.4% 10 59.5%

Character Level Percent of Skill Damage 11 62.6% 12 66% 13 69.5% 14 73.2% 15 77.1% 16 81.2% 17 85.6% 18 90.1% 19 94.9% 20 100%

     Just to run a couple quick numbers - let's say that you're a level 6 Pyromancer with a Fire Magic attribute of 5. Each Flare (22 damage at attribute 5) you cast is going o be dealing:

22 × 2(( 3 × 6 - 60 ) / 40) = 10.63 damage

     If you then gained a level but didn't pump your Fire Magic attribute, your Flares would be dealing more damage regardless, to the tune of:

22 × 2(( 3 × 7 - 60 ) / 40) = 11.19 damage

     Unlike weapon damage, you don't exactly have control over your character's level - you're going to be dealing 100% of listed skill damage once you hit level 20, and that's all there is to it. This makes it incredibly easy to min/max your skills, since the damage they're going to deal in combat is listed right there in the skill description. You'll find this effect most useful when planning an attack against populations of monsters who will quite often have levels that vary considerably from your own. For example, take a level 30 Elementalist boss casting a level 12 Deep Freeze:

76 × 2(( 3 × 30 - 60 ) / 40) = 127.8 damage

     They may just be bots, but respect their ability to dish out damage.

     There's one more category of attacks to consider - attacking an enemy with a wand or staff. Wands and staves aren't weapons in the purest sense - a more apt description is 'skill on a stick', as the damage they deal scales up with your character level, not your weapon attribute. In other respects, they act like all other weapons. Take a customized 6-10 wand, for example - when wielded by a level 10 character, each strike will deal:

6 × 1.20 × 2(( 3 × 10 - 60 ) / 40) = 4.28 minimum damage
10 × 1.20 × 2(( 3 × 10 - 60 ) / 40) = 7.14 maximum damage

     So just use the best wand that your attribute setup will support - there's nothing else you can do to influence their damage.


Critical Hits

     Every time you strike a foe with a weapon of any sort you have a chance of landing a critical hit. Each critical hit strikes for maximum damage, with the additional bonus of hitting as though your weapon attribute were four levels higher. I'd be breaking form if I didn't list an equation, so here you go:

[Effective Damage] = [Max Weapon Damage] × 2(( 5 × [Attribute Level] - 40 ) / 40)

     This translates into each critical strike dealing 141% of your maximum weapon damage, every time. Every critical hit with a customized top (6-28) axe and a level 12 Axe Mastery will deal:

28 × 1.20 × 2(( 5 × 12 - 40 ) / 40) = 47.5 damage per critical

     The frequency with which you will land a critical hit is dependent upon several factors - your character level, the level of the victim, and, if you're using a sword, axe, or hammer, the appropriate attribute level. The exact formula for landing a critical chance is unknown, and due to it being a matter of statistical analysis it is rather difficult to test. In rough terms, your chance of landing a critical hit against a foe of comparable level with a level 12 weapon mastery is roughly 20% - if you're a level 20 character beating up on level 1 targets with a level 12 weapon mastery, your critical chance is very close to 100%.


Weapon Requirements

     Many of the weapons that you'll find require you to have a certain level in a linked attribute to be effective - Swords that require 9 Swordsmanship, Bows that require 7 Marksmanship, and the like. If you do not meet the requirements on a given weapon, your effectiveness with it will be greatly reduced.

     In simplest terms, if you do not meet the requirements on a given weapon it will deal damage like a newbie weapon of the same type. Thus, if you find a 10-20 wand but don't meet the requirements, it'll deal damage like a pathetic 2-4 wand of the same type. However, the weapon will keep all modifiers. Thus, if you were using a 14-20 Sword of Enchanting (+20% enchantment durations) and didn't meet the requirements, it would only deal the 2-4 damage, but you would still get the +20% enchantment duration from the modifier. This includes internal modifiers - some weapons have built-in additions to damage, and these will remain even if you don't meet the requirements. This is most readily apparent on a focus - if you find a normal +10 focus and don't meet the requirement, you will only get +3 energy from equipping it. However, if you get a +12 focus - a +10 focus with an internal +2 modifier - you'll get 5 energy from equipping it, even if you don't meet the requirement. Thus you are always better off using a weapon with a requirement you can reach - unless the only thing you want access to is the weapon's modifiers.


