Why We Do What We Do (An in depth explanation of EHP and ICC 3.3.3 tanking mechanics)

[Aggathon]

Why We Do What We Do:It's finally finished. This atrocity of theory crafting to try and help other people understand the "why" behind stacking EHP. A HUGE thanks to all those that helped me with this, especially Thegreatme, PatrikL, and enricobr


Introduction:
The reason for the gigantic wall of text to follow is to try and get more of the “why” floating around the community. Many people do things such as stacking stam without knowing why they are doing it, other think stacking stam is short sighted. Some think that threat is a huge concern, others contend that it is just in your spec. There are a lot of mantras thrown around that are simply said because people have heard the mantra, but sometimes they never really looked into the WHY.


Another reason why I want to put the WHY out there is to try and give the people reading Tankspot.com a better understanding of game mechanics so that they can understand it as well, not just follow it, and subsequently make their own intelligent gear decisions and advice, thusly making them better tanks. A lot of times people will do something just because they make a HALP! Thread and someone says, “don’t worry avoidance trinkets just use the black heart,” and so they do, but they may not have a full understanding of WHY.


This isn’t to say that people that think opposite me aren’t intelligent or are completely wrong, but rather that maybe they haven’t been exposed to the range of theory craft, or haven’t been in certain situations, or just didn’t know!


A side note when reading this: I would really prefer that if it's a specific contention that you read the entire thread because perhaps the question has already be answered/responded to or send me a PM first. If you have good evidence then I may add your concerns to the thread, but there have been a lot of people that come forth with merely assumptions and don't really have any proof to back up what they say, especially in terms of the EHP vs. Avoidance debate. If you do have data to support your claims than I will be more than willing to listen.

Section 1: Introduction to Effective Health
I’ll start out with the most controversial section first, since it’s also probably the most likely part to be read. There are quite a few tanks out there that will fight tooth and nail about EHP vs. marginal gains in avoidance or threat. Many take a “balanced” approach to tanking. While that is fine in non-progression raids or when you out gear something to the point you might as well gem spirit, for progression tanking it is not optimal, and the following sections will try to show why.


Most subtopics will probably have the math towards the bottom, however with this section I am going to include some terminology and equations at the beginning so that people understand when I throw out forum/WoW jargon.




Section 2: EHP and terminology

First: RNG, which stands for Random Number Generator. This is used to denote anything that is a statistical probability, not a certainty. That is there is a hidden dice roll to determine whether or not you dodge/block/parry/miss or get hit by an attack. This is a random number generator. I’ll discuss this a little more when talking about avoidance and the combat table.

EHP stands for Effective Hit Points. Essentially what this is is converting incoming damage to a raw hit point value. It is a way to relate hit points to damage reduction. The reason is that the number one cause of tank death is running out of hit points, but hit points come in two forms, health and armor. Since armor reduces the damage you take, it acts like hit points in that it keeps you alive and you take less damage, damage that would otherwise need to be healed. So if you convert all incoming damage based on your damage reduction to “raw” hit points, you get how much the boss would hit for if you went from full hit points to zero if none of the damage was mitigated at all. So for a small example if you had 1000 hit points and 50% armor reduction and you were hit for exactly 1000 hit points, the actual hit unmitigated would be 2000 hit points, so a tank with 0 armor reduction and 2000 hit points could take the same equivalent damage as a tank with 1000 hit points and 50% armor reduction.
The variable setup and subsequent equation for EHP is as follows:


Hit Points: This is simply how many hit points you have. If you are evaluating a piece of gear and want the stam conversion then it is:

Health = Stam * 10 * 1.1 * 1.09 * 1.15

The 10 is simply 10 hit points per point of stam

1.1 is the modifier for kings (10% increased stats)

1.09 is for the warrior modifier for hit points (9% increased stam from vitality) and this number changes based on your class. For paladins it is also a 9% modifier, for blood DKs it is 1.03*1.08*1.02(runes) (which is 1.13465), and for druids I’m pretty sure for the number here it is 1.06*1.1*1.02*1.25 (6% from survival of the fittest, 10% from heart of the wild, 2% from imp MotW, and 1.25% from bear form)

1.15 is 15% for the ICC raid buff currently.



Armor: The reduction from armor is given by the following equation
First we need a value “K” which is a static number based on blizzard calculations. The use of it will be more evident in the total armor equation. The equation for K is as follows:

K = (467.5 * L) - 22167.5

In this equation L is the level of the mob you are fighting. Therefore at level 80 a boss (level 83) would yield an L value of 83. Subsequently solving this equation we can get the constant we are looking for:

K = (467.5 * 83) - 22167.5
K = 16635

This is going to be the constant value we use in the armor calculation. Remember the constants 467.5 and 22167.5 come from blizzard set standards, at least as far as I can tell it does and it yields proper numbers so I assume it is correct.


