Calculating the avoidance value of expertise

Warning: Theorycraft inside.

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Re: Calculating the avoidance value of expertise

Postby Awyndel » Sun Aug 30, 2009 9:01 am

Putting this into perspective for gear, It doesn't change a whole lot for me. I like a lot of expertise plate already because personally I don't like the tier. I don't like expertise jewelry coz I am taking the armor ones.

I am in doubt though about some pieces. I could get the crafted wrists for the hit, or the expertise drop for extra avoidance. Kind of a pressure on that decision because a guildy has the expertise wrists for offspec, I could get them ticket to me and have the lootmaster change my points.

Also I was gonna get the emblem shoulders and tier chest, both for full avoidance. Getting 2 set bonus with hands. But I could also keep using the expertise hauberk I already got, and get the tier 9 shoulders. Minor loss of threat, minor upgrade in avoidance.

The fact this somewhat effects my multi target tanking doesn't bother me because I am geared for hard hitting bosses. My threat is also fine atm. The only thing that worries me is wich hard hitters still parry haste these days, and how will it look in the future.
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Re: Calculating the avoidance value of expertise

Postby theckhd » Mon Aug 31, 2009 6:46 am

By the way, Mex was kind enough to link me to the EJ thread where parry mechanics were worked out in detail. The 0.24 number comes from the following simple equation:
Bregonn wrote:I'll leave the formal mathematical proof to someone else, but from the graphs in this threat and the theoretical numbers the average haste increase from parry can be fairly easily deducted:

40% of your swings get the maximum of 40% speed reduction (to 0.6 * weapon speed).
20% of your swings get no speed reduction at all (1.0 * weapon speed).
The remaining 40% get reduced by on average the middle of these (0.8 * weapon speed).

So your average swing speed as a result of a parry is 0.4*0.6 + 0.2*1 + 0.4*0.8 = 0.76. In other words, 1% parry is worth 0.24% haste.

The value is a typo in that last sentence, it's actually 0.24, or 24%.

Anyway, on to a question that I've been meaning to get back to since I read it, but haven't had time for:
Awyndel wrote:Slept on it, but it still makes sence to me. Let's say we're talking about preventing 2 hits in a row here, since that is what a parry haste will mainly cause. Now let's say you have 62% avoidance raid buffed, not at all a weird thought in current gear with libram proc up. Now excuse my newby math but here goes. Let's say 0.38x0.38 so about 14.4% of the incoming attacks will be followed by a second one. Now you said in a 5 min fight we get 4.2 parry hasted attacks ( not sure if that's still up to date for your current code though ), so with 2.0 swing timer that's what, 2.8%. So you can see here wich source of 2 hits in a row is bigger. Now if we start talking about 3 or 4 hits in a row, stuff that kills us, avoidance might get further ahead, although I am not sure coz a parry hasted attack can also be part of longer spikes as well.

Secondly let's not forget that in this scenario 62% of the parry hasted attacks are being avoided by standard avoidance. So if you'd take away 62% of those 2.8% your left with a lot less. I know it doesn't work like that when trying to see what expertise prevents but it is something important to think of when making comparisons between the value of the 2.


That's not a fair comparison. First of all, the newer calculation is 9.88 extra attacks due to parry-hasting, which brings your percentage up to 6.2%.

But even if you used the correct numbers, the chance of a double-hit isn't directly comparable to the percentage of attacks that are caused by parry-hasting, for a few reasons.
  • The attacks "caused" by parry hasting are not independent of the ones coming in normally, because they're the same attacks that you're considering in your "2-in-a-row" probability. They're just coming faster.
  • In addition, those 9.88 extra attacks are spread out more evenly among multiple attacks during the encounter. Since the average amount of parry haste you get on a single parry is 24% of an attack, it's more realistic to say that 9.88 / 0.24 = 41.2 attacks are sped up by parry-haste, or 25.8%.
  • Perhaps an even more relevant way to view it is this: You're making a parry-able attack every 0.5464 seconds on average. With a 7.5% chance for the boss to parry those attacks, that means that the boss will parry an attack on average every 7.2853 seconds, or every 3.64 attacks (which also works out to 41.2 attacks over a 5-minute fight, or 25.8%).

