Diminishing returns and defense vs dodge

[quadturtle]

I've been struggling a bit to work out how the diminishing returns on avoidance should affect my gearing choices and so I worked myself out a little example and I was slightly surprised by the results.

I decided to mimic Satrina's examples on how constant increases in armour will increase your survivability in a linear fashion. The formula's I used to calculate the this example were all taken from Satrina's post on diminishing returns.

For this example I've assumed we have a character with 100hp and no mitigation and no avoidance who is being attacked by a mob who attacks once a second for one damage. The numbers are a very unrealistic I know, but maths is not my strongest point and it made it easy for me to understand.

In the table below each row shows the percentage dodge chance before diminishing returns (A), after diminishing returns (Ad), the amount of dodge rating required to get this dodge % (Rating) and the number of seconds you would expect to live on average if being attacked by the example mob (TTL or time to live). Each row is a 5% increment in raw dodge.

Dodge			
Rating       A          Ad          TTL
0.000000     0.000000   0.000000    100.000000
196.739950   5.000000   4.937126    105.193537
393.479900   10.000000  9.350426    110.314914
590.219850   15.000000  13.319067   115.365625
786.959800   20.000000  16.907028   120.347122
983.699750   25.000000  20.166576   125.260818
1180.439700  30.000000  23.140827   130.108088

If you calculate the deltas between the rows you get the following

Delta A     Delta Ad    Delta TTL
0  -> 5     4.937126    5.193537
5  -> 10    4.413300    5.121377
10 -> 15    3.968641    5.050711
15 -> 20    3.587961    4.981497
20 -> 25    3.259548    4.913696
25 -> 30    2.974251    4.847270

As you would expect increasing your raw dodge by a constant amount increases your actual dodge be increasingly small amounts what surprised me was the fact that the increase in your time to live is also decreasing i.e each point of dodge is actually worth less than the last point added. I had assumed that the changes to avoidance would bring it in line with mitigation from armour so that each point of dodge would be worth the same amount as the last point added.

Here's the same thing for parry

Parry			
Rating       A          Ad          TTL
0.000000     0.000000   0.000000    100.000000
245.924950   5.000000   4.706436    104.938881
491.849900   10.000000  8.556150    109.356725
737.774850   15.000000  11.763553   113.331852
983.699800   20.000000  14.477018   116.927634
1229.624750  25.000000  16.802487   120.195900
1475.549700  30.000000  18.817626   123.179447

Delta A     Delta Ad    Delta TTL
0  -> 5     4.706436    4.938881
5  -> 10    3.849714    4.417844
10 -> 15    3.207403    3.975127
15 -> 20    2.713465    3.595782
20 -> 25    2.325469    3.268266
25 -> 30    2.015139    2.983547

The lower cap on parry means that it is affected by the diminishing returns much faster. Given that its also the most expensive avoidance stat it seems like quite a poor thing to stack for avoidance.

There's been a couple of posts on defense vs dodge so I wanted to see how they compared with each other. The table below shows how much defense skill you get for the same amount of rating from the dodge table above. It shows how much raw dodge (Dd), parry (Dp) and miss (Dm) you would get from this and then the total raw avoidance (A) this gives you. I'm ignoring block rating in this example and again I'm assuming the character has no other avoidance.

Defense					
Rating      Skill       Dd          Dp          Dm          A
0.000000    0.000000    0.000000    0.000000    0.000000    0.000000
196.739950  39.999990   1.600000    1.600000    1.600000    4.799999
393.479900  79.999980   3.199999    3.199999    3.199999    9.599998
590.219850  119.999970  4.799999    4.799999    4.799999    14.399996
786.959800  159.999959  6.399998    6.399998    6.399998    19.199995
983.699750  199.999949  7.999998    7.999998    7.999998    23.999994
1180.439700 239.999939  9.599998    9.599998    9.599998    28.799993

The tables below show the diminished avoidance levels for raw dodge and parry from defense in the above table. Rather curiously at very low levels of avoidance the diminishing returns formula actually seems to increase the amount of avoidance you get.

