Diminishing Returns on Avoidance? Not really.

[Kerg]

First of all, I am usually an EH first guy. But I do value Avoidance. EH saves you from spike damage. But often the reason you take spike damage in the first place is because you get a string of 3 hits in a row from a boss, and fail to avoid at least one. So I think as you focus on high EH, you should also keep stacking avoidance as well, wherever you can pick it up, so you end up taking spike damage less often.

Anyway, that is not the topic of my thread.

My concern is that many people may be getting the perception that, because Avoidance in WotLK is now subject to diminishing returns, stacking Avoidance is bad now. Like the more they get, the worse it is to get more. I’ve seen people say that avoidance trinkets suck because after you factor in diminishing returns, the effect you get is not nearly as good as you would think. This is very flawed thinking, because what often fails to get mentioned is that, although Avoidance is subject to diminishing returns, it is also at the same time subject to significant increasing returns.

For example, when you are at 0% Avoidance, 100% of all blows are hitting you, so adding +1% Avoidance reduces the chance that you avoid an incoming blow by 1/100 = 1%. But when you are at 50% Avoidance, only 50% of all blows are hitting you, so adding +1% Avoidance reduces the chance that you avoid an incoming blow by 1/50 = 2%. Effectively, adding +1% Avoidance at 50% is worth TWICE as much as adding it at 0%.

At 75% Avoidance, adding +1% Avoidance reduces the chance that you avoid an incoming blow by 1/25 = 4%. Effectively worth 4 TIMES as much as adding it at 0%.

This is basically the reason for the WotLK Avoidance nerf, IMO. Blizzard was worried that tanks (or rogues ) would be able to exploit the game by stacking Avoidance to ridiculous levels by popping trinkets, etc., and thus would be getting hit so infrequently it would make keeping the tank alive trivial.

Anyway, I ran some numbers based upon Satrina's formula here for calculating Avoidance.

It is not a perfectly linear balancing out effect like Armor is, as Satrina's famous example way back when showed, but in my example, it still canceled out the diminishing returns effect. In fact, as you’ll see, I am still getting slightly increasing returns from adding any form of Avoidance.

I used the values currently on my gear. I added base 9.96% Dodge, 10% Parry, and 5% Miss for a naked warrior with the two +5% avoidance talents.

Agility from Gear: 44
Defense Rating from Gear: 717
Dodge Rating from Gear: 512
Parry Rating from Gear: 176

Dodge: 26.48345%
Parry: 18.13830%
Miss: 9.41599%
Total Avoidance: 54.03774%

I then calculated the Avoidance added by hybrid gems (X & 12 Stam.) at my current gear level (54.03774% Total Avoidance)...

+8 Agility: +.07505%
+8 Defense Rating: +.12693%
+8 Dodge Rating: +.14012%
+8 Parry Rating: +.11597%

Then I calculated how much Avoidance those same gems would add to a naked warrior (24.96000% Avoidance)...

+8 Agility: +.11366%
+8 Defense Rating: +.20373%
+8 Dodge Rating: +.21216%
+8 Parry Rating: +.16952%

There you can see the Diminishing Returns effect. Adding +8 Dodge Rating to a naked warrior you get +.21216% Avoidance. Adding it to my Naxx/Ulduar gear only nets me +.14012%. So adding Dodge Rating now nets me only about 66% of the Avoidance I got from my first Dodge Rating pieces added when naked.

HOWEVER, when I compare the Effective Avoidance gained, i.e. the % reduction in blows that will actually hit me, given my current level of Total Avoidance, the differences are MUCH MUCH smaller. And in fact, not only is there not a decreasing return, there is actually a slight increasing return.

Effective Avoidance added from those gems at my current gear level (54.03774% Total Avoidance)...

+8 Agility: +.16329%
+8 Defense Rating: +.27616%
+8 Dodge Rating: +.30486%
+8 Parry Rating: +.25232%

Effective Avoidance added from those gems to a naked warrior (24.96000% Avoidance)...

+8 Agility: +.15147%
+8 Defense Rating: +.27150%
+8 Dodge Rating: +.28273%
+8 Parry Rating: +.22591%

As you can see, comparing Effective Avoidance, current gear vs. naked, the numbers are very close, and in fact, not only are there no diminishing returns in terms of Effective Avoidance, there are slightly increasing returns. Adding Dodge Rating now nets me about 108% of the Effective Avoidance I got from my first Dodge Rating pieces added when naked.

