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举报邮箱：&edit& It’s been brought to my attention that, unlike in Cataclysm, the mastery rating to mastery percent conversion factor isn’t the same for all classes anymore.
I’ve used the paladin value here (600 mastery rating = 1% mastery), but for Prot warriors it’s actually 272.6 rating to 1% mastery.
This will have three major effects on the simulation:
1) Mastery’s value will be about 2.2x larger than the simulation predicts, putting it on-par with dodge and parry
2) Dodge and parry will get slightly weaker.
It’s not clear what the exact order will be, but I’ll try and run some sims and get results posted soon.
3) Hit and Expertise should lose even more value due to the extra rage income from more critical blocks.
I’ve just run the simulation again, fixing the mastery conversion as well as a few other bugs (my Sword and Board proc rate was too low, Berserker Rage is now included), and got the following results:
N=50, \tau=10000, stat=1500
exp | mastery |
mean | 0.0072 | 0.0072 | 0.0001 | 0.0000 |
std | 0.0004 | 0.0005 | 0.0004 | 0.0004 |
| std_mean | 0.0001 | 0.0001 | 0.0001 | 0.0001 |
mean | 0.9730 | 0.9740 | 0.0114 | 0.0031 |
std | 0.0562 | 0.0644 | 0.0562 | 0.0558 |
| std_mean | 0.0079 | 0.0091 | 0.0079 | 0.0079 |
Note that this is with 0 hit/exp natively, so this is as good as hit/exp get. I’ve increased the amount of stats to 1500, as you can see, which will make them a little less valuable because we’re averaging their contribution from 0 hit to 1500 hit rating. Those values will be lower than if we averaged from 0 to 500, for example, but the upside is that the simulation is more stable.
These results aren’t that surprising. With Berserker rage and significantly increased Rage generation from critical blocks, we’re already at or above the shield block rage threshold without hit/exp. In essence, we don’t need any hit or expertise to improve our Shield Block uptime, so they give us no significant TDR effect.
Also, as you can see, mastery has caught up and slightly surpassed dodge/parry now that the coefficient has been corrected.
The question of hit/exp’s value is something I’d like to investigate further. Right now I’m working on another set of simulations modeling short fight durations (i.e. 30-second windows), hopefully I’ll have those for you by Friday. After that, we’ll see what we can do about incorporating Shield Barrier. Though to be honest, I could use your help on that – what’s the logic that a warrior uses to determine when to use Shield Barrier?
It’s been a while since I last covered
and many of the spec’s core mechanics have changed since then, so it seemed like a good idea to revisit that calculation now that we have a more finalized implementation.
At first this seemed like it would be a pretty simple problem.
The avoidance calculations are similar to paladins (though the
$C_p=237.1$ and $C_d=90.4$, with $k=0.956$ – but that’s a trivial issue).
One of the major complications in the previous calculation was the , which was completely removed.
Unfortunately the rotation, which was , got significantly more complicated.
Revenge was tweaked to a 9-second cooldown that resets when the warrior dodges or parries, which throws a wrench into the works.
Whereas before we could so some simple 4-sequence modeling and solve the matrix, now that sort of modeling requires ~32 sequences, and the randomized cooldown on Revenge means that each sequence has up to 6 potential variants (based on the cooldown of Revenge at the beginning of the sequence), leading to 192 potential states to consider.
I think it’s possible to write the transition matrix for that system analytically, but the result will be complicated enough that it’s unlikely to simplify to something useful.
I’m still considering various approximation approaches so that we can find an analytic form, because it would be useful for the sorts of
I’ve done for paladins, or to put together a .
But that will probably have to wait until several other projects are finished.
In the meantime, however…..
A good friend of mine once said “If at any point a brute force strategy isn’t working, you’re not using enough brute force.”
And since I’ve been updating the paladin
simulation, it was pretty fast to rewrite it to create a warrior version.
So, brute force it is.
I was pleasantly surprised by the results.
One significant “feature,” if you will, of the new warrior mechanics is that there’s absolutely no haste scaling.
You don’t get rage from auto-attacks, nor are any of your ability cooldowns reduced by it (ala
for paladins).
This is significant for two reasons:
Warriors will be at a distinct disadvantage on fights with a big haste buff (, ) because their resource generation doesn’t scale with it.
This is an odd reversal of sorts, since Paladins were in this position in Cataclysm while rage tanks and DKs all gained a benefit from the raid-wide buff.
I’d also argue this is a big problem for Warriors, just as it was a
(looking at that old post… wow, it’s eerie how accurate Mel was).