Defense Dodge	
A           Ad
0.000000    0.000000
1.600000    1.642448
3.199999    3.224797
4.799999    4.750284
6.399998    6.221922
7.999998    7.642512
9.599998    9.014665

Defense Parry	
A           Ad
0.000000    0.000000
1.600000    1.616096
3.199999    3.124755
4.799999    4.536346
6.399998    5.859946
7.999998    7.103532
9.599998    8.274148

Since the cap does not appear to be known for miss at the moment I calculated it with two values. The first cap I set at 100% miss and the second I set at 0.000001% miss (I would have used 0 but it made the spreadsheet unhappy).

Defense Miss (Cap 100)	
A	        Ad
0.000000    0.000000
1.600000    1.646090
3.199999    3.238866
4.799999    4.780875
6.399998    6.274508
7.999998    7.722006
9.599998    9.125473

Defense Miss (Cap 0.000001)	
A           Ad
0.000000    0.000000
1.600000    0.000001
3.199999    0.000001
4.799999    0.000001
6.399998    0.000001
7.999998    0.000001
9.599998    0.000001

Summing the totals together we get

A           Ad-m100     Ad-m0.000001  Dodge Ad
0.000000    0.000000    0.000000      0.000000
4.799999    4.904634    3.258545      4.937126
9.599998    9.588417    6.349553      9.350426 
14.399996   14.067506   9.286632      13.319067  
19.199995   18.356376   12.081869     16.907028
23.999994   22.468050   14.746045     20.166576
28.799993   26.414286   17.288814     23.140827

The first column is the total raw avoidance from defense, the second the total diminished avoidance from defense assuming a miss cap of 100% and the third the total diminished avoidance from defense assuming a miss cap of 0.000001%. For reference the final column contains the diminished amount of dodge you get if you stacked an equivalent amount of dodge rating.

From this you can see that dodge is more initially more effective at providing avoidance but as the diminishing returns kick in it moves into the range of avoidance values that defense can provide. Which stat is more effective to stack depends on the cap rate on miss and while we don't know this we can calculate at what point defense becomes more effective.

Rating	     Miss Cap Crossover
196.739950   92.5000%
393.479900   37.0079%
590.219850   26.0874%
786.959800   21.3868%
983.699750   18.8643%
1180.439700  17.3840%

From this table we can see that for a low amount rating the miss cap would need to be quite high for defense to be more effective than dodge but as you increase the amount of rating the crossover point for the miss cap drops. By the time you have ~1180 rating to spend (about 30% raw dodge) the miss cap only needs to be ~17% for it to be more effectively spent on defense.

I'm not really 100% sure what conclusions you can draw from all this but my gut feeling is that amount of dodge rating you'll get from gear will mean that if you want to gem or enchant for an avoidence stat defense will probably be your best choice.

[quinten]

Just a few things:

1) You failed to factor in block rating, and you acknowledge this. Is this intentional because you are looking at real avoidance as opposed to an avoidance/mitigation hybrid? It just seems odd to me to compare the two stats without including block.

2) You are assuming that miss has the same coefficient as dodge and parry. This is likely a safe assumption but it could be wrong.

3) Your miss cap choices seem rather odd. You want what is effectively a 0 cap on miss (your spreadsheet didn't like a true 0 because then you are dividing by 0) and a 100 cap on miss. Why you even bothered calculating the 0% confuses me because if the cap is 0 then you will never be missed and you can ignore the stat. So really you just chose 100% as an arbitrary cap that could either be far to high or far too low. (Some stats, druid and rogue dodge for example, have caps above 100%)

I'm not saying your work is all wrong, but I get nit picky around things like this.

[Kazeyonoma]

interesting, and as expected, dodge is better for pure avoidance up to a point (I can't find this exact point in your findings) then defense wins out in pure avoidance not to mention defense provides an undiminished return on Block Rating.

Good post, i'll leave the math crafters to fix any problems or what not but I find your stuff quite readable =] thanks!