Or to put it another way: You have a 14.4% chance to see two un-avoided attacks in a row. But you have a 25.8% of those two un-avoided hits to occur even faster due to parry-haste.

I think the only fair way to compare the two is based on some incoming damage metric, which is exactly what I've done in the calculation that shows expertise is 70% as effective. Now that's on average, over a long fight, and not for small bursts of unavoided hits. To look at that, we'd want to think of DTPS in a worst-case scenario, I think.

Worst case scenario for two un-avoided attacks:
Boss damage X = 20k, Boss attack speed T=1.884 seconds. Spike damage goes from < X/2T (avoiding every other attack) to X/T. So incoming DTPS spikes from 5.3k to 10.6k (doubled). This happens with a probability of 14.4%, as you calculated.

Worst-case scenario for two un-avoided attacks with parry-haste (40% swing speed reduction):
Boss DPS spikes from X/2T to X/(0.6*T), or up to 17.7k, over triple the "dodge every other attack" value. Of course, on average this would really only be X/(0.76*T), or 14.0k, but that wouldn't be a "worst-case scenario."

The chances on this: Well, the boss's second attack has a 25.8% chance of being sped up by parry-haste. 40% of the time that's going to give the full 40% speed reduction (resulting in the 17.7k DTPS value). So we first of all have the base 14.4% chance of not avoiding two in a row, multiplied by 40% of 25.8%, or 0.144*0.258*0.4= 1.49%.
If we use the "average" value of 0.24 (the 14.0k DTPS value), it's 0.144*0.258, or 3.72%

Now, those percentages looks small, and they are. But that's slightly misleading, and here's why.
Using the numbers from the calculation, it takes 3.03594 expertise to remove 1 extra attack, equivalent to a 1/0.24=4.166% reduction in that 25.8% chance that the boss will have a parry-hasted attack. That amount of expertise, if converted to dodge rating, would give a bonus 0.341584% chance to be avoided. That reduces our 14.4% chance of getting two hits in a row to 14.18% - not a very significant reduction.

If we use a metric that multiplies the max "worst-case scenario" DTPS by the reduction chance, we'd get the following for equal amounts of dodge rating and expertise rating:
Dodge:
10,600 * (0.144-0.1418)= 23.3 DTPS
Parry:
17,700 * 0.144 * (0.04166) * (0.4) = 42.5 DTPS (worst-case)
14,000 * 0.144 * 0.04166 = 84.0 DTPS (average)

So whether you look at it in a "worst-case" or "average" scenario, expertise rating is 2x-4x more effective than dodge rating at reducing spike damage, and will only get better relative to dodge as your dodge rating increases due to diminishing returns.

In addition, as has been pointed out several times already, it spends all of that reduction on the largest spikes in DTPS that we receive from auto-attacks, meaning that it "hits them where it hurts (us) the most."

TLDR summary: Provided these calculations are correct (hopefully someone will error-check them for me), expertise is actually 2x-4x better than dodge at reducing spike damage intake from boss melee attacks, with the added benefit that it preferentially reduces the largest spikes, thus smoothing out our spike damage by reducing the maximum spike size.



Awyndel wrote:I will honesly admit I can't copy the math, but my feeling tells me dodge/parry will always rule over expertise by a long shot. Dodge/parry as 1 and expertise as 0.7 would give an expertise itemized item a slight benefit over a def/dodge/parry item, but I don't believe the numbers are like that when it comes to preventing spikes.

This is why it's not a good idea to trust gut feelings. Often they turn out to be wrong due to psychology - we tend to "seed" our gut feelings too much with what we expect to be true based on our current understanding (i.e. we don't like to be wrong) as well as what we "wish" was true. It's very hard to shelve that instinct (even I do it), but we have to in order to properly analyze something without bias.

On the other hand, it's always good to re-check the calculations if they go against your gut feelings. Calculations can also be wrong, and it always pays to be skeptical until something's thoroughly proven. I still don't feel 100% about these results, but I've been feeling more and more confident the more I work with them. I think all the calculations are correct so far, but I don't want to rule out the possibility that I'm excluding or neglecting something in the base assumptions.
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Re: Calculating the avoidance value of expertise

Postby Thalastor82 » Mon Aug 31, 2009 8:39 am

Thank you for the amazing work Theck,

I like very much the feeling of abandoning the "choice by feeling" for "guiding myself with exact numbers"
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Re: Calculating the avoidance value of expertise

Postby Awyndel » Mon Aug 31, 2009 4:53 pm

Thanks for that lengthy answer Theck. I will admit that while you where ocupied with other stuff I had another try at reading all the math, and I did see that I have to view all the attacks on the same page, except for the hasted ones being, well hasted.

