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tamral
03-31-2008, 01:23 PM
The Mechanics Approach to Tanking:

So you’ve read all the tanking guides out there, plugged some numbers around, and you’ve come to the discovery that well, nothing seems definitive. Unlike other classes, no one seems to have the ideal gear set, many great tanks seem to have different opinions on things, and worst of all, it actually seems impossible to effectively min/max the way some DPS classes can.

Welcome to the world of being a protection warrior.

This guide is going to be a bit different. There is no basic introduction, and this guide, unlike others, is not meant to cater to the typical person who just wants the quick answer. What we will be doing, however, is exploring, mathematically and structurally, what it takes to be a good tank, and precisely why the min-maxing approach does not work for us all of the time, and how the min-maxing approach must be done on a boss by boss basis, or a gearset by gearset basis, not on an overall one.

For those of you with a sound fundamental understanding of math, my goal is to help you improve yourself and think about the APPROACH to tanking, as opposed to simply hitting buttons and stacking certain stats.

As a prerequisite to this guide, please read Quigon’s guide which is up on Elitist Jerks. From a standpoint of all the basic formulae and the basic approach to tanking, it is without a doubt the most comprehensive and best written guide you can read.

This guide will be broken up into 4 segments:

I. Raid utility and approach
II. Mitigation and Threat Analysis
III. Protection Warrior DPS
IV. Advanced Theorycrafting Mechanics, a boss by boss approach

I. Raid Utility and Approach

One of the primary reasons tanks are not performing at optimal levels is because they do not fully understand their purpose in every fight. They are each their own entity, and synergizing a raid for tanks is not a priority. Just because you are a tank doesn’t mean you need to be in the same tanking gear for every fight, nor that you will even be tanking on every fight.

Example 1: For pure mitigation, the 8/5/48 build is generally regarded as having the strongest overall mitigation with a viable solo tanking build. 5/5/51 has higher general mitigation, but the lack of imp thunderclap does it in unless someone else can keep it up for them. 11/5/45, 12/5/44, or 12/4/45 can generate higher levels of threat, but usually at the expense of a mitigation talent or three. But most guilds with 3 protection warriors have 3 tanks with variations of the above builds.

While it may seem that having the above builds makes for strong tanking, the fact is, the above builds have no synergy whatsoever. On a fight that requires 3 tanks, 3 warriors with the above builds will never perform as well as having 3 warriors, one with 11/5/45, 12/5/44, or 12/4/45, one with 17/3/41, and one with 8/10/43. Why is that you may ask? The first tank has a build designed for strong threat generation, the second tank can viably DPS, and the third one can ensure that thunderclap and improved demo shout are always up on the target. More mitigation, more threat, and more damage all rolled into one little package of tanking synergy.

Example 2: I see lots of tanks who end up being off tanks on a fight like Mother, who don’t switch any gear to do it. That has always confused me. Personally, off tanking Shahraz, in addition to being a 5 minute break, is also a prime opportunity to throw on all of your avoidance gear enjoy the show. While I typically tank with about 28 dodge, 20 parry, and 520ish defense, I have a set of gear, for that fight only, where I can get my raidbuffed dodge over 35%, my parry over 21, and my defense close to 550, with a stopwatch to pop every 2 minutes. The difference in the amount of damage I take (and need to be healed) vs the other tank is often astounding, averaging in most cases a 1.7 to 1 ratio.

Would I use the same set to tank Gorefiend? Of course not, but that’s the point. It is the very reason why any guide that gives a hard rank or score for tanking gear is by definition false. It automatically assumes a certain role, or that the objective is singular and modal.

The objective of this section is to help you think about, from a role perspective, what specifically you will need to do, on a boss by boss basis, to optimize performance in your specific role. I do not score gear, I do not allow myself to employ tunnel vision in my approach, and in fact, I freely admit that often at times, I believe that the approach to tanking is just as important or more important as the button mashing itself.

Let’s take an example approach, and as the example, I will use a complicated example, Gurtogg Bloodboil, and go through my own approach. Later on, in section 4, it will be explored in more detail mathematically, but simply understanding that the approach is based on the encounter is often more than enough to get an idea of what needs to be done.

Gurtogg Bloodboil:

1. Mitigation:

Q. How hard does the boss hit?
It depends on the number of debuffs present on the tank. For low numbers of debuffs, melee hits for about 4k, crushes for 6-7k, at 10 debuffs, melee hits for 6k, crushes for 9-10k. The debuff can be avoided.