Of course, I’m sure this is nothing new to the administrators of this site or those who understand how it works (and btw, I have been a big fan of this site for a while ). But I thought I’d share my example, because there is a lot of misinformation out there, and nowhere did I find this explained clearly.

Note: If anyone wants to see my math, I can post it. And if anyone sees any flaws in my calculations, feel free to point it out so I can correct it. Thanks.

[Xav]

I think a lot of people were aware of this, indeed, as you said. What I would be curious to see, though, is a full table with a wide range of numbers. Rather than the two polar ends: Zero gear on, and fully geared, perhaps with gear equipped, and then an ever increasing amount of avoidance on the tank, to see if when you're at 60-70% avoidance, how much worth the same +8 dodge rating is worth at that point, etc.

I know blizzard designed it to stay somewhat equal on all ends of the scale, but I'd still like to see it!

[Kerg]

I think a lot of people were aware of this, indeed, as you said. What I would be curious to see, though, is a full table with a wide range of numbers. Rather than the two polar ends: Zero gear on, and fully geared, perhaps with gear equipped, and then an ever increasing amount of avoidance on the tank, to see if when you're at 60-70% avoidance, how much worth the same +8 dodge rating is worth at that point, etc.

I know blizzard designed it to stay somewhat equal on all ends of the scale, but I'd still like to see it!

Hmmm... I was playing with the numbers again, and going from naked to fully geared and then comparing Dodge then vs. Dodge now is a little misleading now that I look at it. The reason is because Total Avoidance also takes your other types of Avoidance into play. So part of the reason the benefit of +8 Dodge Rating goes up when you jump from naked to full geared is because of the fact that your Parry and Miss are also going up significantly. Basically, the more Parry and Miss you have, the more your Dodge is worth, and vise versa. So it's a little more complicated than I thought.

I'll play with it some more, though, and try to come up with a graph.

[MrDuck]

Thanks for the post, i was trying to explain this to plenty of people, but there's tons of tanks out there who just state "avoidance sucks because of diminishing returns killing it" or such, might just point them to this instead of hours of explaining why it just is so. I'm rather interested in these graphs if you can push some, i'm already at over 30% dodge with raid buffs, would be nice to see how is the curve looking like at such avoidance levels.

[jere]

I think a lot of people were aware of this, indeed, as you said. What I would be curious to see, though, is a full table with a wide range of numbers. Rather than the two polar ends: Zero gear on, and fully geared, perhaps with gear equipped, and then an ever increasing amount of avoidance on the tank, to see if when you're at 60-70% avoidance, how much worth the same +8 dodge rating is worth at that point, etc.

I know blizzard designed it to stay somewhat equal on all ends of the scale, but I'd still like to see it!

This isn't exactly what you are looking for probably, but it is in that direction: If you take the derivative of the DR equation with respect to the avoidance type, you get "how effective" your "next" dodge % is. I.E. your "next" 1% dodge really ends up being roughly __% dodge after DR. Now it won't tell you exactly how much dodge you would get from the next 8 rating points, but it gives you an idea of where you are at:

               (C_d^2)*k
dodge_ratio = -----------
              (d+C_d*k)^2

I called it dodge_ratio for lack of better wording. C_d is the C constant for dodge, k is the k constant for your class, d is your dodge % before DR but without talents and base/naked values, which aren't affected by DR (I have an equation for using the "after DR" value on your char sheet, but I need to hunt that up).

So if you have like say 30% dodge from dodge rating, defense rating, and extra agility (before DR), then your dodge ratio is

(88.129021^2)*0.956/(30+88.129021*0.956)^2 = 0.5688

So, roughly, you are at the point where your next 1% dodge will yield 0.56% dodge after DR. I say roughly because if the slope of the DR function is changing quickly, it will be less than that. But it gives you a ballpark of where you are currently.

We were talking about it at maintankadin a couple of weeks ago. Here is a graph for the dodge curve. The blue line is this "dodge ratio" I am talking about, the green line is simply dodge after DR, and the pink line is just there for visualization (it's dodge before DR plotted against dodge before DR so you can see how DR deviates from "before" DR).

[Kazeyonoma]

sexy graph =D

[Kerg]

Okay, here's what I came up with...

I took Agility out, giving it a value of zero from gear. I set Defense Rating from Gear = 689, which the minimum needed to reach 540 Defense, and then I froze it there.

Then I looked at how you would gear from there. My Dodge Rating to Parry Rating on my current gear is about 3 to 1. So starting with these stats...