After this we can solve for the total armor reduction equation:

Armor EQ1..JPG

Where M is the mitigation percent in decimal form and A is the armor value you have on your toon. So for an example if you have 35000 armor the equation yields:

Armor EQ2..JPG

Therefore at 35000 armor the percent reduction against a level 83 mob is 67.78%
A side not here, there is an innate diminishing return to the value of M. As A approaches ∞ it because significantly greater than K, but since K is still a part of it technically never reaches A/A or 1, so as A is really big, each increase in A is a smaller increase in total mitigation. To further illustrate I’ll give an example in 5,000 armor increments:

Armor Table..JPG

As you can see by the 5,000 step increase in armor, the change in reduction gets smaller as you approach 75%. Now 75% is the actual cap, so the “Armor Cap” is 49905 armor.


These values lead us to the final EHP calculation which in its simplest form is:

EHP EQ1..JPG

However the denominator of this equation is a little tricky because you see the damage reduction values are not “Additive” they are “Multiplicative.” If it were additive then if you had 67% reduction from armor and 10% reduction from inspiration and 3% reduction from blessing if sanc you would add those up and have 80% damage reduction. However, this is not so, instead the damage is multiplicative meaning that each modifier gets multiplied by the last. Therefore for every reduction effect you add (1-%reduction) to the bottom. Essentially all classes have a set number of reduction effects when raid buffed: reduction from armor, some talent or presence, 10% from inspiration effects, and the 3% reduction on buffs like blessing of sanctuary or vigilance. If there are any other modifiers, you simply multiply them to the end of the denominator in the same (1-%reduction) form.


Therefore the EHP calculation for a warrior raid buffed would be:

EHP EQ4..JPG

Therefore the effective hit points of a prot warrior with 50,000 health and 35,000 armor fully raid buffed is 195,918. Meaning that if a boss was to hit the tank unmitigated and you turned the tanks mitigation into hit points, the tank could take 195,917 damage without dying.

Now for some more terminology:

I think one of the biggest misconceptions is “mitigation” versus “avoidance.” Mitigation is defined as a reduction in damage when hit. Therefore armor mitigates a hit because it still lands, but hits for less. Block is another form of mitigation in that when you do block the attack lands, but usually for just a decreased amount. However, since unless you have a warrior with shield block up or a paladin manages to get to 102.4% dodge+block+miss+parry (or warrior in an unhittable set), block still is somewhat RNG based since it is a % chance. I guess I’d call it probable mitigation, not guaranteed mitigation.

Avoidance is simply completely avoiding an attack and subsequently taking no damage from it. Thusly things on the avoidance category include dodge, miss, and parry.


Since we’ve already talked about armor reduction, I guess I’ll briefly cover magical resist. Now, when gearing for a max EHP fight, armor doesn’t mitigate magic or bleed damage. Nothing can really mitigate bleed damage except for base reductions like inspiration. However magic damage can be mitigated by resistance. Basically it comes down to calculating EHP for a fight is divided into two halves, the physical half and the magical half. You calculate PEH or Physical Effective Health by the formula above, then you do the same thing for MEH or Magical Effective Health using the above formula only instead of %reduction from armor you use %reduction from resistance. You then multiply each by the %damage intake (so if there is 40% physical and 60% magical damage then you calculate .4*PEH+.6+MEH = TEH or Total Effective Health). However the percentages you use may vary depending on fight mechanics. For example if melee damage isn’t a huge factor, but the magical damage is very bursty but relativisticly infrequent (like Sindragosa) then you may choose to use a full to dead calculation instead of a total damage over the course of the fight in which case say you have 50k health, a boss does a magical attack for 40k and then the rest of what could kill you would be physical, then you’d have .2*PEH + .8%MEH = THE instead of the overall damage intake which might be closer to the aforementioned 40%/60% split.
For more on this Total Effective Health I highly recommend reading TheckHD’s thread on maintankadin.com: http://maintankadin.failsafedesign.c...rb_v=viewtopic

[Aggathon]

Section 3: general arguments for gearing for EHP
Now, why we do EHP gearing in current content has several reasons. But to further investigate those I think we should look at the arguments FOR avoidance. There are some logical conclusions that could be made, and they are seemingly obvious, however the reality is not so.

1)If you have too much EHP, you simply become a mana sponge and the healers have to just pour mana into you until they go OOM. Without mana there are no heals and subsequently tanks die.

2)Avoiding an attack is far better than getting hit by an attack, so avoidance is necessary

3)If you have an EHP Minimum, and you have enough hit points to survive 3 maximum attacks in a row, then you don’t need any more hit points and you should start focusing on avoidance.