[Sparan]

So the "Miss Cap Crossover" column illustrates what % the miss DR cap would have to be in order for defense rating to break even with dodge rating in terms of the avoidance it provides at a given amount of dodge rating, right? If that's the case, this feels pretty uninformative as it is, seeing as everyone is going to have enough defense to be uncrittable, and that defense would be effecting the dodge DR as well. Maybe you already have that factored in, but if not it would be useful to assume 540 defense as the question would really be whether or not defense outranks dodge from the minimum value a raid tank would have as a minimum. I'm pretty sure this will make defense look even better even earlier, on the plus side.

Also, how important is parry rating going to be to this discussion? Obviously it would negatively effect the value of def relative to dodge (more DR on parry makes defense give you less avoidance/point), but is the lowish cap on parry going to make it largely a nonissue except when people are stacking rating? And what about for our DK friends who get much more parry from strength conversions and itemization? Is defense a worse stat than dodge for them through and through?

Edit: Oh, and I forgot to say: thanks for putting in the time and energy on this. We all really appreciate it.

[Satrina]

I also would not have included block in the analysis. It's irrelevant to the problem being examined, and isn't subject to DR anyway. The coefficient is by class, so I'd guess it unlikely that there would be a separate one just for miss (but yes, not impossible).

What we need is for someone nit picky to put in the time to derive the stuff for miss, really.

Nice work, quadturtle. It's very nice to see new people taking initiative and diving into the theory side of things.

[quinten]

I also would not have included block in the analysis. It's irrelevant to the problem being examined, and isn't subject to DR anyway.

I disagree. Near the beginning of the post he ventures into the realm of ttl values where block is certainly reliant. I admit that you could easily say that is a separate argument and thus excluding block is ok; however, I would argue that block is now similar enough to avoidance to be treated as such.

As we are all agreed upon, block doesn't suffer from diminishing returns, but it is given to us via defense. While a block will not negate 100% of an attack it now negates what I would argue is a sizable percentage of an attack. If you are blocking for 2k on a boss that hits for 5k then 1% block chance is effectively worth 40% of a dodge or miss chance. Some people prefer to look at block as mitigation but I view it as partial avoidance until the hit table is full.

Because of this, when comparing dodge itemization to defense itemization you really need to factor in blocking. You can't simply say that you are better off with dodge below x threshold for avoidance when ignoring the partial avoidance blocking gives you.

Is it fair to call blocking purely mitigation when block values are so high?
Is it fair to consider it a form of avoidance when you still take some damage?
Are we looking at what gets hit harder by DRs or what is better to budget for?

I think a lot of the disagreements come from people differing on one of these 3 points. To me, tanking equations were a lot more linear in the past but the new mechanics are making things a lot more multi-factorial.

[Satrina]

So grab a calculator =)

[Shortypop]

It's too early in the morning for me to make sense of maths but I'm confused about the differences in parry and dodge calculations as I understood that despite the cap being lower the curve for diminishing returns for both are meant to be the same and therefore the calculations should be the same - or am I missing something? ie. if you had equal amounts of dodge and parry adding in 1% or 1% would give you the same diminishing return % of less than 1%, or does it have something to do with the rating -> % calculation for them both?

[quadturtle]

Hey, thanks for the replies. To answer some of the questions:

Quinten there are a couple of reasons I didn't factor in block rating while doing this. My primary reason was that I was trying to work out what would be the most effective form of gemming for an avoidance set. I'd asked in another thread about whether people though an avoidance set would be necessary in Lich King given the removal of crushing blows and one of the responses suggested it would be handy for bosses like Bloodboil who have an avoidable stacking debuff. I was, for my own benefit, looking just at pure avoidance stats.

Beyond that though my gut feeling was what I stated at the end of the post; that we'll get enough dodge through gear to not make it worth stacking in gems. In fact I wanted this to be true as it would make my gearing decisions a bit easier. So I partly didn't include block so that if the fudge factor was small I could always say to myself that defense is also giving me block rating and thus is still better. Not very scientific I know...

The choice of values for the miss cap is a little arbitrary, I was trying to catch the boundary conditions and while I may not have done that terribly well I would be surprised if the actual miss cap did not fall within this range. For the lower bound you're right it is a divide by zero error but, partly due to laziness, I wanted to have the formula's in my spreadsheet for this boundary so that if I wanted to increase it later (based on some experimental data) it would update automatically. For the upper bound, yes I realised afterwards when looking over Satrina's diminishing returns post that Druids had a cap higher than 100%, I guess I missed that due to a bad habit of skipping over non-warrior info.