Q. What additional damage is taken?
Every debuff causes a loss of 500 armor and causes 250 nature damage every 2 seconds for 1 minute. The boss has a frontal cleave every 8-10 seconds, hits for normal boss damage, and the cleave CAN refresh the debuff. The boss targets someone in the raid randomly every 15 seconds for a nature breath, doing around 7k damage to everyone in the cone, cannot be avoided, can be resisted.

2. Threat

Q. What are the threat concerns, and is it a burn fight?
It is not a burn fight, however, threat is a concern for the fight, as the boss has a knockback, and will also periodically disorient the tank. The knockback reduces threat by 25%, and the disorient uses normal tanking mechanics (will take a new target, will not switch back unless another target in melee range is 10% over the new target in threat OR a ranged target is 30% over). 2 or 3 tanks need to be able to stay at the top of the threat chart for the entire fight. This requires time spent off tanking as still being capable of building threat, increasing the value of expertise, hit rating and possibly SBV for that purpose. With 8 debuffs, rage is plentiful and can sustain any cycle.

Q. What is the parry risk for the fight?
A. Low, the boss hits are not that hard.

Fight mechanics:

Q. What are the benefits and weaknesses of avoidance in the fight?
In addition to the basic benefits, avoidance will reduce the rate of incoming debuffs. However, the knockback can also be avoided, and standing in too long will result in too many debuffs to be healed through. At 15 or more debuffs, it is impossible for a single healer to effectively heal through the debuff without a significant mana drain. From an off tanking perspective, high avoidance is neither needed or desired.

Q. What can go wrong in the fight, and is any of it under my control?
Four major things can go wrong in this fight.
First, the bloodboils are not properly absorbed, and people start dying from them. There is nothing a tank can do to prevent this.
Second, too many consecutive knockbacks can be avoided, resulting in a mounting stack of debuffs on the current tank target. For this, only a bubble can help, waiting inside the bubble until the next tank has passed you in threat.
Third, the bottom tank who still has a lot of active debuffs takes aggro because of a combination of knockback/disorient on tanks 1 and 2. Once the disorient has worn off, a bubble can help this situation as well, however, the debuffs may have been refreshed in the meantime.
Fourth, the fel rage target can die. If that happens, the current tank target must use some form of cooldown to survive the incoming damage, which will be double the normal damage, and with BB swinging up to 40% faster. Shield wall is generally the best form of survival for this.

Q: What is my specific role in the fight?
When I am main tanking, my job is to build threat until I have a set number of debuffs, say 5 or 6, and then back down and change to survival mode, letting the other tanks catch up on threat, while waiting to get punted (not too high passive avoidance, but enough to prevent quick debuff stacking), hopefully before I have more than 10-12 debuffs. Once punted, while the debuffs are active, I have to build threat quickly to get back to third overall on threat (therefore, threat building stats, like expertise and hit, are critical), but careful not to pass the second tank until my debuffs are going away. Once the debuffs are off of me, I stand on top of the tank to eat the cleaves for some additional rage to keep my threat high. (not too much passive avoidance)

If I am the target of aggro when the fel rage occurs, I stand on the tanking spot like a dumbass the entire time. If I am NOT the aggro target, I stand behind him and keep up TC to counteract the boss’s self haste buff, and keep up demo shout to help with mitigation. If I have a number of debuffs on me (and therefore, essentially an infinite supply of rage, I can also LOLdps during this time. If I have no debuffs, I also use this time to melee him and build rage to use at the start of the next normal phase. Either way, with 3-4 seconds left in the phase, I return to my normal phase position.

If I am the target of aggro and the fel rage tank dies (LOL mages and priests), I am to first use shield wall. If it happens a second time ( *()_*)_(ing healers ), I need a better out than just last stand, preferably something that provides a lot of help with mitigation (meaning Moroes Stopwatch is the extra oh shit button I need).

So what does all of this mean?

Based on my role ON THIS FIGHT, the basic strategy of stacking stam and armor doesn’t really apply. In fact, EH (effective health) doesn’t really play ANY PART in determination of what gear is optimal for the encounter.

I need to be able to build threat quickly in this fight while standing behind him and maintain threat production while standing on top of the tank. While parry-gibbing is a low risk, it is still a risk. Therefore, I want lots of expertise and hit, with maybe some SBV thrown in for good measure.

High passive avoidance is a bad idea, as is too low passive avoidance. Therefore, about 55% passive avoidance is plenty for the fight. (25 dodge, 20 parry, 530 defense). However, there is a clear and present need for extra oh shit buttons. Therefore, trinket wise, Moroes stopwatch can provide that help, as can the Scarab of Displacement from Hydross, or even the commendation of Kael’Thas from the new heroic Magisters Terrace.