Defense Rating from Gear: 689
Dodge Rating from Gear: 0
Parry Rating from Gear: 0

... I raised Dodge Rating and Parry Rating in increments of +30 Dodge Rating & +10 Parry Rating, all the way up to 75% Total Avoidance. And I calculated the Avoidance gained from the last increment of 30/10 Dodge/Parry Rating. Then I calculated the Effective Avoidance gained from last increment of 30/10 Dodge/Parry Rating. And then I graphed it.

As expected, Avoidance gained goes down each increment, reflecting Diminishing Returns. But Effective Avoidance gained for each increment is pretty flat, with a gradual upward slope reflecting net increasing returns.

[jere]

Did the math quickly at home

So this is the general equation if your input avoidance value is before DR:

               (C_d^2)*k
dodge_ratio = -----------
              (d+C_d*k)^2

Which is what I posted earlier.

And here is the general equation if your input avoidance is after DR:

              (C_d - d)^2
dodge_ratio = -----------
               (C_d^2)*k

As an example, according to the armory, Xav has 24.66% dodge after DR. Using the second equation:

              (88.129021 - 24.66)^2
dodge_ratio = ---------------------  =  0.542535
               (88.129021^2)*0.956

So right now, Xav is at the point where his dodge is about 54.2535% efficient, for lack of better wording.

You can extrapolate this to find equations where you compare parry to dodge, but for all practical purposes dodge is better to stack (I did the comparisons over on maintankadin if anyone is interested).

EDIT:
Here are the equations for both parry and dodge graphed versus dodge and parry after DR is considered:

So taking the previous example, if Xav has 24.66% dodge after DR, then the graph shows he has a little over 54% efficiency in dodge. Additionally, if Xav has 18.71% parry after DR, then he has a little over 38% efficiency in parry. Those are "percentages" mind you and not ratings. If you wanted to look at how efficient a rating point is, then you would need to adjust using the combat rating coefficients (parry takes 25% more rating per pre DR % than dodge does)

EDIT: As a side note, when I refer to efficiency, I mean the efficiency of the "next" amount you add, not the efficiency of what you have already. Basically, how good is the "next" point of dodge or parry going to be.

[Satorri]

Your point is a good one, many people who don't understand the nature of the mechanics eschew avoidance as no longer a valuable stat (missing the fact that it has brought down to linear value like armor always has been, zomg armor diminishes?? Stop getting gear with armor!!).

You lost me on your math though. Did you pull out the EJ formula for diminishing avoidance and just plot lots of values as if you were adding rating to gear?

And "effective avoidance gained" what do you mean by that? When you say "total avoidance" you're counting these diminished dodge+parry? Are you factoring miss? Are you factoring the sliding scale as the avoidance value from defense is also reduced as you have more avoidance from direct ratings?

Forgive all the questions, I get the sense we're not using common terms for expressing values.

[Xav]

I think I get your graph, kerg, but I'm not so sure. The pink line represents the % you're gaining. The blue is what your actual avoidance is.

The pink curve going up represents that your overall value (real avoidance gained) from each point of dodge is still getting more valuable, all the way up to 75%.

So in layman's terms, even when you're adding "1%" dodge from gear and it only actually adds ".4%" dodge to your character screen, doing that was significantly more beneficial than the previous "1%" you added.

I suppose it's all simple math, and a formula, and they indeed designed it all just dandy - avoidance DR's, but it's always good, and still ends up being extremely efficient at the high end.

[Magnuss]

Avoidance works like armor, the more you get the better, but the less each bit is worth on its own.

^That is what I extracted from you math guys

Edit: Just out of curiosity, does this change our line of thinking in terms of gearing/geming at this point? Or is it still Effective Health is king until you are farming Algalon kind of stuff?

[Trondhjem]

Hey, wow, that's genious... so if I understood this right it means about that:

from 0% to 1% Avoidance you get 1% damage reduction without diminuishing returns.
from e.g. 50% to 51% Avoidance, you get 2% damage reduction but with -- let's say 0.6 -- diminuishing returns.
given that the diminuishing returns are always 'cheaper' than your effective damage reduction, it's still no 'waste' to go for avoidance.

However I'm more the EH guy, but with that JC nerf I have to go for some socketboni, so looking for some alternatives can't hurt

[Darksend]

Avoidance works like armor, the more you get the better, but the less each bit is worth on its own.

I have been trying to say this since i saw the OP yesterday without all the fancy math but i fail at putting things into words.