It means that the Monte-Carlo simulation will run a lot faster.
If everything happens in lock-step with the GCD (1.5-second boss swing timer, everything else changing in 1.5-second intervals), I can use very large time steps in the simulation.
Whereas in the paladin Monte-Carlo, I need to use time steps of 10 milliseconds, the warrior one can function properly with 1.5-second time steps.
I’m using 500 milliseconds just to be safe, which means the simulation runs 50x faster than the paladin version (and then some, since I don’t even need to sim a haste stat weight).
I could get another factor of ~3 by boosting the time steps to 1.5-seconds, but when the simulation runs in minutes to the paladin version’s hours, it seemed unnecessary.
The first is bad news for warriors.
The second is good news, because it means I can run the simulations quickly and efficiently.
So without further adieu, let’s look at how we do it.
Simulation Set-Up and Methodology
The code for this section can be found in the
repository:
I’ve covered the details of how the Monte-Carlo works , but in short: we simulate a limited combat environment, generating random numbers (just like the game does) to determine the outcome.
By running the simulation long enough and accumulating enough data, we get statistically significant results.
We track a number of variables: the boss swing timer, the global cooldown, the durations of the , , and
buffs, the cooldowns of , , , , the internal cooldown on rage generation from , and the cooldown/charge mechanics for Shield Block.
At each time step, the simulation checks these variables to see if something should happen.
For example, if the boss swing timer is up, it’ll generate a boss attack and check to see if it’s avoided, blocked, critical-blocked, or taken as a full hit.
If we avoid, it’ll set the cooldown on Revenge to zero.
If the GCD is up, it’ll choose a spell to cast based on availability and priority, generate rage if appropriate, apply the appropriate buffs and debuffs, and so on.
The priority I’m using is:
Shield Slam & Revenge & Battle Shout & Thunder Clap & Devastate
This should be the maximum rage-generating priority queue, at the expense of slightly lower Weakened Blows uptime.
That’s probably fair, since in a two-tank situation you may not be responsible for keeping it up alone anyway.
I’m using the , so Shield Slam is generating 20 (25 if Sword and Board is up), Revenge is generating 15, and the internal cooldown on gaining rage from Critical Blocks is 3 seconds.
I’m also including the passive rage generation from
(1 every 3 seconds, i.e. every other GCD).
We start out with a warrior geared in LFR-level (483) Mists of Pandaria gear.
These are actually the stats of a paladin gear set I came up with for my MATLAB simulations, but they should work just as well for a warrior.
The point is to get a reasonable estimate of the amount of Strength and ratings at that gear level.
The stats I’m using are:
buffed Strength: 9208
parry rating: 4834
dodge rating: 4892
mastery rating: 6758
hit rating: 1521
expertise rating: 1777
haste rating: 0
On the character sheet, these translate to:
Mastery: 28.86%
Parry: 17.86%
Dodge: 10.45%
Block: 19.56%
Critical Block: 28.86%
and of course, against a L93 boss we lose 9% of that avoidance and 4.5% of the block.
To determine how effective each stat is at reducing damage, we calculate the total damage taken at those stat levels after 500,000 minutes of combat (50 “encounters” of 10,000 minutes each).
We then do it again after adding 500 points of whichever rating we’re interested in.
By comparing the two results, we can determine the amount by which we’ve reduced damage intake.
Simulation Results
With the stats I described above, here are the results.
I’ve normalized everything to mastery for clarity, so mastery gets a 1.000 stat weight and everything else is represented as it’s effectiveness at reducing total damage taken in comparison to one point of mastery.
exp | mastery |
mean | 2.6734 | 2.6448 | 0.2882 | 0.2120 |
std | 0.5378 | 0.5073 | 0.3969 | 0.4399 |
| std_mean | 0.0761 | 0.0717 | 0.0561 | 0.0622 |
As I discussed in , the mean (often represented as $\mu$) tells us the effectiveness of each stat.
($\sigma$, or “std”) tells us how much variation we’d expect to see in one “encounter.”
The ($\sigma_{\rm mean}$) tells us how much variation we’d expect to see in the mean if we ran the entire 500,000-minute simulation again.
Thus, we’d expect our mean values to be accurate to ~$\pm 2\sigma_{\rm mean}$, as that’s the 95% confidence interval.
In this simulation, that means we’re pretty confident in our values to ~$\pm 0.1$.
The results here were a little surprising.
Dodge and Parry are excellent, and hit and expertise appear to be abysmal.