So in the end, I need to gear for high threat with extra panic buttons, meaning, I will sacrifice some stamina and armor for more threat building stat aligned gear.

Compare this, for example, to a fight like Azgalor, where survivability is the number 1 issue, and you will see clear differences in up to 4 or 5 pieces of gear (as I would stack more passive avoidance, and potentially 2 panic button trinkets for a bad silence in a rain of fire (I would use the 10 rage I save every 5 seconds from not hitting shield block to assist with making up the threat difference).

Tanks need to ask themselves what stats are important for their particular role in a fight. The answers to these questions becomes obvious in say, resistance fights, but sometimes, the answer is not so obvious, and the pitfall is to think that the same set of gear is equally optimal for everything. As a result, while you may be optimized for some fights, on other fights, you may find yourself not performing very effectively. To provide an analogy: It is entire possible to eat soup with a fork, but you’re going to do a much better job with a spoon.

But the bottom line is: There is no single gear set that is optimal for every fight. You may be able to get away with using it, but you are not doing the raid any favors by doing so.

This brings me to the next topic. How many tanks blindly gem everything with +15 stam? The answer is most. Once again, they are doing this based on either instinct, or reading some scoring report for tanking which scores everything based on a very arbitrary role of main tanking a tank and spank fight. It is important to socket based on what you intend to use the piece for, and what the socket bonus is.

If the piece of gear is part of your avoidance set, chances are you may want to see what the socket bonus is. If the socket bonus provides some avoidance, why not take advantage of it, or at least see if it can? If it is a piece for a burn threat set, by general rule, unless the socket bonus is something ridiculous (+hit, + expertise), chances are that 15 stam gems are the way to go.

Rather than score pieces of gear, ask yourself what role it could help you perform, and judge how much of an upgrade it is IN THAT PERSPECTIVE only. There are several pieces of gear in the game that don’t seem all that great, but in fact, are ideal pieces for a specific type of role.

II. Threat and Mitigation Analysis

The first thing to understanding threat and mitigation is to understand what benefits each stat provides:

I’m specifically not covering the basics (490 def, 102.4% with SB active) because this is not meant to be a beginners guide, and I assume you have them anyway.

For itemization points, here is the equivalencies:
1 Str = 1 Agi = 3.9 SBV = 1 dodge rating = 1 parry rating = 1 defense rating = 1 expertise rating = 1.5 Sta = 1 hit rating = 12 armor = 1 crit rating

Strength: Each point of strength provides 2.42 attack power raidbuffed (vitality and kings). 14 attack power translates into 1 DPS. Every point of strength (on a 1.6 speed weapon), provides an increase of effectively .25 damage on melee hits, and .13 damage on devastates, meaning an increase of roughly .18 DPS at 20% parry (an average of 4.14 melee swings per 6 seconds and 2 devastates, where the extra .34 melee attacks reflect the benefit to YOU from your parries, also assuming 24% of white hits glance, and 10% of attacks are misses). This provides an effective threat bonus of .27 TPS. Strength provides 1 SBV for every 20 points of strength, a value that is so low, its not even worth calculating (.0046 TPS)

Agility: Every point of agility provides 2 armor, .03 crit, and .03 dodge. So from a tanking perspective, this means that every point of Agi provides an increase in threat generated by: [(1 + current crit rate + .03)*(Current TPS)/ (1 + current crit rate)] – (Current TPS). Assuming 12% crit and 1000 TPS, it provides about .26 TPS. However, the extra dodge has a negative effect on threat which we will go into in more detail in the more advanced Theorycrafting section (it turns out to be significant as well). While also providing .03% more avoidance.

Stamina: Every point of stamina provides 11.5 health for non Taurens and 12 health for Taurens.

Hit Rating: Every point of hit rating increases the chance to hit a target by .0633%. While the static marginal rate of return for threat/hit rating is a decreasing one, as the first few points of hit rating actually provide more relative percentage benefit to TPS than the last few, the rate of return is constant due to the increases in scale. The effective hit cap for bosses is 9% (142 hit rating), or 8% (126 hit rating) if a melee Dranei is in the group.

Basic talents provide 6 expertise, meaning at 0 hit rating, the hit chart looks as follows:

Parry: 9.1%
Dodge: 4.1%
Miss: 9%
Glancing: 24%
Hit: 53.8%

Glancing only affects white melee swings, and for the purpose of the delta in TPS, will be ignored. Also, due to the nature of expertise (it is the first stat to cap under any and all circumstances, being the new best tanking stat in the game), I am going to redo the hit chart to reflect that.