These are all seemingly logical arguments, however they are simply not true in current content. The rebuttals are as follows:

1)This assumes that healer mana is an issue, and this is actually a false assumption, especially if you take a gander at overhealing meters. In fact, healer mana is not an issue and most of the time healers will be spamming you with their biggest heal possible, expecting you to take that hit. This means that if you don’t take the hit, that heal usually just turns into overhealing. Paladin overhealing is usually around 70% and is often as high as 80-85%, but this is a necessary precaution to ensure that when it’s not over healing it is both timely and effective. In ICC if you start casting a spell after the tank has taken damage, sometimes it might not be enough time to get the tank healed up again. Healers are always spamming, and regen rates are crazy right now, mana is NOT an issue.

2)Obviously, not taking an attack is really nice, but the fact of the matter is you ARE going to be hit, and you can’t rely on RNG to save you from a boss. I’ll get more on this with some Gaussian curves and probabilities later.

3)The minimum hit point argument dates back to the days of brutallus, and on face it makes a lot of logical sense. If you can’t stack enough EHP to survive 1 more hit, then what’s the point? However, the reason why it is wrong today, but was sort of right in Sunwell, has to deal with the nature of the mechanics of that fight, and how WoW mechanics in general have changed today. On Brutallus it was very possible to take 5 hits in a row, but completely impossible to survive 5 hits in a row. What you wanted to do was minimize the number of hits because healer mana was an issue! If healer mana wasn’t an issue then pallies could have just spammed max rank holy light and the tank could have stacked EHP and it would have been a cake walk. And that would have worked for a little bit, until the healers ran out of mana. Instead tanks geared for a certain hit point level that was equal to the amount a healer could heal in 2 seconds with a down ranked heal to ensure that the healer could heal constantly for all 6 minutes of the fight. After that a tank stacked avoidance to ensure that heals weren’t always needed and burst damage situations were less frequent. The healers almost never had to move and heals had to be constant and reliable.

However, mechanics have changed. Healing and tank movement has increased a lot in fights, cooldown usage becomes much more necessary, and healer mana is not a problem. Boss mechanics are also very different and the DRs on dodge and parry make this a very different ballgame. Druids used to be able to get up to 70% avoidance even with sunwell radiance. That’s not even remotely possible to do in ICC. Furthermore, end game progression fights typically hit harder per hit than brutallus. H-TLK25 hits for 55k(or so I’ve been told), it’s impossible to take 2 hits without a cooldown and/or heals. Therefore the only way to guarantee that you don’t get hit more than twice would be to stack as close to 102.4% pure avoidance as possible, and frankly, it’s not possible and 1 hit would kill you. The tradeoff, especially with DRs, is just too steep.

However, this is also assuming that there IS such a thing as minimum EHP line. In reality, there isn’t. If you notice from the brutallus example, the minimum EHP line wasn’t based off of total health of the tank, it was based on what the healers could heal effectively without stopping heals. Now it is the same, and healers can heal for a LOT. What happens more frequently when a tank dies is that they don’t have a big enough EHP “buffer.” That is when a tank at 50k hps is hit for 30k, 1 more hit kills the tank, but if they get only 10,001hp worth of healing before the next hit, the tank lives. That buffer that you give for healers to catch up is what keeps you alive. Cooldowns also factor into it. Using a cooldown like shield wall or shield block increases your EHP further so that you could take more hits.

There are statistics about odds of getting hit, and I’ll get to that more in following sections.

[Aggathon]

Section 4: The stochastic nature of incoming damage

***DISCLAIMER***
This following section is somewhat speculative on the details. No one has done an in depth analysis on boss damage output to determine the exact distribution of hits. I hypothesize that it is a gaussian curve here, but it very well could be a uniform distribution. Either way it is distributed the theory remains the same, just the shape of the curve is shifted.

It has always been my opinion that the key to winning a boss fight is the minimizing of RNG. A quick example would be doing 6 minute malygos back when it was hard. You could keep wiping until you got 3 sparks that come from the same corner of the map to get malygos down in 1 vortex, or you could learn to kite malygos around so all sparks land in the middle of the room, thusly removing the RNG from the fight. Kiting putricide away from a raid dpsing green blobs when he is about to drop gas bombs helps minimize RNG (the odds that the gas bombs will explode near raid members before they’ve killed an ooze). Etc. Etc. The list is long but the point is you beat boss fights by getting rid of variability.