That dodge and parry would be strong could have been predicted, perhaps, given the symbiotic nature of avoidance and Revenge.
Not only does avoiding reduce damage, it also gives you more rage to reduce even more damage with Shield Block.
Even so, I was surprised at just how far ahead of mastery both are.
Mastery has a small negative interaction with Shield Block, insofar as during the Shield Block uptime the block chance portion of mastery (~15% chance against the boss) is useless.
However, during that uptime the Critical Block portion (28.86% chance of a crit block) gets a lot stronger, which should more than compensate for the loss.
What surprised me more was the hit and expertise values – why are they so low? After all, the recent buff to rage generation was presumably to help correct rage starvation issues.
If warriors were rage-starved, wouldn’t resource generation stats like hit and expertise be very valuable?
Well, sort of.
The answer lies in some of the data I haven’t shown you yet.
Here’s a plot from a single 50,000-min simulation that shows the break down in hit size, along with some other statistics:
Histogram of our sample warrior’s damage events. 40% corresponds to a critical block, 70% to a regular block. 100% is a full-sized hit, 0% is an avoid.
The breakdown itself is interesting – only ~34% of their events are regular hits and many more are blocked or crit-blocked.
That’s some nice damage smoothing.
But the stats of interest to us right now is in the title.
S=65.9% is the uptime of the Shield Block buff, which is nearly at the cap of 66.6%, as enforced by the charge mechanic.
$R_{\rm rage}$ is the rate of rage generation, an amazing 6.713 rage per second.
That explains why hit and expertise aren’t valuable – we’re generating rage fast enough that we could cast Shield Block every 8.9 seconds, but we’re artificially limited to every 9 seconds by the cooldown.
This is reflected in the “average time between Shield Blocks” statistic, $T_{\rm sb}=9.1 s$.
Adding more rage generation is mostly worthless at this point, because we can’t actually use that rage.
In fact, the only reason the stats have any value at all is because this is a Monte-Carlo.
In an analytical calculation, we’d be capped and they’d drop off in value to zero.
But since this is a statistical simulation, there will be cases where we miss a few generators in a row and end up in a rage drought, leading to slightly lower Shield Block uptime (and thus more damage taken).
That’s why $T_{\rm sb}$ is 9.1 and not 9.0.
Hit and expertise just smooth out those droughts, giving us a marginal but low value.
If I track rage generation in the simulation, I find that out of the total ~20 million rage generated, we “waste&# thousand of it because we can’t use it (technically we can convert that to Shield Barrier usage, but the sim doesn’t know it – we’ll talk about this more later).
To test this, let’s artificially set the hit and expertise on our gear to zero.
In theory, that should drop our rage generation and Shield Block uptime, which should give us a better feel for how much value hit and expertise have when we’re not bumping into the “rage cap” (nominally 60 rage every 9 seconds, or 6.6667 rage per second).
If we do that, we get:
exp | mastery |
mean | 2.4602 | 2.3382 | 1.0757 | 1.1019 |
std | 0.5113 | 0.5616 | 0.5172 | 0.4833 |
| std_mean | 0.0723 | 0.0794 | 0.0731 | 0.0684 |
And here’s the associated histogram:
Histogram of our sample warrior’s damage events at 0 hit/exp. 40% corresponds to a critical block, 70% to a regular block. 100% is a full-sized hit, 0% is an avoid.
So if we’re below the rage threshold (and we are, at ~6 rage per second), hit and expertise are quite valuable because they have a measurable effect on Shield Block uptime.
Note that our Shield Block uptime has dropped to 60.6%, and the corresponding time between Shield Blocks has increased to 9.9 seconds.
This is reflected further in the rage generation analysis: we generate about 18 million rage and “waste” only a little over 8 thousand.
We can run a simulation to see exactly how these stat weights scale with expertise.
Using , I run the simulation at expertise values from 0% to 15% in 1% steps.