As for the diminishing returns class coefficient for miss you're right I did assume it would be the same as it was for dodge and parry, I probably should have mentioned that in the post.


Kaz the point at which defense becomes more effective to stack for pure avoidance is dependent on what the cap is for the miss rate. For low amounts of dodge the miss cap would need to be fairly high for defense to be the better stat (e.g with 5% raw dodge the miss cap would need to be around 90%) as you increase the amount of dodge you have the value the miss cap needs to be for defense to be the better stat drops (e.g with 30% raw dodge the miss cap would need to be around 17% for defense to be better).


Sparen, that's quite a good point, I'd not considered calculating the values with the +140 defense to get crit immune. If I get a chance today I'll make a new table, as you say this should make defense a more effective stat even sooner. About parry rating, yep this will reduce the value of defense rating but I'm not sure exactly how, I may need to play about with that a bit. As for DK's I'm not sure, I guess trying to work out how parry rating affects the value of defense should give an indication of what is the better stat for them to stack.

In the end we really need to know the value of the miss cap for this to be useful. I think I'll probably hit 80 this weekend so I may try and work out at least a rough value then.


Shorty I think I see what you're saying and I'm not sure why its turned out like that. I've used the same formula for both dodge and parry and just varied the caps so I think it should be right. I'll see if I can find something to graph it with, it may make more sense if we can see it.

[quadturtle]

Ok I was going back over this again to try and work out how the +140 defense rating to be crit immune would affect things and I noticed I'd made a mistake in my spreadsheet for calculating the point at which defense becomes better. The amended table looks like this

Rating      Miss Cap crossover
196.739950  -568.497947
393.479900  28.997143
590.219850  20.482438
786.959800  17.279523
983.699750  15.388468
1180.439700 14.025643

The first value rather confused me so I plotted the formula and there seems to be an asymptote at around 200 item rating points and the graph tends to minus infinity as it approaches this from the left hand side.

ImageShack - Image Hosting :: crossoverpointca1.jpg

I'm not sure if this is a quirk of the diminishing returns formula or if I've made a mistake somewhere. The functions I used to work this out are:

Calculate raw dodge for X rating
A(x)=x/39.34799

Calculate raw avoidance from defense for one stat for X rating
S(x)= (0.04*x)/4.9185

Calculate diminished dodge for a given raw dodge percentage
D(x)=1/(0.011347 + (0.956/x))

Calculate diminished parry for a given raw parry percentage
P(x)=1/(0.021275 + (0.956/x))

For X rating work out what the diminished miss percentage would need to be for dodge rating to be equivalent to defense rating
M(x)=D(A(x)) - (D(S(x)) + P(S(x)))

Calculate for X rating what the miss cap would need to be for dodge rating to be equivalent to defense rating
C(x)=(M(x)*S(x))/(S(x)-(0.956*M(x)))

The final function was worked out as follows:

1/Md = 1/C + K/M
1/C = 1/Md - K/M
1/C = (M*Md)/(M*Md) (1/Md - K/M )
1/C = (M - K*Md) / (M*Md)
C = (M*Md) / (M - K*Md)

Where M is the raw miss and Md is the diminished miss. I made a bit of a poor choice with the names of my functions as M(x) actually corresponds to Md while S(x) corresponds to M.

If it is correct though Dodge seems to get worse faster than I had originally expected.

I've plotted the graph of how Parry and dodge diminish for shorty and you can see that parry does diminish faster

ImageShack - Image Hosting :: finalmc1.jpg

I've run into a little trouble when trying to work out how reaching +140 defense affects this though as I think I would need to know what your diminished miss percentage was at this point to be able to work out the miss cap crossover. My thinking had been to modify the function M(x) as follows:

M(x) = D(A(x)+5.6) - (D(S(x)+5.6) + P(S(x)+5.6) + Md(5.6))

Where Md(x) takes a raw miss percentage and converts it into the diminished amount. I can't really look at this much at the minute as I'm at work and my lunchbreak ended about half an hour ago I'll see if I can progress this any further when I get home.

