Calculating mitigation

To begin with, I understand that this has probably been beat to death already, but sifting through all the threads is a poor way to spend an afternoon or two.

From my understanding of the damage that is mitigated by a tank, it is a three tiered system.

First, you remove the defended damage.

Secondly, your shielded damage is calculated from what is left over, modified by your absorb.
Whatever shielded damage that is not absorbed is mitigated by armor.

And finally, whatever is not defended or shielded is mitigated by armor.

So, for the interest of displaying just how much I DO NOT understand about this, I will give some BS numbers.
50% DR
14% defense
50% shield
50% absorb

14% of damage is taken from the top
Of the 86% that is left over, your shield will mitigate 43%. Your shield's mitigation is only 21.5%, so your armor mitigates an additinal 10.75%, leaving your shieldy bit at 32.25%.
Now, 43% of incoming damage is mitigated solely by your armor, for a whopping 21.5% of the total.

What I come up with is 67.75% total mitigation.
Am I wrong in my calculations?

P.S.
This is a completely theoretical exercize assuming some bloke is going to hit you with a baseball bat a million times, for exactly 1000 weapon damage each hit, to give you a baseline sample.

You misunderstand the defense rating. Your armor gives you damage reduction, not the defense. An incoming hit can be avoided (chance defined by your defense rating) or reduced by the shield (amount definded by absorb, shielding chance by shield rating). Whatever goes through will be migated my your armor (unless it's a damage type which bypasses the armor.

So in your example in 14% of the incoming hits you completely avoid the damage. 50% will hit the shield an be reduced by 50% absorb and the rest again halfed by your armor (overall 75% reduce on shielded hits). All other hits (36%) will hitnyou, won't get shielded and only reduced by your armor for 50% ( which is the only completely unrealistic value in your example).

Ok its a dual roll system.
First roll is Accuracy vs Defence.
Roll 100 + Accuracy - Defence. If <=0 : Attack misses.

Second Roll is Shield cs Crit*.
Roll 100. If >= 100-Crit : Crit. If <= Shield : Shield.

Armour is applied last.

Assuming Accuracy is 0 you can get:
Expected Average Mitigation ~= 100 - ((100 - Defence) * ((1 - (Shield/100) * (Absorb/100))) * ((100-Armor)/100))

So using your numbers of:
50% DR
14% defense
50% shield
50% absorb

You will take 32.25% of the base damage. Leaving you're number if 67.75% mitigated on average. It is important to note however that on any given attack you could take 50% of base damage, 0 (parry) or 37.5% of the damage. Also Elemental/Internal damage is mitigated differently and ignores Defence/Shield.

50% DR is completely reasonable (and possbily low) for a Guardian or Vanguard tank. Not so much for a Shadow.

*It is important to note that Crit trumps shield. If the attacker has 100% crit you can never shield.

What would be the equation to compute the average damage taken given the incoming damage?

I assume we get the defensive roll first, then the shield roll.

if we take d as the defense chance (between 0 and 1) then the first roll would give 0 damage d*100% of the time.

the remaining 1-d time you take either the full amount of damage (call it dmg) or (1-ab)*dmg sh*100% of the time we dont get the defense roll (where ab is the absorb amount and sh the shield amount, again between 0 and 1).

Does this mean that our average damage would look something like:

dmg*(1-d)*(1-sh+sh*(1-ab))

?

For 1000=dmg, sh=0.6, ab=0.5, d=0.25 we get 525, which seems like a higher mitigation than most reported

Warning: Incoming wall-o-text, with maybe a little math in the middle.

The two-roll system has already been covered, so I'll just focus on the main bits of the question: how to calculate overall mitigation.

The first thing you need to realize is that there are two different attack types and two different damage types, any combination of which is valid (though not necessarily realized in the game). The attack type determines whether or not your shield and defense come into play, while the damage type determines whether or not your armor comes into play. As I implied, one of the attack/damage combinations doesn't exist, so there are really only three that we need to be concerned about:

• weapon/kinetic
• tech/kinetic
• tech/elemental

Weapon attacks are shieldable and defendable, kinetic attacks are mitigated by armor. There *are* other mitigation stats to consider, but we are rarely able to affect them except through talents and set bonuses, so they're usually ignored. All in all, the full set of defensive stats are as follows:

• Endurance
• Defense %
• Shield %
• Absorb %
• Damage Reduction %
• Resistance %
• Resistance DR % (oddly, the game uses the same word for these two things)

Resistance % is basically defense % for tech attacks, and it's usually very low (and not affected by defense rating). Resistance DR % is effectively armor for elemental damage. It is also usually quite low, though the exact value varies by class (e.g. shadows get 23%, while vanguards get 19%).