Someone was talking on the forums about the stochastic nature of incoming damage (that is the randomness of boss hits), and was using it to say that stamina is RNG. Just because you have 20,000 hit points remaining, a boss might hit for more than 20,000 or less than 20,000, if it’s more, then the RNG screwed over your stam and it wasn’t reliable. I pondered this for a while, for it is a seemingly logical and very intriguing concept. I mean it’s really hard to nail down exactly the hardest hit a boss can do to help determine how bad your incoming damage is going to be. So I took a long shower and was fairly flustered by the idea that stamina could be RNG, and I had an epiphany. Stam isn’t RNG, it diminishes RNG. Stacking stam is a way to decrease variability in a boss fight! If stam were RNG, in my opinion, that would mean that stam could give you 8 hit points, or it could give you 12 hit points. It doesn’t, it gives you a base of 10 hit points every time (times modifiers, obviously). To demonstrate what I mean by decreasing variability by stacking stam, some graphs are the best way to demonstrate this.

Gaus1..JPG

Figure 1: The Normal Gaussian Distribution

Thanks to the power of Google images, I’ve found this standard Gaussian curve. The numbers under each section are the probability that an event will occur between each standard deviation. Thusly the further away from the center the less likely it is that something will happen. If you are going from a 1 tailed assumption, then you simply add .5 to each of the values , which is what we will be doing here shortly. Assuming damage is normally distributed (which due to the central limit theorem, if we have enough samples the curve will be normal even if the original curve isn’t, so we can use this as a good baseline) where the center point X is the mean damage a boss does, then the odds a boss hits for more HP or less HP than you have can be determined. So as with the previous example, let’s say you have 20k HP and the boss hits for an average of 20k HP. You would then have a 50% chance of dying or living. That is to say that the left half of the curve is your chance of dying and the right half of the curve is your chance of living. Now as you increase the hit points you have the further away from the middle you get, thusly if you increase your HP from the mean by 1 standard deviation, then you have increased your chance of survival by 34.13% (up to 84.13% chance, .5+.3413). And it gets better as you continue to push your HP value further away from the mean. This is graphically shown here:

Gaus2..JPG

Figure 2: The Gaussian curve shifted

In this figure, the red is now the chance that you will die from the attack and the blue is the chance you will survive. What you are essentially doing is pushing off RNG, and diminishing it, which is how you consistently make attempts at bosses.

So no, even due to the stochastic nature of incoming damage, it does not mean that stam is RNG, or that the marginal benefit of stam doesn’t matter, quite the opposite, it decreases the RNG chance that you will die and gives your healers a bigger buffer range to heal you back up before the next hit lands.

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Section 5: The Avoidance Table


***DISCLAIMER***
A table like this is NOT how it is ACTUALLY programmed. In programming they run a random variable on the range 0 to 1. The calculations get reduced based on level difference as explained below. This is merely a VISUAL REPRESENTATION that helps me visualize how this all works in a way that is, in my opinon, easier to understand. It is probably easier for me to understand because I've played DND a lot and generally think of things like this in terms of lookup tables, not actual programming random variable decimals. If it helps you to understand it better as a random variable and what it REALLY is, then just know that it's not an ACTUAL table with 10s of thousands of spots, it's really a bunch of decimals on the range 0 to 1 but it is still a sort of lookup table. If you have any more questions feel free to PM me and maybe I'll add more detail to this.


Before we move on to more debates there is something that is kind of critical to know, but not necessarily related completely is how avoidance is calculated or the EHP debate, that I would like to cover. Some have had the misconception that each attack goes through a series of checks: does it miss -> dodge -> parry -> block, etc. However this is incorrect. What happens is there is a table and almost literally a dice roll. The table is filled (I imagine it has 10240 values in order to carry avoidance to 2 decimal places like in tooltips, but I don’t know for sure) with values for block/dodge/parry/miss. For each level above you that the mob attacking is, it gains an extra .2 per section that it avoids. That .8 per level is where the 102.4% comes from for being “unhittable” against a level 83 mob. To sort of explain in example form what I mean, assume the following stats for a tank in ICC:
Miss: 5+6.2 = 11% (bosses have a 5% base chance to miss)
Dodge: 26.55% - 20% = 6.55%
Parry: 20.57%
Block: 13.2%
Therefore there would be a table similar to this:

Avoidance Table..JPG

Then obviously a hit would fill in the rest of the values on the table. A random 1-10240 (against a level 83 mob) is then rolled. Whatever number range that correlates to is the result of the attack. So if the RNG roll ends up being 987, then it’s a miss. If it’s 6013, then it’s a hit, if it’s 2400, then it’s a parry, etc. etc. etc.
If you manage to get to 102.4% unhittable and add more to that number, then block is the first thing bumped off the table. If somehow you were to manage to get even block off of the table (which I don’t think is even possible currently) I have no idea which ability comes off next, but it also doesn’t really matter since you won’t be getting hit at all, who cares if it is a miss dodge or parry, avoidance is avoidance.