Armor and Defense

     Once you know how much damage an attack is supposed to do, you can start thinking about the defense of the target, to figure out how much damage you will ×actually× do - or how much damage you will take when someone swings back. This means talking about how your armor works, as well as any defense boosting skills.


Hit Locations

     Each player can use up to five pieces of armor - a chest piece, leggings, boots, gloves, and a hat. Any given attack on a player will hit one of these five locations, and only the armor at that location is considered - all the other pieces are ignored. Additional defensive measures, such as an armor-boosting enchantment or a shield, are added to the target's defense, regardless of hit location.

     Exactly where a given attack will strike depends on the orientation of the target, and the type of attack that you are making. Skills that automatically hit the target, such as Lightning Strike, always use the target's chest piece when calculating the target's armor level. Weapon attacks and projectile spells, however, have a chance of hitting the target at any of the five hit locations. The exact percentages are dependent upon the orientation of the target - if you're attacking from above, you are more likely to get a head shot - but under most circumstances, the rough hit locations break down as follows:

Hit Location Percentages

Hit Location	Chance to Hit
Arms	12.5%
Head	12.5%
Feet	12.5%
Legs	25.0%
Torso	37.5%

     Clearly, your chest piece is your most important piece of armor, as it receives the highest percentage of hits as well as being the target of auto-hit spells. At the same time, your non-torso armor will be subject to a proportionally higher percentage of physical attacks, as many of the elemental spells that you'll encounter go right for the body. Mixing up your armor sets might not be such a bad idea after all.


Effect of Armor

     All that needs to be done now is to figure out what effect the target's armor is going to have on your damage. The relationship is a simple exponential function, the form of which you should be familiar with by now - simply add up all of the target's defenses and calculate:

[Defensive Adjustment] = 2((60 - [Armor Level]) / 40)

     Or, in tabular form:

Effect of Armor on Damage

Armor Level Defensive Adjustment 0 282.8% 5 259.4% 10 237.8% 15 218.1% 20 200% 25 183.4% 30 168.2% 35 154.2% 40 141.4% 45 129.7% 50 118.9% 55 109.1% 60 100%

Armor Level Defensive Adjustment 65 91.7% 70 84.1% 75 77.1% 80 70.7% 85 64.8% 90 59.5% 95 54.5% 100 50% 105 45.9% 110 42% 115 38.6% 120 35.4% 125 32.4%

     As you can see from the table, the target will take exactly the amount of damage they are expected to when they have an armor level of 60, and adding or subtracting 40 defense will result in them suffering half or double damage, respectively.

     So, for example, a 30 damage physical attack directed at a Warrior wearing Knight's Armor (80 AL, +20 AL vs. Physical) using a 15 AL shield, would deal:

30 × 2(( 60 - (80 + 20 + 15) ) / 40) = 11.6 damage

     While that same attack on an Elementalist in Master's Armor (50 AL + 24 AL vs. Physical) would deal:

30 × 2(( 60 - (50) ) / 40) = 35.7 damage

     Note that the Elementalist actually suffered more damage than what the attack would have naturally dealt. This is the case whenever a target has a net armor rating below 60 - sometimes you ×want× your attacks to be affected by armor, so that you can reap the benefits of their pathetic AL!


Armor Penetration

     Armor Penetration is incredibly straightforward - it simply allows you to ignore the listed percentage of the target's total defense. If a given attack has 10% armor penetration and the target has an armor level of 100, he will only have an armor rating of 90 for the purposes of defending against this attack. Thus the armor penetration equation:

[Armor Rating] = [Armor Level] × (1 - [Armor Penetration] / 100)

     To give you an idea of how this works, here's a piece of the armor effectiveness table, adjusted for various levels of Armor Penetration:

Effect of Armor Penetration

Armor Level	0% Penetration	5% Penetration	10% Penetration	25% Penetration	50% Penetration
40	141%	146%	152%	168%	200%
50	119%	124%	130%	148%	183%
60	100%	105%	111%	130%	168%
70	84.1%	89.3%	94.9%	114%	154%
80	70.7%	75.8%	81.2%	100%	141%
90	59.5%	64.3%	69.5%	87.8%	130%
100	50%	54.5%	59.5%	77.1%	119%

     So while our 30 damage attack from the previous examples dealt a paltry 11.6 damage to a well-defended Warrior, that same attack with 50% armor penetration would deal:

30 × 2(( 60 - (80 + 20 + 15) × (1 - (50 / 100)) ) / 40) = 31.3 damage

     An attack with 50% armor penetration cuts through Knight's Armor as though it were nothing more than Master's Robes. If you're having trouble taking down a target with an exceptional armor rating, attacks with armor penetration are exactly what you're looking for.