Parry: 5.1%
Dodge: .1%
Miss: 9%
Hit/Glancing: 85.8%.

The first point in hit rating provides a TPS increase of:
[(85.863 * 1000)/(85.85)] – 1000 = .73 TPS

The last point in hit rating provides an increase of:
[94.8 *1104]/[94.737] – 1104 = .73 TPS

Where 1104 is the new TPS generated by being one point under the cap, and the reflecting the overall effect of stacking the previous hit rating.

Expertise Rating:
3.924 expertise rating reduce the chances that an attack is dodged or parried by .25%
From the same math as used in calculating hit rating threat, 3.924 expertise provides:

Parry: 5.1%
Dodge: .1%
Miss: 9%
Hit/Glancing: 85.8%.

[86.30*1000]/[85.8] – 1000 = 5.9 TPS, meaning 1 expertise provides about 1.5 TPS

Expertise also provides passive avoidance of the best variety in that it reduces burst damage incoming. It is important to know that once every 60 seconds, regardless of the situation, you will be subject to a crush simply because one swing every 2.4 = 25 swings per minute, while your shield block covers only 24. One 5 second gap per minute is naturally uncovered.

Assuming Weapon Speed = 1.6:
Parry Effect Chart:
12.8% no effect
25.6% chance swing time reduced by .4 seconds
61.6% chance swing time reduced by .7 seconds

Just multiplying that out, each parry results in the boss getting a swing time reduction of about .55 seconds.

Assuming your typical 6 second cycle: (4.14 melee/HS, 2 devastate, 1 revenge, 1 SS), you attack 8.14 times. Over the course of 1 minute, that is 81.4 attacks. At 6 expertise, 7.3 of those attacks will be parried, resulting in an average of 4 extra incoming attacks per minute. At 22 expertise, that number is reduced to 4.1 extra attacks, and 2.3 extra incoming swings per minute. This is positively critical to avoiding burst damage.

Shield Block Value:
1 itemization point, or 3.9 SBV provides 3.9 additional blocked damage per block (for EH calculation) and also provides additional threat in the form (on a fully sundered boss) of: (3.9*.7) = 2.73 extra damage per shield slam. The additional threat provided by this is based on your current expertise, hit and crit rating. At 0 hit, 22 expertise, and 12% crit, it provides a TPS increase of:

[2.73 * 1.12 * .858 * 1.52]/6 = .66 TPS

At 142 hit and 22 expertise:

[2.73 * 1.12 * .948 * 1.52]/6 = .73 TPS

Therefore, SBV actually scales to your hit and expertise. The more hit and expertise you have, the more threat benefit you gain from SBV. Once you get near the hit cap, SBV becomes relatively interchangeable with hit rating, and for effective health purposes, SBV becomes superior.

Dodge Rating:

Dodge Rating provides approximately .051 avoidance per rating.

Parry Rating:

Parry Rating provides approximately .041 avoidance per rating. Furthermore, parries speed up your attacks and generate rage.

Defense Rating:

Each 2.4 points of defense rating provides 0.12 avoidance in the form of .04 dodge, .04 parry, and .04 miss. Furthermore, it also adds .04 to your chance to block as well.

Crit Rating:

Each point in crit rating provides .04% chance to crit, or roughly .35 TPS

Analysis:

It is important to note that the above threat calculations are based on having applicable raidbuffs (kings, might, sharpening stone, battle shout), and is actually based off of 983.9, not 1000, which is the value of TPS maintained at 6 expertise, 0 hit rating, 12% crit, 1300 AP, with 2.6 HS, 1.54 melee, 2 devastates, 1 SS, and 1 revenge every 6 seconds. The lack of taking extra swings due to your own parries into account is the reason all current TPS spreadsheets are in essence, undercalculating threat, and underestimating the importance of expertise, since they calculate a fewer number of attacks.

If a windfury totem is active, or other buffs are active, most of these values scale multiplicatively. So for example, with a windfury totem active, the base 1000 TPS is really 1100 TPS, so 1 hit rating actually gives .8 TPS instead of .73 TPS, etc..