Anyways, so you know a little bit more about avoidance now.

MOVING ON!

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Section 6: The Avoidance Vs. EHP Debate

I’ve touched on these subjects a little bit, but really haven’t covered the full extent of the avoidance side of the debate. This side gets very complicated, especially from the math angle. I mean… at what point do we prefer an amount of avoidance over an amount of HP. I know I’d take 90% avoidance over a loss of 100HP any day, but that is obviously an unrealistic tradeoff. So a lot of the time the question becomes, “what is the tradeoff?” Using math to figure this out is damned near impossible. However, I would say that from a raw experience perspective, for most fights and gear the tradeoff doesn’t exist in a good amount. There is one obvious exception: H-Anub’Arak25. The fight mechanics of that encounter are very unique and actually do require add tanks to build an unhittable set and MTs to favor resistance and armor above stam. But I digress…

The “if you only can take 2 hits and you can’t stack enough stam for a 3rd, why stack stam?” argument is also extended here. And that is people tend to argue whenever they are proposing avoidance over EH, that the added avoidance you gain from gemming/enchanting/gearing for it will magically make you dodge that dreaded 3rd attack. However this is often argued without the backing of numbers to prove it, and in fact the math proves otherwise.

When you are looking at a tank death in a vacuum, it usually occurs over a few seconds. A tank doesn’t die over a long period of time like a boss does. Avoidance factors into this somewhat, but really avoidance shines with damage reduction over time and average damage taken per second, not over a small parcel of time.

Now, from an experience perspective, I know for a fact that stacking EHP is the right thing to do. I have told others to do the same thing and I have never once had someone come back and say they tried my advice and started dying more. Quite the opposite actually. In fact, I was once a huge proponent of “balancing” EHP and RNG for progression. I liked the idea that you need to balance hit and expertise and avoidance and stam and armor. However, the reality is quite the opposite. I went into the high end, and I became the squishy tank. I was benched for EHP fights like Hodir Hardmode, Thorim Hardmode, and Steelbreaker Hardmode, and I was an officer! I was forced to completely re-evaluate my gearing situation. That’s when I personally converted to the “Stack EHP” camp. I immediately saw a boost in performance and stopped getting benched and was lucky enough to tank our server’s first Algalon 25. I kept this mantra for ICC when it came out and my healers loved me for it. We proceeded to progress at a blistering pace and even got a top 10 US kill. EHP works, and every high end tank knows it.

But why? Why does EHP work? A lot of the arguments for avoidance seem very logical, but in practice EHP wins. Thegreatme and I set out to tackle this problem. We literally spent over 10 hours in front of excel, debating and calculating what we need to do to prove this. We tried coming up with a survivability quotient but ended up not including it in our final results. I will be posting another thread with this discussion to see what other people think.

Instead we came up with our “Survivability Spreadsheet.” It’s not terribly pretty but it gets the job done. It is currently setup for a warrior, so you will need to change the base HP, the misc. mitigation coefficient, and the stamina modifiers for your class. This spreadsheet calculates pretty much everything you need to know. It is attached to this article.

Our conclusions are that avoidance modifies damage taken over time, and average damage taken per second. However, it does not modify the max damage taken per second (maxDTPS), the minimum time to die (minTTD), or the minimum time to die with heals (minTTDwHeals). How it is pretty blatantly obvious that if HPS > maxDTPS, then as long as you have enough hit points to survive 1 or 2 hits and your healers are on the ball and can heal you up, then nothing else really matters. However, if HPS becomes lower than maxDTPS, then you have to start worrying about minTTD and minTTDwHeals. maxDTPS obviously occurs when you get an unlucky avoidance streak. Therefore, while avoidance doesn’t modify minTTD or minTTDwHeals, it does modify the frequency at which minTTD or minTTD with heals occurs.

Okay, so I know how fast you’ll die, roughly. But what does it really mean?! minTTD and minTTDwHeals are buffers. They are the time periods that your healers need to be able to react or you need to be able to hit a cooldown to in order to ensure that you get healed up in time. The minTTDwHeals, in my opinion, is probably the more important of the two in terms of actually dying, and minTTD has to do more with the granularity of healing and the stochastic nature of incoming damage. When you’re taking damage and still being healed, minTTD calculates (roughly) when you will die if HPS is < DTPS. Even if you only get a fractional increase in this, it means when the healer goes to heal again, or pops a cooldown, or you pop a cooldown, you increase the probability of surviving it. If HPS < DTPS for longer than that period of time, then the tank is probably going to die, no matter what their avoidance is at. Therefore minTTDwHeals is the buffer that a healer has to heal the tank back to full. The longer this is, the better the odds of survival.