And as xav said this is something most of us already new but i dont know if any of us actually knew the specifics.

wonder how the numbers work for druids since we cant parry

[Satorri]

Aye, for those just tuning in, avoidance has been given the same sort of mechanic as armor for the same reason.

Because these work on a limited scale (i.e. 100% is complete immunity) they are given a diminishing return so that the effect on time-to-live is actually linear, just like straight health.

In BC avoidance was less common directly from gear and you had to work to add it, but the lack of diminishing returns meant that a very well geared tank would actually get significant survival value from high avoidance (though people were also very wary of it when it wasn't required since you had to give up more static damage reduction leaving the shots between avoids the opportunity to do really painful damage and/or crushing blows of doom).

[Kerg]

I will post the math behind the graph I did this afternoon, when I get some time.

[CKaz]

Maybe its clear from the above but I thought a simple 3 swing analysis would do the trick, lets look at the percent chance you'll get hit 2 and 3 times in a row

At 30%/40%/50%/60% avoid

30% -> 2 hits 49% -> 3 hits (.7)^3 = 34.3%
40% -> 2 hits 36% -> 3 hits (.6)^3 = 21.6%
50% -> 2 hits 25% -> 3 hits (.5)^3 = 12.5%
60% -> 2 hits 16% -> 3 hits (.4)^3 = 6.4%

This paints a lot of pictures and lends even more analysis
[Again apologies if covered above, I made my own avoidance check on the graph ]

Each step is quite impressive, and frankly where it looks like it might be losing some of its edge, its gaining. Lets take 50% to 60% for example.

50% -> 2 hits 25% -> 3 hits (.5)^3 = 12.5%
60% -> 2 hits 16% -> 3 hits (.4)^3 = 6.4%

Not quite 10% better in either category right? Oh contraire.
Comparing the two the guy at 60% will take 2 hits in a row and 3 hits in a row almost half as much as the 50% guy. This is actually just picking up steam.

60% -> 2 hits 16% -> 3 hits (.4)^3 = 6.4%
70% -> 2 hits 9% -> 3 hits (.3)^3 = 2.7%

2 Hits is even closer to half, and 3 in a row? We've surpassed it now.

70% -> 2 hits 9% -> 3 hits (.3)^3 = 2.7%
80% -> 2 hits 4% -> 3 hits (.2)^3 = .08%

And scene.
You can clearly see the need to 'diminish' returns as returns get silly good.
This last iteration shows the 80% guy avoiding the double-hit more than twice as much and the triple hit less than a third of the time the other guy needs to take it.

Definately there is a need to 'run the numbers' in scenarios, e.g. that helps you figure out where Crit is good IMO too. Going from never crit to every 10th/5th/3.x swing is a pretty significant for each 10% crit jump. But it continues to tail off as you go.

Here is the reverse - it starts pretty slow if you have little avoidance, but it ramps up, getting stellar at high levels

edit - part of the core of the 'DK drama' as the avoidance tank right, possibly even moreso for Death Nights
[not a typo - Death Knight Night Elves ]

[Satorri]

As much as I loathe probability in real life it is a great point in the game, as that is what we want to rely on Avoidance for most, the holes in which we don't take damage that gives healers time to restore our health.

Thanks for sharing that.

[Kerg]

The Graph



The Math Behind It

First of all, I tried to simulate a simple gearing method that many people might use. I think the majority of MT’s first focus on getting to 540 Defense to become uncrittable (689 Defense Rating). And then focus on Effective Health (Stamina and Armor), while picking up Avoidance where they can get it without sacrificing much Effective Health.

Most pieces have Defense on them, and rarely do Effective Health-focused tanks go too far above the 689 needed for uncrittable. So to simplify things, I froze the amount at 689 for this model. I also took Agility out of the equation for simplicity, since there is not much available on tanking gear anyway.

That leaves Dodge Rating and Parry Rating as the variables. All else being equal, pieces with Dodge are favored over pieces with Parry, because Parry yields significantly less Avoidance per rating point. Parry gems are never used. But 8 Dodge/12 Stam. gems are often used in red sockets to pick up socket bonuses, especially since many socket bonuses yield more Stamina. However, many of the best tanking pieces have Parry on them, so you inevitably get some of it on your gear. My current gear has about 3 times as much Dodge Rating as Parry Rating, so I chose this ratio as my gearing up model.

So... I started out with these variables...

Agility_from_Gear = 0
Defense_Rating_from_Gear = 689
Dodge_Rating_from_Gear = 0
Parry_Rating_from_Gear = 0

And these constants...