I also noticed that for the second point I raised, if you had avoidance avoid some of the extra attacks, had already been answered by you before. You did.

I was working on a new model on how this all relates to spike damage, coz the way I put it before was quite simplistic, and well wrong. I wasn't really able to come up with something I was satisfied about, but with you going deep into spike damage in your reply, wich was the topic I am most interested about, it makes my model irelevant.

The whole, "hits us where it hurts the most" idea is what brought me to use blood-draining, and it's what is gonna make me gear for more expertise right now. And if blizz nerfs some of them, then well, I won't be totally wrong geared for the imune ones. Same goes for multiple targets or trash, I really am not worried.

theckhd wrote:On the other hand, it's always good to re-check the calculations if they go against your gut feelings. Calculations can also be wrong, and it always pays to be skeptical until something's thoroughly proven. I still don't feel 100% about these results, but I've been feeling more and more confident the more I work with them. I think all the calculations are correct so far, but I don't want to rule out the possibility that I'm excluding or neglecting something in the base assumptions.


Well you possibly noticed by now that I always wanna see something proven before I adjust my ways. And I should really tank :wink: you for the patience and explanations. Been a big help on multiple subjects.

As far as gear goes, I know it might partially belong in a different section but this topic does have implacations on it. This lucky boy missed a raid once and decided to pug it, where he won the expertise legs and chest. Gonna get the expertise emblem head as well, and the t9 shoulders. I had my buddy ticket those wrists to me, they arived today :D. Prolly skipping the jewelry to take armor ones.
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Re: Calculating the avoidance value of expertise

Postby theckhd » Mon Aug 31, 2009 7:43 pm

I took the T9 shoulders as my first tier piece since I got the off-set legs and chest already. I worked out the difference between the tier shoulders and the badge shoulders for someone else earlier today:

Theck wrote:With the T9 shoulders, I get about 300 more health and some defense skill at the cost of 0.23% avoidance. But the 43 expertise should more than make up for the avoidance; the 67 Dodge Rating on the shoulders gives me 1.03% dodge, so even at 70% effectiveness 43 expertise should be around 0.46% avoidance - I've been using the rule of thumb that every point of expertise is worth about 0.01% avoidance, which is a little bit of an under-estimate, but is good for quick-and-dirty estimates.
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Re: Calculating the avoidance value of expertise

Postby badgermonkey » Mon Aug 31, 2009 8:35 pm

theckhd wrote:I took the T9 shoulders as my first tier piece since I got the off-set legs and chest already. I worked out the difference between the tier shoulders and the badge shoulders for someone else earlier today:


Were they the T9 + token shoulders? (They're all named the same, it's hard to tell).

Otherwise I'd be right in assuming the Vendor Shoulders are better than the lowest T9 shoulders?
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Re: Calculating the avoidance value of expertise

Postby Iselian » Mon Aug 31, 2009 8:46 pm

"of Triumph" indicates the 245 equipment, or trophy+emblems turn-in. Conquest is the emblems only and Triumph(Heroic) is.. well, from heroic. 25man, of course.
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Re: Calculating the avoidance value of expertise

Postby Doxa » Mon Aug 31, 2009 10:29 pm

Iselian wrote:"of Triumph" indicates the 245 equipment, or trophy+emblems turn-in. Conquest is the emblems only and Triumph(Heroic) is.. well, from heroic. 25man, of course.


Well of course. Oh.. and don't forget the horde vs. ally distinction. I love searching for T9 sets and seeing 20 of them. It's cool.
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Re: Calculating the avoidance value of expertise

Postby Doxa » Mon Aug 31, 2009 10:49 pm

Regarding the embedded discussion of avoidance, stamina, etc (and a bit off topic)..

I think that this is the main reason that avoidance doesn't matter too much any more and everyone is stacking stamina is because mana doesn't matter anymore.