Here are the associated values, along with the mean rage generation rate (“RPS”) and Shield Block uptime (“SBU”):
std | std_mean |
1.0000 | 0.3813 |
0.0539 | 6.0611 | 0.6155 |
0.8916 | 0.3962 |
0.0560 | 6.1278 | 0.6220 |
0.8908 | 0.4396 |
0.0622 | 6.1986 | 0.6284 |
0.8274 | 0.4781 |
0.0676 | 6.2652 | 0.6345 |
0.8182 | 0.4092 |
0.0579 | 6.3311 | 0.6401 |
0.7261 | 0.3817 |
0.0540 | 6.3990 | 0.6457 |
0.6671 | 0.3972 |
0.0562 | 6.4667 | 0.6505 |
0.5626 | 0.3531 |
0.0499 | 6.5332 | 0.6548 |
0.3836 | 0.3612 |
0.0511 | 6.6002 | 0.6584 |
0.3374 | 0.4133 |
0.0584 | 6.6653 | 0.6611 |
| 10.0000 |
0.1987 | 0.3614 |
0.0511 | 6.7291 | 0.6633 |
| 11.0000 |
0.0835 | 0.3768 |
0.0533 | 6.7937 | 0.6647 |
| 12.0000 |
0.0599 | 0.3666 |
0.0518 | 6.8561 | 0.6656 |
| 13.0000 |
0.0254 | 0.3691 |
0.0522 | 6.9206 | 0.6661 |
| 14.0000 |
0.0513 | 0.3546 |
0.0501 | 6.9834 | 0.6663 |
| 15.0000 | -0.0373 | 0.3645 |
0.0515 | 7.0452 | 0.6663 |
And here are the associated plots:
The expertise stat weight (normalized to its value at 0% expertise) as overall expertise value is increased.
As expertise is increased rage generation increases linearly, but Shield Block uptime saturates to its asymptotic value of 0.6667. Datatips are shown to highlight certain values.
The data is a little noisy, but it’s pretty clear that the value of expertise drops off pretty steadily up to around 9%.
From the bottom graph, we see that 9% is where we hit the 6.667 rage threshold, and correspondingly we see a “knee” in the Shield Block uptime curve.
Shield Block’s uptime is saturating to it’s asymptotic value of 66.67%, and this “knee” is where we start really feeling the effects of that cap.
Since warrior skills are equally affected by hit or expertise, we can surmise that hit behaves similarly, and what really matters is that the sum of hit and expertise is 9% (though of course, hit over the 7.5% cap doesn’t count!).
Beyond the 9% threshold, SB uptime only increases slowly and the value of hit or expertise is primarily due to the smoothing of rage droughts that I described earlier.
The bulk of the rage generation beyond this point is what we consider “wasted” in this simulation (more on that in a second).
So we can reasonably say that any hit or expertise beyond a total of 9% (i.e., hit+expertise=9%) has little to no effect on Shield Block uptime.
It’s also worth noting that the TDR value doesn’t see a sharp drop-off at 9%.
The TDR value steadily decreases up to around 11%, after which it hits rock-bottom.
The steepest descent seems to occur from 6% to 10%.
But by 7.5% expertise, the TDR value is at half of its original value.
As a brief aside, I’m wondering if the recent rage generation buffs were even necessary.
At zero hit and expertise, warriors already get 60% coverage with Shield Block, and they can’t realistically improve that very much with hit and expertise.
Perhaps that’s intended though, and the excess rage is supposed to go into Shield Barrier usage.
I’m not as clear on what the design goals are for warriors.
In fact, it’s about time we addressed the issue of Shield Barrier.
The simulation doesn’t do anything to try and model Shield Barrier, partly because it’s so variable.
I could probably implement it under a few circumstances.
I’d have to track Vengeance somehow, which isn’t impossible, but would slow the simulation down a bit.
We’d also have to approximate Shield Barrier’s absorb according to our normalized Vengeance value, which I haven’t had time to do.
However, we can come up with a conceptual estimate based on the assumption that all of the “wasted” rage can be converted to Shield Barrier casts.
There would obviously be a little attrition, in that you might push Shield Block back a few seconds if you use a 60-point Shield Barrier at the wrong time, but overall the result shouldn’t be that far off from the approximation.
Even in our near-capped case, where we “wasted&# thousand rage (which is about 1.8% of all rage generated), that’s still only an extra Shield Barrier every ~5000 seconds, or every ~83 minutes.
We can do the same analysis using our expertise scaling data.
At 14% expertise we’re generating 6.983 rage per second.
If we assume that everything over the 6.667 RPS we need to maintain Shield Block gets turned into Shield Barriers, that’s 0.316 rage per second, or another Shield Barrier every ~190 seconds (about every 3 minutes).
So that extra 6% expertise beyond the 9% threshold buys one Shield Barrier every fight, maybe two on a 6+ minute encounter.
That’s doesn’t seem like a lot of survivability for a 2040 rating investment (then again, it only takes one sometimes…).
Conclusions
I think these simulations were quite fruitful.
It’s clear that the results differ enough from the paladin ones that a separate analysis was warranted.