Ignoring Armor

     Some skills, such as Illusionary Weaponry, allow you to deal damage that ignores the target's armor entirely. Figuring out the damage actually dealt by a skill that ignores armor is trivial, as they work exactly as advertised - they literally do ignore the target's armor, dealing the damage listed in the skill description directly to the target. Mathematically, this means that an attack that ignores armor always considers the target's Defensive Adjustment to be 1, and the original equation reduces to:

[Actualized Damage] = [Base Damage]

     So if a skill's description says it will deal 40 armor ignoring damage, it'll deal 40 damage every time. Not only does ignoring armor help immensely when attacking someone with a lot of armor - it makes your calculations easier as well.


Consolidation

     So far we've been looking at different combat effects individually. While it makes each aspect of combat easier to understand, you lose sight of the so-called 'big picture' and how all of these aspects are intertwined. So, now that we understand how each effect works, it's time to put them all together into a single, generalized equation that shows what's ultimately going on.

     Way back at the beginning of this article I mentioned that all of the numbers, equations, and charts were normalized for level 20 characters. While this was necessary to present the data in a reasonable manner, it's important to note that the choice of basis is arbitrary. Indeed, you might have picked up already what happens when a character with a level 0 attribute attacks a target with 0 armor:

2(( 5 × 0 - 60 ) / 40) × 2(( 60 - 0 ) / 40) = 100% of listed damage

     The result is exactly the same as what would happen if your level 20 character with a weapon attribute of 12 attacked an opponent with 60 defense! This underlies the truth of the matter - the damage equations we've been talking about so far are just different aspects of a single, self-normalizing equation. With no further ado:

[Actualized Damage] = [Base Damage] × 2(( [Strike Level] - [Armor Level] ) / 40)

     Where, for weapons:

[Strike Level] = 5 × [Attribute Level]

     For skills:

[Strike Level] = 3 × [Character Level]

     Consequently, characters will actually be dealing base damage to enemies as long as their Strike Level is close to their Armor Level - which, if they're following the normal progression of the game, will be pretty much all of the time. Game mechanics don't revolve around characters at level 20, but at whatever level you happen to be at the time!

     Damage affecting skills and situations work by manipulating one of these three values. Weapon customization increases your Base Damage by 20%. A critical hit increases your Strike Level by 20, while 50% armor penetration reduces their Armor Level by 50%. Attacks that ignore armor forget about Strike and Armor Levels entirely and just deal base damage. Every damage effect in the game can be represented in a similar fashion.

     To run a couple examples that put everything together - let's say that you're a level 15 Elementalist butchering packs of level 7, 21 defense Charr with 69 damage, level 8 Fireballs. Each one you cast is going to deal:

69 × 2((3 × 15 - 21 ) / 40) = 104.6 damage

     You're a level 1 Warrior, fresh out of Ascalon city with a level 2 Hammer Mastery and a 5-10 Hammer that a friend gave you. After customizing the hammer, you peek outside the gates and engage the level 2, 6 defense Flash Gargoyle standing there. Your first swing scores a critical hit, dealing:

10 × 1.20 × 2((5 × 2 + 20 - 6 ) / 40) = 18.2 damage

     Finally - you're a level 20 Ranger hunting level 28, 118 defense Rift Wardens in the Tombs. You have a customized, 15-28 Half Moon Bow, and a level 12 attribute to go with it. Knowing that your target has a disgusting amount of armor, you take a shot with Penetrating Attack, dealing:

15 × 1.20 × 2(( 5 × 12 - (118 / 2) ) / 40) = 18.3 minimum damage
28 × 1.20 × 2(( 5 × 12 - (118 / 2) ) / 40) = 34.2 maximum damage

     That's all there is to it. You now have the tools needed to figure out the damage from any attack in the game, and choices can be made based upon actual effects experienced in the game world, not just numbers shown on your character screen.

Peace,
-Charles Ensign
ensign@pyen.com


     This article is a living document and will continue to be updated as information becomes available. All tables and formulae discussed herein are based upon data gathered and verified during the monthly Beta World Events. Questions, comments, and criticisms are welcome.