In terms of burst mitigation and expertise, On average, absent expertise rating beyond that provided by talents, the boss will get one extra swing at the player every 29 seconds. That extra swing is guaranteed to provide one extra attack over that 5 second period. The likelihood that this attack will crush is 15% of the chance that the other two attacks were successful hits, or

(1 – Avoidance)^2 * .15 = .054 at the base level. About 1.84 times every 30 seconds, you’re running a 5% chance of potential catastrophe against a harder hitting boss by stacking nothing but stam (hit, hit, crush in a 5 second gap). Yes, this can be healed through and survived MOST of the time, and you may survive, but congratulations, everyone else’s life is harder because you’re stacking your gear poorly.

At 60% avoidance and 22 expertise, the one extra swing in time is every 64 seconds, and you’re only subject to .61 swings per 30 seconds that fall outside your coverage gap.
(1 - .6)^2 * .15 = .024. So now, instead of a 5.4% chance of catastrophic damage 2.34 times per 30 seconds, it’s a 2.4% chance 1.08 times per 30. it may not look like much, but try asking the guy underwriting life insurance how different those two sets of numbers are. How it affects odds of success over a 5-7 minute fight is absolutely astronomical.

Where tanks fail is in assuming that blindly stacking stamina and armor does the job. It doesn’t. In fact, with expertise now being a factor, stacking just stam, armor, and SBV NEVER does.

The objective is not to avoid damage, or be pure avoidance, or have the big health pool and high SBV, the objective is to avoid burst damage, and avoid spike back to back hit/crush damage.

III. LOLProt DPS

Coming Soon to a theatre near you. (still under construction), but its not a myth.

IV. Theorycrafting FTL

The section is titled for a statement at an old tank class lead I had that who thought theorycrafting was for noobs. Yes, it’s a potshot and I don’t care, because I think theorycrafting has legitimate value.

Initial Value Assumptions:

500 defense = Df0
15% dodge = Dg0
15% parry = Pa0
25% block = BL0
400 block value = BV0
6 expertise = EX0
0 hit rating = HT0
12% crit = CT0
200 Strength = ST0
150 AGI = AG0
550 AP = AP0
15000 Health = H0
17000 Armor = Armor0

Other values:
BP = boss parry rate
BD = boss dodge rate
BB = Boss Block rate
MI = Static Miss Rate of incoming attacks
MO = Miss rate of your attacks

Marginal Values:
Value of increasing each value by 1
dDFrating = +.016Dg0 + .016Pa0 + .016BL0 + .016MI
dDgrating = +.051Dg0
dParating = +.042Pa0
dBL0rating = .128BL0
dExrating = - int(Exrating/3.924)*BP - int(Exrating/3.924)*BD - int(Exrating/3.924)*BD
dST = +2 AP + (ST/20)BV
dAG = .033Dg + .033CT
DSta = + 10.5 Health
DST = + 1.1 ST

Again, using the basis of the following for comparison

1 Itemization point = 1.5 stam = 1 agi = 1 str = 1 crit rating = 1 dodge rating = 1 hit rating = 1 parry rating = 1 block rating = 3.9 block value = 1 expertise rating = 1 defense rating = 12 Armor

Without pages of math, I concluded that the full complement of all BT epics (all level 141/146 items) provides a grand total of 3600 itemization points, including gems. Considering the very base of these numbers consumes roughly 2300 of the itemization points (a good 900 of them in stam alone, 300+ in defense rating, +150 or so across other stats)

Therefore:
67*DSta + DAg + DST + DCT + DDg + DHT + DPa + DBL + .26*DBV + DEx + DDf + .08*Armor = 1100
Subject To
All >= 0
All <= 550 (Simply limiting you from stacking any one stat past this point (its impossible to do anyway, but it provides a force cap to give better relative values)

Remember, using the initial value assumptions, that’s our baseline 0 stats. The wiggle room we have is with where we allocate the other 1400. And more importantly, we can’t put all 1400 in one place either, because there is no gearset that allows it. What instead we are trying to do here is assign relative weighting to each stat, so we can mold our gearsets around getting those primary numbers.

The above subject equation provides us the basis for what stats we can get and how much “purchasing” one of those stats will cost us in itemization points. Remember, if you are wearing lower level pieces (like many people, including myself), do, in order to maximize a certain stat (in my personal case, expertise), you DO effectively lower the total number of itemization points that you have.

The basic subject equation is now based on need for 2 other calculations. First, is the need for high threat generation in the fight. How important is threat, and how much of a burn fight is it? For threat generation purposes, expertise, hit and SBV are critical. Even though SBV catches up (and will eventually pass) hit rating in terms of raw TPS generated, hit rating provides a more steady rate of return, whereas SBV is a hit or miss proposition, where part of the overall threat gen is determined from spike threat generated by a critical hit.