I’ve seen a lot of ambiguity regarding stat weights for prot warriors, and these simulations bring statistical evidence into those discussions.
For one thing, the value of hit and expertise seems a lot lower than many assumed, and their value at level 90 will end up being directly correlated to how important extra Shield Barrier casts are.
That may even be a player-specific decision rather than a hard-and-fast rule.
In addition, I think everyone will be a little surprised at how strong avoidance is for warriors.
The synergy between Revenge and dodge/parry is a fearsome thing, effectively turning dodge/parry into rage generation stats.
The warrior that eschews hit/exp and stacks dodge/parry may not suffer much as far as rage generation is concerned simply because of Revenge.
Rather than having to choose between a resource generation stat and an avoidance stat, like a paladin does, the warrior can stick with avoidance and get both.
I’d say that this devalues hit and expertise, and that every warrior will be stacking avoidance to the sky, but hit and expertise still give an element of control that’s worth considering.
High avoidance may give you enough resource generation to maintain a fairly high Shield Block uptime, but that avoidance-based rage income is just as stochastic as the damage reduction it grants.
You could easily get doubly screwed by a string of missed attacks occurring simultaneously with a string of unavoided attacks, leaving you high and dry.
At least with respectable hit/exp values, you have more control over your rage income.
That said, Shield Block’s 2-charge mechanic and the ability to pool 120 rage mitigate this problem quite a bit.
That might be enough control for a skilled warrior, making the sheer rating investment of hit- and expertise-cap unattractive for such a small benefit.
And we have shown that it only takes about 9% total hit+expertise to reach the point where Shield Block is available as often as it can be.
It wouldn’t surprise me if avoidance-stacking became the dominant gearing strategy for warriors even while the hit/exp/mastery/haste strategy becomes dominant for paladins.
As a brief footnote: The warrior diminishing returns equations are a little different than the paladin ones, and seem to have been balanced pretty well given our gear set with roughly equal amounts of dodge and parry rating.
Remember, the stated design goal was for our rating values to be approximately equal after the extra parry-&strength conversion.
I find myself wondering if that’s why the warrior equations were changed again.
They were identical to paladins at one point in beta, but the most recent change adjusted the scaling and the dodge cap.
I also wonder whether the paladin dodge cap will be adjusted similarly before L90 raiding.
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Several people have commented on how much they liked my WeakAuras setup for protection, and asked if I had any ready-made setups for other classes.
As it turns out, I do.
Over the past week, I’ve been setting my UI up on various alts.
Unfortunately, I only have versions set up for classes and specs I play, so it’s not a comprehensive list.
Most of them have a similar structure to my protection UI, with slight variations.
The ones that differ the most tend to be caster classes (Warlock, Mage, and Moonkin) because they get built around a DoT timer (NeedToKnow), so they break the mold.
I’ll include them nonetheless, so that if people want to fool with them, they can.
Unlike Theck’s UI, I’ve set up most of the other classes to be modular.
For example, both Blood and Unholy care about disease timers, so I have a “DK-Any” group that includes those.
The “DK-Blood” and “DK-Unholy” groups only contain the parts specific to one spec or the other.
Thus, my DK’s blood UI is the “DK-Blood” group plus the “DK-Any” group.
Also, fair warning, I’ve set most of these UIs up but I haven’t played or raided with most of them.
There may be bugs/errors/etc. that I missed.
I probably won’t be keeping these as up-to-date as my protection paladin setup, so use at your own risk.
Updated 9/18/2012 with additional changes/fixes/etc.
DK-Any: Disease timers/reminders, Outbreak cooldown, Horn of Winter reminder
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DK-Blood: Presence reminder, Death Strike, Rune Strike, Heart Strike, Vampiric Blood, Rune Tap, AotD, Icebound Fortitude, Dancing Rune Weapon, Bone Shield, AMS, and Death Pact.
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DK-Unholy: Dark Transformation ready indicator, DnD, Scourge Strike, Festering Strike, Death Coil, Gargoyle, Unholy Frenzy, Horn of Winter (rotational), Dark Transformation buff tracker, and Presence/Ghoul reminders
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Druid-Balance: Spellsurge, Moonfire/Sunfire trackers, Treants/Starfall indicators. This is one that’s barebones because it’s built around NeedToKnow timers and an eclipse bar.
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Druid-Guardian: Rage tracker, Savage Defense indicator, Mangle, Lacerate, Thrash, Swipe, Berserk, Barkskin Might of Ursoc, Survival Instincts, and Weakened Blows tracker
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