However, with minTTD normally, this represents the buffer in granularity. You see, all these calculations assume a continuous system, however the system is not continuous, it is discrete and random. Heals heal in individual time slots at intervals (even HoTs). Therefore the exactness of this science is a little off. Instead, minTTD gives the buffer for granularity. That’s the timeframe that the healer has to ensure at least one heal is landed, whereas minTTD gives the timeframe that the healer needs to increase their HPS to be greater than the incoming damage.

These two values combined are therefore the most important values for determining a tanks GUARANTEED survivability. Because, after all, the number one cause of tank deaths is running out of hit points.
But wait! What about avoidance, you said that avoidance minimizes the possibility of maxDTPS to occur! This is correct. However, the problem with this is, that while for a single string it may minimize the possibility (for example, say you have 40% avoidance total in ICC, therefore a 60% chance to get hit, then the probability that you will get hit 5 times in a row in 5 strings is .6^5, which is only 8.93% about) relatively significantly over a short time frame, as time approaches infinity, the probability of taking maxDTPS and dying is almost guaranteed. And actually, having more EHP subsequently increases the number of swings you can take before you die in the minTTD and minTTDwHeals calculations, thusly, EHP also minimizes the odds of the “worst case scenario” from occurring.

I’d also like to quote someone that was writing about this matter on tankspot, because it said it very well:
Don't forget that if health can take you from 5 to 5.2 hits, this means healer needs to heal you for 0.8 hits less between the 5 hits. The extra hp is not completely wasted.

Taking 4.9 hits to die instead of 5.1: If somebody can land a very quick heal, or a tick of HoTs happen between the hits... you got healed for 0.2 hits, you survive 5 hits. If you had hp to survive exact 4 hits, you would not have survived (with the hot/fast heal you would have survived 4.2 hits).

If you put 5 hits on the table, you have quite a lot of room. This reaches a point where you can almost assure that there will be heals coming in - instead of like 2-3 hits gibs.

-enricobr

But, avoidance isn’t COMPLETELY wasted. Indeed, it still does minimize the chance of the worse case hit string from occurring.
For the next part a very special and significant credit MUST be given to enricober and PatrikL. They were instrumental in figuring out the simulation needed in order to truly figure out this someone mind boggling statistics problem. You can read the whole thread here: http://www.tankspot.com/showthread.p...istics-problem

I will first define the variables of the simulation so that you can understand what we did better.
X = Hits taken in a row
Y = number of times a boss swings at you
N = number of times the simulation is run

Basically what we did is setup a simulation that ran a random binomial variable equal to my current avoidance in ICC (38.32%) through a simulation that ran that random variable 100 times. If that simulation had at least 1 streak equal to or greater than the number of attacks X we were looking for, then we increased a counter. When then divide that number that we get from the counter by N to get the % times it occurred throughout all simulations.

For the setup of 38.32% avoidance, 100 swings, and a streak of 5, the simulation then gives a result of 98.5% chance of taking a 5 hit swing.

As you increase or decrease avoidance, the odds of that 5 string streak change. Here is a graphical representation, the table is attached:

&#65;&#118;&#111;&#105;&#100;&#97;&#110;&#99;&#101;&#32;&#67;&#117;&#114;&#118;&#101;&#46;.JPG

As you can see, at about 40% avoidance, the graph starts to shift the other way. However, the probability of a 5 string hit still occurs at 60% avoidance just under half the time. If you were somehow able to increase your avoidance by 10% in ICC (which is a pretty hefty task, I might add), then you would still take that 5th hit in the string that kills you over 80% of the time. While it is a significant increase to 98.5%, it’s still definitely not a number I would gamble on.

And remember, EHP is what determines how long this streak is, so if you tradeoff hit points with that 10%, then you might drop your minTTDwHeals time from 5 hits to 4 hits or even 3, thusly making your odds of survival even worse.

Avoidance in and of itself is not bad, it is the tradeoff of EHP that us “EHP stackers” are against. I might even be for avoidance more if Chill of the Throne wasn’t in place, but as you can see by this graph, there’s reason to why blizzard cut off our avoidance by 20% base.

I’ll add the simulator to this, however we have a slight kink in that it only calculates 5 hit strings at the moment, I will add it when it is finished.

Further reading on this is a good post by Aragwaen3: http://www.tankspot.com/showthread.p...528#post423528

Ending Conclusions:

When all is said and done, it is obvious from the math, the graphs, the experiences, etc. that in ICC for progression fights, EHP is the way to go. However, it is also obvious that this is not an exact science like it is with DPS. With DPS there is a lot less variability, a lot less probability involved, and the DPS charts appear far more continuous (especially since they're usually over longer time periods than 12 seconds) than the discrete nature of tank damage and healing. In the end, it comes down to what I have in my signature:

DPS is Science, Healing is Art, Tanking is Strategy.