Naked Dodge = 5 from Anticipation talent + 4.96 base = 9.96
Naked Parry = 5 from Deflection talent + 5 base = 10
Naked Miss = 5
Defense/Defense Rating = 4.9185
Dodge/Defense = .04
Parry/Defense = .04
Miss/Defense = .04
Base Dodge Rating/1% Dodge = 39.34799
Base Dodge Rating/Agility = .01360
Base Parry Rating/1% Parry = 48.18499
Warrior Diminishing Returns coefficient = k = .956
Warrior Dodge cap = c = 88.129021
Warrior Parry cap = c = 47.003525
Warrior Miss cap = c = 16

And then I calculated Total Avoidance using Satrina’s step-by-step formula in Diminishing Returns - Avoidance.

Total Avoidance = Dodge + Parry + Miss

Dodge = Ad = 1/(1/c+k/A) = 1/[(1/88.129021)+.956/[((Defense_Rating_from_Gear/4.9185)*.04)+(Dodge_Rating_from_Gear/39.34799)+(Agility_from_Gear*.01360)]]+9.96 = 15.45572%

Parry = Ad = 1/(1/c+k/A) = 1/[(1/47.003525)+.956/[((Defense_Rating_from_Gear/4.9185)*.04)+(Parry_Rating_from_Gear/48.18499)]]+10 = 15.21138%

Miss = Ad = 1/(1/c+k/A) = 1/[(1/16)+.956/((Defense_Rating_from_Gear/4.9185)*.04)]+5 = 9.28977%

Total Avoidance = 15.45572+15.21138+9.28977 = 39.95687%

That’s where I started. Then what I did was add increments of 30 Dodge Rating and 10 Parry Rating (reflecting the 3 to 1 ratio I chose for the model) all the way up to 75% Total Avoidance.

For each one of those increments I calculated the actual Avoidance Gained...

Avoidance Gained = (Total Avoidance +30 Dodge & 10 Parry Rating) - (Total Avoidance)

Going from 0 Dodge Rating and 0 Parry Rating to 30 Dodge and 10 Parry Rating, you get...

Avoidance Gained = 40.81958-39.95687 = .86271%

For the next increment, you go from 30 Dodge Rating and 10 Parry Rating to 60 Dodge Rating and 20 Parry Rating, and get...

Avoidance Gained = 41.66935-40.81958 = .84977%

This number for each increment reflects the Avoidance Gained by adding 30 Dodge/10 Parry, including Diminishing Returns. This represents the lower downward sloping blue curve in the graph.

Then I calculated Effective Avoidance Gained, which represents the reduction in blows that are actually getting through given your current Total Avoidance. (See the above example in the OP, where I showed that at 50% Avoidance, adding 1% Avoidance actually raises your Effective Avoidance by 2%.) This is calculated like this...

Chance To Be Hit = 100-Total Avoidance

(Avoidance Gained/Chance To Be Hit) = (Effective Avoidance Gained/100)

Plug and solve and you get...

Effective Avoidance Gained = (Avoidance Gained*100)/(100-Total Avoidance)

Going from 0 Dodge Rating and 0 Parry Rating to 30 Dodge and 10 Parry Rating, you get...

Effective Avoidance Gained = (.86271*100)/(100-39.95687) = 1.43681%

For the next increment, you go from 30 Dodge Rating and 10 Parry Rating to 60 Dodge Rating and 20 Parry Rating, and get...

Effective Avoidance Gained = (.84977*100)/(100-39.95687) = 1.43515%

This number for each increment reflects the Effective Avoidance Gained by adding 30 Dodge/10 Parry. This is the reduction in blows that are actually getting through given your current Total Avoidance, and is what really matters when looking at how avoidance benefits you. This represents the upper violet curve in the graph.

Clear as mud? Sorry, I made up terms where I didn’t know what to call them, but hopefully I defined them well in this explanation. And I am not a mathematician. I love playing with numbers, but I haven’t taken any math classes since Calculus and Stats in college, and don’t really do any math at work unless I’m playing with WoW stats.

Here are the plot numbers from the graph....

[Agmar]

So in a nutshell this means - defense and dodge are still very good, but half as good as they were, but we just won't need sunwell radiance.

[shez]

Effective Avoidance Gained = (Avoidance Gained*100)/(100-Total Avoidance)

I don't understand this, how do you get this formula? And what exactly is effective avoidance?