From a mitigation standpoint, expertise, stamina, defense, dodge, and parry become more important. For a very bursty fight, or a fight where short term burst is an issue, you must also consider the need for items that have: “oh shit help me god” on use abilities, which can provide needed short term buffers.

To illustrate this, consider the state equation matrix that is tanking:

21000 raid buffed health
Assuming HPS

State 1 Initial: 21000 health
State 2 = {A0 min (State 1 health +2.4 HPS, 21000)
1- A0 min (State 1 health +2.4 HPS, 21000) - Boss Damage + BV
}
State 3(A0) {A0 min (State 1 health +4.8 HPS, 20000)
1- A0 min (State 1 health +4.8 HPS, 21000) - Boss Damage + BV
}
State 3(BL) {A0 min (State 1 health +4.8 HPS, 21000) – Boss Damage + BV
1- A0 min (State 1 health +4.8 HPS, 21000) – 2 * Boss Damage + 2 *BV
}
State 4 (no 3 swings)
State 4 (chance of cycle having 3 swings)
Subset 1:
(A0 * A0) ) {A0 min (State 1 health +4.8 HPS, 20000)
1- A0 min (State 1 health +4.8 HPS, 21000) – Boss Damage + BV
}
Subset 2:
2* (A0 * (1 –A0)
) {A0 min (State 1 health +4.8 HPS, 21000) – Boss Damage + BV
1- A0 min (State 1 health +4.8 HPS, 21000) * Boss Damage + 2 *BV
}
Subset 3
(1 – A0)^2
{A0 min (State 1 health +4.8 HPS, 21000) - 2 * Boss Damage + 2 *BV)
BL min (State 1 health +4.8 HPS, 21000) – 3 * Boss Damage + 3 * BV
Max(0, 102.4 – A0 – BL - 15) min (State 1 health +4.8 HPS, 21000) - 3 * Boss Damage + 2 *BV
Max(0, 102.4 – A0 – BL) min (State 1 health +4.8 HPS, 21000) - 3.5 * Boss Damage + 2 * BV + 4.1 HPS
}
If health = 0 then, end

Boss Damage in this state is the mitigated damage taken based on armor, I simply didn’t write it in because I didn’t feel like typing it out and clustering it with something that is already obvious if you’ve been following along so far anyway.

Your chance of the cycle of having 3 swings is simple. Divide the base number of 3 swing cycles per 30 by 6. That is the chance of any given cycle putting you into one of the subsets of State 4. Subset 3 is basically known as the catastrophic state, because that is the state at which you are in danger of taking more damage per that 5 seconds than you are normally getting healed for, meaning either
a) you just died, congratulations
or
b) you are going back into state 1 with significantly less than full health, so either catch up time/Loch/etc. is needed before you are relatively safe again.

Assuming you are receiving in the ballpark of 4500 HPS (2.5 direct healers)
Plug in the numbers. For a boss hitting for 8k per hit, 500 block value, and plug into the 4x4 matrix, and knock yourself out. 20,000 health with 22 expertise, 60% avoidance is SAFER (less likely to reach health = 0) than 25,000 health, 6 expertise, 45% avoidance. In fact, at 6 expertise and 45% avoidance, you would need almost 27,800 health to get to that same level of relative safety, due to the sheer number of times spent over 7 minutes in subset 3 of State 4.

Take the above matrix and subject it to the following:

Min P(health = 0, (21000 + sta * 1.15, 39.2% + .033Agi + .051Dg + .042Pa + .047Df , 25% + .128BL)
Max: 1000 + .21Agi + .27 Str + 1.5 Ex + .73 HT + .17BL - .31Dg - .12Pa - .24Df +.35CT - .00003 Armor - .005 BL
Subject to:
+.67 DSta + DAg + DST + DCT + DDg + DHT + DPa + DBL + .26DBV + DEx + DDf + .08Armor = 1100
All >= 0
EX <= 43
Hit <= 142
All <= 550

What this shows is that mitigation tanking has positively nothing to do with how hard the boss hits when he connects, or how deep the health pool is past a certain amount, but rather, how much time are you going to spend in the catastrophic state (subset 3 of state 4) over a 5-6 minute fight, how many times will you be in that subset, and how often will you be in that subset in 2 consecutive 5 second cycles?