Back when we started Ulduar we were having tons of tank deaths from big melee hits back to back. Each time I'd look up after the tank death the healers would be sitting 3/4 to full mana.

I got a little pissy about it so I put on my Holy gear and spec'd into it just for Hodir. The druid tanked it and I healed. No tank death. Why? Because I only pressed one button the entire fight - Holy Light. Back then I was casting Holy Lights at 1.6 seconds for crits of around 18-20k if I recall correctly. It doesn't take long to fill up a tank and it didn't matter if he dodged anything because I was chain casting regardless of whether or not he needed the heal. His avoidance didn't matter because my mana didn't matter.

And of course, I out-healed our regular healers.

A bit off topic but I'm pretty convinced that avoidance won't matter until something happens with healers' mana regen / pool.

EDIT: Or to put it another way the symbiotic relationship between DPS, tanking and healing is broken. It used to be that the tanks had to minimize damage to keep themselves alive and not drain the mana pool of the healers while the DPS was primarily concerned with racing against the draining mana pools of the healers as well. DPS now races against enrage timers instead and healers primary concern is spike damage.

I learned this lesson on the Heigan fight one night when all of our DPS died to the lava burst and a disc priest and I finished the fight after 22 minutes.
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Re: Calculating the avoidance value of expertise

Postby Awyndel » Mon Aug 31, 2009 11:49 pm

EH deals with spike damage. Avoidance prevents it.

Now it really gets complicated when you take into account non perfect healer behaviour. Or to be more precise, realise that perfect healer behaviour is not always possible. There will often be lapses in healing, maybe not if you're tanking in ensidia, but in most guilds there will be many times. Now some extra health will mean you can take that extra hit in that time the healers are busy, or you will need a heal less to be topped off again. Avoidance lowers the chance of that spike damage happening while that healer is busy, it's a gamble. It's a bit of a delicate issue. This is why ppl have been asking for simulations regarding different incoming hit sizes and healer behaviours.

In tbc most tanks would say stack EH untill you're not dying on spikes anymore, then take avoidance to prevent them. Nowadays most tanks say stack EH because ulduar hard modes kill me. It all depends a bit on your healers and wich hard modes you're currently progressing on.

My advice would be, and for a part i'm quoting theck here, if your AD is not proccing, get the best oppertunity cost. The cheapest stats. That way you mostly have a good balance, and quite possible more stats in total. And you will only have to swap out a few items in a specialized fight. If you're running 25 mans I would go ahead and overwrite that with making 2 stamina trinkets mandatory though.

To put expertise into perspective into this, hasted attacks are when the shit hits the fan, even during perfect healer behaviour. So protect yourself from it.
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Re: Calculating the avoidance value of expertise

Postby Awyndel » Tue Sep 01, 2009 7:16 pm

Got another question. How would weapon speed affect all this. How much better would a slow weapon be, in terms of reducing hasted attacks. How would the value of expertise change. And would it be possible to give the weapon speed a value as well, let's say expressed in expertise rating. My feeling tells me it's gonna be a big influence, since most things we do that can be parried are done on swing.

Edit: just read the thread again seems you mentioned getting it done sometime soon, can't wait.

Slow weapons already have high threat, and the new arena weapons have high stamina, then there is the dps weapons, wich have stam and agility, and possible more expertise, they could be very interesting. Of course I can't deny the armor weapon, but it's ilevel and fast speed will limit it.

Woaw seems there is lots to consider for chosing a weapon. Knowing where speed stands would really help, and we can prolly dig out the values of weapon dps and speed for threat in the mathlab thread somewhere. Then we'd just have the EH and AV to consider.
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Re: Calculating the avoidance value of expertise

Postby Serv » Wed Sep 02, 2009 7:35 am

I don't know if my logic is correct, but if a mob got a 5% chance of parrying your attacks, then he will parry 5% of your attacks in average. Am I right? The only concern i can imagine is having a attackspeed auf 1.3 and the chain-parries 3 oder 4 of your attacks, which would give him a serious haste buff compared to an attackspeed of 2.0 post raidbuffs.
Or I am just missunderstanding the mechanics.
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Re: Calculating the avoidance value of expertise

Postby theckhd » Wed Sep 02, 2009 9:25 am

Serv wrote:I don't know if my logic is correct, but if a mob got a 5% chance of parrying your attacks, then he will parry 5% of your attacks in average. Am I right? The only concern i can imagine is having a attackspeed auf 1.3 and the chain-parries 3 oder 4 of your attacks, which would give him a serious haste buff compared to an attackspeed of 2.0 post raidbuffs.
Or I am just missunderstanding the mechanics.