I also recommend that people read this post by Satorri:
http://www.tankspot.com/showthread.p...742#post423742

[Aggathon]

***EDIT*** I AM TEMPORARILY TAKING DOWN THE SURVIVABILITY SPREADSHEET!!!! THERE ARE SOME INACCURACIES THAT WERE POINTED OUT TO ME AND I WILL FIX IT AND PUT IT BACK UP SOON***

Here's the Survivability Spreadsheet

The Yellow Squares are inputs.
The light gray-blue squares are simply the names of the inputs.

[Aggathon]

Here are the results from the test in section 6

[Aggathon]

Okay, all done, I'll post the Simulation file here once we're done with it.

A HUGE thanks again to everyone that helped:
Thegreatme especially for the creation of the survivability spreadsheet
PatrikL for helping with the initial simulation creation
enricobr for the exel simulator
Everyone in the statistics thread that helped me figure this out, even those I disagreed with at times =P, in the end, truth was found. That's what science is all about!!!
All those in the shoutbox that randomly helped me.
And I'm sure there's a lot of other people I'm forgetting, this was a huge effort and I definitely could not have done it alone.

I've edited this thing down several times, so if I made any errors or say something that I don't end up talking about later, let me know.

***UDPATE***

The avoidance simulator is now fixed and up and running for excel. It doesn't generate pretty graphs, but it will take the inputs of times you are hit in a fight, the number of swings you are looking for, and how many times you want to run the simulation and spit out the likelihood of you being hit that many times in a row in 1 fight (hint: for 3 hits in a row it's pretty big). I suggest 100,000 or 1,000,000 runs for the most accurate results. While waiting for excel to simulate, I suggest reading this webcomic: http://www.xkcd.com/303/

[Kazeyonoma]

I dunno what's worse, the fact that this could probably be someones dissertation, or that I"m giddy in excitement to read it.

[Aggathon]

I dunno what's worse, the fact that this could probably be someones dissertation, or that I"m giddy in excitement to read it.

The sad part is, this is about half of what I wanted to do initially.

[krc]

Sweet Jesus this is going to take me a long time to read.

Great job as always Agg.

[felhoof]

Very nice set of arguments. Well done.

I think the only quibble I'd have is that while this is absolutely true if your goal is to maximize survival currently, that's not always the tank's optimal goal for a fight. It's also a shame that you don't mention that minimizing worst-case scenarios often comes down to defining said scenarios well, and isn't about simply how much EH you can pile on in a vacuum; the implication about the PEH + MEH = TEH is that you simply look at the total damage you'd take in a fight and then figure out what percentage of each is what, when in reality tank deaths happen nowadays on very specific sets of events that can be treated differently.

It would also be interesting to look at what stamina provides vs what armor provides, and where they specifically are strong. And not just on MEH vs PEH; armor has significant strengths when HPS is critical or there are disruptions in healing, and it strengthens that 'buffer' (the 5.2 case) more than stamina can.

The final note is that right now, the problem isn't a string of 5 hits; it's 3. What's interesting about that is that it's virtually guaranteed that you will be hit 3 times in a row in a fight, but how often this occurs is significantly reduced by avoidance. Buffers do help significantly, of course.

[Edgewalker]

Thank you for this.

[Aggathon]

Very nice set of arguments. Well done.

I think the only quibble I'd have is that while this is absolutely true if your goal is to maximize survival currently, that's not always the tank's optimal goal for a fight.

For progression and namely ICC progression it is. I really don't want to go into this argument further because well... it's been hashed out time and time again on forums. This is also 13 f***ing pages long with 3 spreadsheets all in all (once we get the sim working correctly). Is it really not enough? Can you imagine how long this would be if I were to include that? I was going to have sections discussing threat and specs but... F that. It's long enough as it is, someone else can post that.

It's also a shame that you don't mention that minimizing worst-case scenarios often comes down to defining said scenarios well, and isn't about simply how much EH you can pile on in a vacuum; the implication about the PEH + MEH = TEH is that you simply look at the total damage you'd take in a fight and then figure out what percentage of each is what, when in reality tank deaths happen nowadays on very specific sets of events that can be treated differently.

Please read the paragraph more thoroughly

However the percentages you use may vary depending on fight mechanics. For example if melee damage isn’t a huge factor, but the magical damage is very bursty but relativisticly infrequent (like Sindragosa) then you may choose to use a full to dead calculation instead of a total damage over the course of the fight in which case say you have 50k health, a boss does a magical attack for 40k and then the rest of what could kill you would be physical, then you’d have .2*PEH + .8%MEH = THE instead of the overall damage intake which might be closer to the aforementioned 40%/60% split.