Computing your total mitigation requires taking all of these stats into account and then weighting the result by the percentage of damage that falls into each attack/damage type. Those percentages (as determined by my combat logs and a lot of math) are as follows:

• weapon/kinetic: 76.35%
• tech/kinetic: 15.04%
• tech/elemental: 8.61%

In computing each individual mitigation coefficient, we can ignore boss accuracy and crit chance, not because they're irrelevant (see: two-roll system), but because their effect is quite small and it makes the numbers simpler. Each mitigation coefficient is computed as follows:

• weapon/kinetic: 1 - (1 - (DR% / 100))(1 - (defense% / 100))(1 - ((shield% / 100)(absorb% / 100)))
• tech/kinetic: 1 - (1 - (DR% / 100))(1 - (resistance% / 100))
• tech/elemental: 1 - (1 - (resistanceDR% / 100))(1 - (resistance% / 100))

Thus, the full, weighted formula for mitigation, considering all attack/damage types and forms of mitigation, is as follows:

0.7635 * [1 - (1 - (DR% / 100))(1 - (defense% / 100))(1 - ((shield% / 100)(absorb% / 100)))] + 0.1504 * [1 - (1 - (DR% / 100))(1 - (resistance% / 100))] + 0.0861 * [1 - (1 - (resistanceDR% / 100))(1 - (resistance% / 100))]

Time for concrete numbers. My shadow tank, in full augmented Black Hole gear (in Rakata shells) has the following defensive stats:

• Endurance = 22958
• Defense % = 27.81%
• Shield % = 65.19%
• Absorb % = 57.54%
• Damage Reduction % = 40.25%
• Resistance % = 2%
• Resistance DR % = 23%

My weighted mitigation is exactly 64.1020%. (note: this includes my shield proc relic)

However, there are other things to take into account for shadow and guardian tanks: specifically, self-heals. Shadows get a constant self-heal from Combat Technique, as well as a proc'd self-heal from Telekinetic Throw. Guardians get a proc'd self-heal (effectively) from Blade Barrier. The self-heal on my shadow tank is ideally 145 HPS (about 135 HPS in practice). When I include my heal proc relic, it goes up to 165 HPS.

Back-converting self-heal values into DR percentages that can be added to mitigation is…non-trivial. Basically, you need the post-mitigation DPS of every individual boss, from which you can compute the average HPS / DPS, which is effectively the value of your self-heal as a DR percentage. This can be added to your weighted mitigation to determine your weighted survivability.

On my shadow, the average HPS / DPS value of my self-heal is 16.5717%, meaning my weighted survivability is 80.6737%. This is to say that any healer trying to keep me alive during a fight with x raw, pre-mitigation DPS would need to have an average HPS of the following: x(1 - 0.806737). For the record, pre-1.3, I was looking at a weighted survivability of 90.5489%. Nerfs: justified!

Note that none of these numbers consider cooldowns, which are too situational to include in a smoothed statistical model.

I hope this answers your question!

Thanks Ninja and Grall for your replies.

I was trying to figure out a way to estimate the best returns on an augment stat budjet. Being kinda stupid, I wanted to do it myself, but wanted to make sure the math was correct, in it's simplest form.
I am nowhere near heavy DR, and complete accuracy on my augmentations was not necessary. What it came dowwn to was, that while, as Ninja has said before, absorb was a better place to spend my augment budjet for mitigation, the increased shield chance proccs more rocket punches/stockstrikes. I needed a rough way to look at it and decide if that .05% mitigation was worth that extra .05% rocket punch. I ended up simply splitting it.

So, once again, thanks.

Oh, and Ninja, I copied your formula to Excel. I'll figure out how to get it to work about two days after I need it.

Procrastination is a way of life.

For the purposes of deciding among defense, shield, and absorb augments, you can ignore most of KN's wall-o-text, and just look at this number:
(1-D%)*[1-(S%*A%)]
Where D%, S%, and A% are the on-character-sheet reported Defense%, Shield%, and Absorb%. Smaller is better!

The full expression that KN gives is more useful if you want to estimate how much benefit you're getting from D/S/A, if you're considering additional Endurance or possibly offensive stats like Crit/Surge.