The reality is that tanking is nothing but 5 second cycles over and over again, with chances each time that something bad will happen. Limiting that chance is far more important than having an extra 1k armor and 1k health. People can talk all they want about health and armor and EH being static amounts, but avoidance/expertise just playing with chance, but the math shows otherwise. It is precisely reduction of that chance that helps to ensure success, while the total health pool and armor are actually absolutely meaningless unless they are ridiculously and unacheivably high numbers. This is not to say stamina and armor are bad, but rather than expertise dominates them even from a mitigation standpoint.

Actually doing the math will bring you to a borderline comical conclusion in terms of pure mitigation. The leather belt from Vashj provides almost as much mitigation (due to helping reduce spike damage), as the Girdle of Stability from BB. The difference being the damage reduction from the Vashj belt is all spike and burst damage, while the Girdle of Stability provides a constant reduction.

The next step on the chart is to provide dominance reduction:

Expertise dominates hit, crit, strength, SBV, and agility for Threat generation, AND provides the highest reduction in P (health = 0). Therefore, Expertise caps first at all times. This reduces the equation to:

Min P(health = 0, (21000 + sta * 1.15, 39.2% + .033Agi + .051Dg + .042Pa + .047Df , 25% + .128BL)
Max: 1000 + .21Agi + .27 Str + .73 HT + .17BL - .31Dg - .12Pa - .24Df +.35CT - .00003 Armor - .005 BL
Subject to:
+.67 DSta + DAg + DST + DCT + DDg + DHT + DPa + DBL + .26DBV + DDf + .08Armor = 955
All >= 0
Hit <= 142
All <= 550

Expertise = 43 (145 rating)

Now we need to provide reduction for dominated stats. From the threat side, it is clear that strength and crit are completely dominated by SBV and Hit. Therefore, 0 them out.
Furthermore, we find that armor is dominated by every other stat. 0 that one out too.

Min P(health = 0, (21000 + sta * 1.15, 39.2% + .033Agi + .051Dg + .042Pa + .047Df , 25% + .128BL)
Max: 1321 + .21Agi + .73 HT + .17BL - .31Dg - .12Pa - .24Df - .005 BL
Subject to:
+.67 DSta + DAg + DDg + DHT + DPa + DBL + .26DBV + DDf = 955
All >= 0
Hit <= 142
All <= 550

Expertise = 43 (145 rating)

What we find next is that due to scaling density and a sort of economy of scale, hit rating caps next. This conclusion is jumped to for the following reasons:

Avoidance for this calculation is soft capped at 70% passive, at which point, around 400 points would remain for stamina, hit rating, and SBV). Hit takes precedence over SBV initially, and stamina provides little initial return since it provides very small marginal rate of return in P(H=0). Furthermore, SBV also dominates stam for reasons of scaling, up to a certain point, as it provides 26% return in effective health as well as threat scaling up to around 80 itemization points.

Min P(health = 0, (21000 + sta * 1.15, 39.2% + .033Agi + .051Dg + .042Pa + .047Df , 25% + .128BL)
Max: 1334.6 + .21Agi + .17BL - .31Dg - .12Pa - .24Df - .005 BL
Subject to:
+.67 DSta + DAg + DDg + DHT + DPa + DBL + .26DBV + DDf = 733
All >= 0
All <= 550

Hit = 142
Expertise = 43 (145 rating)
BV = 80

What does this mean?
Basically, in a nutshell, that under just about any and all circumstances, you will want 142 hit rating, 145 expertise rating, and 430 SBV for any kind of bossfight. Most of the remainder of the stat point allocation will go towards avoidance stats and more SBV, depending on the desirability of threat or avoidance for the fight. For a mathematical answer, just find the determinant of the 6x6 matrix with the top 4 rows being the state equations and delta probabilities of P(H=0) by stat, and the bottom 2 rows being the maximization statement variables, and the bottom row being the subject equation variables. Multiply that by the X by 1 matrix of (Probability of State 1, 2, 3, 4, 0, 733), (using the determinant is simpler than calculating forward by plugging in boss damage).

When you get your resultant 6 variable equation, you can either try to solve it mathematically (which will be a pain in the ass), or you can solve it by testing endpoints (test out each variable at 733) to give you an idea of relative benefit.

veneretio
03-31-2008, 03:42 PM
I see you decided to leave out the bad mouthing of fortifications in the version you posted here.

That being said now that the Expertise values that this is based off are attainable this is actual useful information. Before all this theorycrafting made absolutely no sense when all of the mathematics promoted a way of gearing yourself that couldn't possibly be attained since there wasn't 43 Expertise Skill worth of gear.