I do cover this.

It would also be interesting to look at what stamina provides vs what armor provides, and where they specifically are strong. And not just on MEH vs PEH; armor has significant strengths when HPS is critical or there are disruptions in healing, and it strengthens that 'buffer' (the 5.2 case) more than stamina can.

Mess around with the survivability spreadsheet, it covers this.

The final note is that right now, the problem isn't a string of 5 hits; it's 3. What's interesting about that is that it's virtually guaranteed that you will be hit 3 times in a row in a fight, but how often this occurs is significantly reduced by avoidance. Buffers do help significantly, of course.

The reason I don't mention 3 hits specifically is 3 fold: 1) the information in the spreadsheet when you download it is slightly modified data from my actual armory and a parse of H-Blood Princes 25, and in the minTTDwHeals calc, 5 hits is what you really want to avoid, I can take 3 hits (the 3rd killing me) as it is without ANY heals or cooldowns. 2) I don't have an exact statistic for 3 hits yet, I plan on posting these various results once we get the simulation parse ironed out and I post it. 3) If at 5 hits avoidance doesn't help much, it sure as hell won't help much at 3. I'm giving avoidance the benefit of the doubt at 5 hits, and the results still say stack EHP in ICC.

[Satrina]

Nice work! A few comments:

Remember the constants 467.5 and 22167.5 come from blizzard set standards, at least as far as I can tell it does and it yields proper numbers so I assume it is correct.

The extraction of those constants is here: http://www.tankspot.com/showthread.php?40760-Mitigation

I think one of the biggest misconceptions is “mitigation” versus “avoidance.”

Also covered in the Mitigation article, not necessarily in more detail, but with different words.

For those interested in more in-depth discussion of the combat table: http://www.tankspot.com/showthread.p...e-Combat-Table
Also here: http://www.tankspot.com/showthread.p...Table-Analysis

Another way to express the concept of "80% of a hit" is to find a way to help people visualise how damage and healing are not lockstepped with each other. It's an obvious concept, but easily overlooked in the "you can't stack enough stamina to survive one more hit" argument. You get hit for 26k, then an 8k heal lands and a 10k heal lands, then a rejuvenation tick before the next hit lands. You don't need to stack enough for an entire extra hit. Every bit of EH that you add works with the healing you do receive to make it more likely that you will be able to eat that extra hit in the worst case.

[Aggathon]

Added the ending conclusion I forgot about whoops.

[Maèl]

You put my gut feeling into numbers. This also explains why avoid was more important at the beginning of WotLK than it is now.
It also explains the mysterious "EHP thresholds" I witnessed.
My warrior went from being squishy to being a brick wall by adding 3k armor(which at face value wouldn't make such a difference; what's the difference between 32k armor and 35k armor unbuffed) and 2k life. Both are not that huge leaps when you put them in relation to the absolute figures. But it makes the difference wether you can take 3 hits from Festergut or 2. And that is HUGE!

The mitigation/avoidance game has changed from preserving healer mana to buying the healers some time.

Avoidance is indeed a gamble. You are betting against 3 straight hits from Festergut. With EHP stacking you start betting against the likelyhood of getting 4 straight hits. EHP is lowering the stakes, avoidance reduces the likelyhood of the worst case scenario. So it's never EHP OR avoidance. It's just that we have reached the point where avoidance from gear is enough, while EHP isn't.

Those magic "EHP thresholds" do exist. The closer you are to one, the bigger the benefit of absorbs(think Disc priests) and shield block become. trouble is, these differ from fight to fight. And you also have to assume you are not at 100% health, but rather 90% or lower due to raid splash damage that also affects you.

You could run the figures of all bosses thru the excel sheets, but that is only fun for a very small subset of players. So here we arrive at the rule of thumb:
Avoid from gear is enough if you are geared for ICC. EHP is something you never can have enough of.

...and we have been promised come Cataclysm we will need to theorycraft from scratch.

[eazo]

This has been bookmarked so I can continue to read it! I'm actually kind of excited to read the Gaussian curves concerning the EHP vs. Avoidance debate. Been trying to convince a guildy that EH is the way to go. Will Probably just link him to this page! Great work man

[Satorri]

Nice bit of work.

Wanted to post before I start digging into it as it will probably take a while for me to get all the way through. =)

[Krenian]

I think I need to get credit in your work cause I reminded you of your end notes.

But otherwise, a pretty solid piece of work here. Definitely something for people who wanna look at this and understand why it is that most take the EH route. Well done Agg.