Each time you attack, the boss has a chance to parry (let's say 5%, since that's the figure you used). So yes, if you average over a large number of attacks, it will end up at 5%. But with a small sample size, it could be much larger or much smaller. For example, there's a 21% chance that the boss won't parry any of your attacks over any given minute ((1-0.05)^30=0.2146=21.46%)

The reason it's a concern is that every time the boss parries, his next melee attack comes up to 40% faster than normal. This leads to cases where you might take a larger-than-average amount of damage in a shorter-than-average time, a "damage spike." Expertise helps prevent these (and in fact, can eliminate them completely if you hit the expertise hard-cap).
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Re: Calculating the avoidance value of expertise

Postby theckhd » Wed Sep 02, 2009 10:25 am

Ok, so a poster at EJ pointed out that HotR is deflected rather than parried, and that deflects don't cause parry haste. That means that the numbers are a bit off, because we need to change player_attack_speed to:
Code: Select all
player_attack_speed=1./((2-boss_avoid./100)./player_swing_speed + (1-boss_avoid./100)./6);


Doing so changes the results very slightly. For a 1.3 base swing speed and 2.0 boss swing speed, assuming 7.5% parry (dodge soft-capped):
Code: Select all
player_swing_speed              1.25975
player_attack_speed            0.601311
boss_swing_speed                1.89276
player_parryhaste               1.03195
boss_parryhaste                 1.05666

redux / exp skill            0.00188863
redux / equiv dodge          0.00264229
redux eff (%)                   71.4771

# extra attacks                 8.49885
expertise to rem 1 att          3.52989
equiv dodge rating              28.9363
equiv dod % post-DR            0.371571
equiv # att dodged (vs 0.4)    0.588936
# att eff (%)                   67.9191


For a 1.3 base swing speed and 2.4 boss swing speed, assuming 7.5% parry (dodge soft-capped):
Code: Select all
player_swing_speed              1.26581
player_attack_speed            0.603936
boss_swing_speed                2.24922
player_parryhaste               1.02701
boss_parryhaste                 1.06704

redux / exp skill            0.00223456
redux / equiv dodge          0.00264229
redux eff (%)                   84.5692

# extra attacks                 8.37961
expertise to rem 1 att          3.58012
equiv dodge rating               29.348
equiv dod % post-DR            0.376828
equiv # att dodged (vs 0.4)    0.502612
# att eff (%)                   79.5842


For a 2.3 base swing speed and 2.0 boss swing speed, assuming 7.5% parry (dodge soft-capped):
Code: Select all
player_swing_speed               2.1817
player_attack_speed            0.976557
boss_swing_speed                1.93125
player_parryhaste               1.05422
boss_parryhaste                  1.0356

redux / exp skill            0.00118657
redux / equiv dodge          0.00264229
redux eff (%)                   44.9068

# extra attacks                 5.33956
expertise to rem 1 att          5.61844
equiv dodge rating              46.0572
equiv dod % post-DR            0.589454
equiv # att dodged (vs 0.4)    0.915655
# att eff (%)                   43.6846


So it looks like as the boss swings slower, expertise gets better (which we expected). Unfortunately, the slower we swing, the worse-off expertise is, dropping to 44% effective at a 2.3 swing speed.

I want to investigate how this varies with all of those parameters a little further, but first I want to re-work the spike reduction figures and incorporate that into the simulation as well. Hopefully pretty plots inc, if I get some time today and tomorrow.
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Re: Calculating the avoidance value of expertise

Postby jere » Wed Sep 02, 2009 12:14 pm

theckhd wrote:Ok, so a poster at EJ pointed out that HotR is deflected rather than parried, and that deflects don't cause parry haste.


Are they sure about that? I saw that post too, but they used wowwiki as the source for it I thought. I don't know if anyone has ever tested whether the deflections of HotR cause parry haste or not.
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