It has been very clear for a while that Expertise represents a potential change in the "how to gear yourself" paradigm that may trump that of pure Effective Health. However, your assumption that acquiring maximum hit rating and a particular level of block value is greatly flawed under the circumstance that if your raid cannot dps hard enough to pass your threat then sacrificing hit rating in favour of armor, stamina and avoidance is the superior route.

Ciderhelm
03-31-2008, 04:21 PM
Appreciate the post. The way you think out boss fights is logical and it's good advice.

Where tanks fail is in assuming that blindly stacking stamina and armor does the job. It doesn’t. In fact, with expertise now being a factor, stacking just stam, armor, and SBV NEVER does.This is a two-parter.

First, what are you responding to? Since Expertise was released it has not been common to find tanks who think it isn't critical.

Second, absolutes work in math, but you suffer from trying to use them everywhere else. When you say something doesn't do the job, then say it "never" does the job, you're casually ignoring the people who it did the job for. Nihilum, for instance. Normally I'm not a progression whore here, but you're 9 months behind in content, saying that what people did 9 months ago absolutely doesn't cut it. We can lament how little gearing choices actually matter in TBC, but the fact is people who play and think in a way you state to be impossible have done spectacularly well for themselves. ;)

Your math is good. The only place it needs serious work that I saw was in determining Shield Block Value's effect on Effective Health. Your Hit and Shield Block section is outlandish, given that people have to make choices between real pieces of gear, not just a pool of itemization points.

I'm sure some of the readers are getting a glazed over look as they check your conclusions. What they may not realize is that the concepts you're presenting are pretty solid and simple, you just chose not to write or organize effectively. Be a little more coherent and use a little less unfounded and useless anger and you'll probably have a very successful guide.

Naka
03-31-2008, 05:22 PM
Very interesting and helpful, thanks. To be honest, I don't undertand all of your math, but one question: You're obviously assuming a flat 10.6&#37; parry rate and a 2.4 swing timer. Well, we know these don't apply to all bosses, so how would different values here effect the result?

Definetely keep it up.

tamral
03-31-2008, 09:11 PM
Appreciate the post. The way you think out boss fights is logical and it's good advice.

This is a two-parter.

First, what are you responding to? Since Expertise was released it has not been common to find tanks who think it isn't critical.

Second, absolutes work in math, but you suffer from trying to use them everywhere else. When you say something doesn't do the job, then say it "never" does the job, you're casually ignoring the people who it did the job for. Nihilum, for instance. Normally I'm not a progression whore here, but you're 9 months behind in content, saying that what people did 9 months ago absolutely doesn't cut it. We can lament how little gearing choices actually matter in TBC, but the fact is people who play and think in a way you state to be impossible have done spectacularly well for themselves. ;)

Your math is good. The only place it needs serious work that I saw was in determining Shield Block Value's effect on Effective Health. Your Hit and Shield Block section is outlandish, given that people have to make choices between real pieces of gear, not just a pool of itemization points.

I'm sure some of the readers are getting a glazed over look as they check your conclusions. What they may not realize is that the concepts you're presenting are pretty solid and simple, you just chose not to write or organize effectively. Be a little more coherent and use a little less unfounded and useless anger and you'll probably have a very successful guide.

I didnt say that doing it the EH way didn't cut it, but rather that doing it with that methodology is never optimal, big difference. As for finding tanks that think expertise is a waste of time may not be common on these boards, where most people do regard it highly, but in general, for your typical whoever, it is quite different.

As for having 43 expertise, correct, not easy at present time, but certainly possible with the new trinket in heroic MT.

tamral
03-31-2008, 09:13 PM
I see you decided to leave out the bad mouthing of fortifications in the version you posted here.

That being said now that the Expertise values that this is based off are attainable this is actual useful information. Before all this theorycrafting made absolutely no sense when all of the mathematics promoted a way of gearing yourself that couldn't possibly be attained since there wasn't 43 Expertise Skill worth of gear.

It has been very clear for a while that Expertise represents a potential change in the "how to gear yourself" paradigm that may trump that of pure Effective Health. However, your assumption that acquiring maximum hit rating and a particular level of block value is greatly flawed under the circumstance that if your raid cannot dps hard enough to pass your threat then sacrificing hit rating in favour of armor, stamina and avoidance is the superior route.

The bad mouthing of fortifications was primary from some pent up anger at people still mouthing it off ad verbatim as the way to go. It is going to be updated soon, and hopefully will have more tangeant and applicable BC solutions.

Ciderhelm
03-31-2008, 09:21 PM
Moved this over to Theory & Articles. More sensible here.