[Shortypop]

Might be worth noting http://www.tankspot.com/forums/f63/4...oidance-2.html

and the post from Kazey in response to my questions. Your graphs are what I originally thought would happen that dodge and parry diminish independently and at different rates to different caps, Kazey's post says otherwise.

[quinten]

Shorty, Kaz is wrong. What used to be hard coded caps are now limits on the avoidance equation and avoidance now curves towards them. He admits that he is wrong here.

Quad, I expect you are dividing by 0 again. In fact I'm sure of it. I just need to go through your work and figure out where. Somewhere in all your subbed functions you have an equation where x can not equal ~200 and that is why you are getting the ± infinity. When I have more time I'll look at it a little closer. Whitetooth's formula is built to have this same thing occur at 0 (which is a value that the game shouldn't really care about) and I suspect you've shifted the formula somehow.

Edit: Ok, so upon further review... the formulas for deminishing returns are set up so that

k/x≠-1/c to prevent a 0 in the denominator.
or
x≠-k/(1/c)

For whitetooth's formula this happens at -44.9 for parry and -84.29 for dodge. I clearly was wrong before when i said it occurred at 0 but I would like to use the coffee break defense for that. Initially I thought you might have flipped a minus sign by mistake but the rating required would be far higher than 200. In conclusion... I have nothing at the moment.

[cjc813]

I'm not so sure I understand all this...

Couple questions...

One, am I understanding this right? The way I get it, each point of a particular rating you stack is worth slightly less skill than the point before it.

And likewise, I'm understanding that each point of skill that you stack is worth slightly less of its particular percentage than the point before it.

Am I understanding this correctly?

Two, is the character sheet telling the truth? What gets me is that is says I have 483 defense and 4.92% decreased crit chance... before diminishing returns.

Am I correct in assuming that the actual value of my decreased crit chance is less than what is displayed? If not, what does it mean "before diminishing returns?"

[Satrina]

One, am I understanding this right? The way I get it, each point of a particular rating you stack is worth slightly less skill than the point before it.

And likewise, I'm understanding that each point of skill that you stack is worth slightly less of its particular percentage than the point before it.

For dodge and parry rating (which convert directly into percentage), each point of rating is worth slightly less percentage than the point before. For defense, each point of rating always converts to the same amount of skill, but the amount of dodge and parry you gain from the defense skill diminishes. See here: http://www.tankspot.com/forums/f63/4...avoidance.html

Chance to block does not diminish whether it comes directly from block rating or defense skill,

Two, is the character sheet telling the truth? What gets me is that is says I have 483 defense and 4.92% decreased crit chance... before diminishing returns.

The defense tooltip is a bit of a mishmash. The dodge, parry, and chance to be missed that is granted by defense is subject to DR as above. Whatever you see on the character sheet for dodge and parry chance is the final number with diminishing returns already factored in, so whatever that tells you is correct. We don't know the actual DR formula for chance to be missed yet.

The reduction in chance to be critically hit granted by defense (and resilience) is not subject to DR. You can see this by taking your defense skill, subtracting the base 400 (assuming level 80) and multiplying by 0.04%. For example, my defense skill is 553. Subtract 400, leaving 153, and multiply by 0.04% gives me crit reduction of 6.12% - exactly what the tooltip tells me I have. (Whether I actually have 6.12% or 5.6% is a different discussion)

[Sparan]

Quad you're really doing tremendous work here. After the holiday I'll try to apply my meager math-skills to the problem, as well.

And don't you actually have the 6.12% reduced chance to be crit, as evidenced by PvP data and fighting against mobs that have increased chance to crit?

[Oghula]

Quite interesting math in here

I've run into a little trouble when trying to work out how reaching +140 defense affects this though as I think I would need to know what your diminished miss percentage was at this point to be able to work out the miss cap crossover. My thinking had been to modify the function M(x) as follows:

M(x) = D(A(x)+5.6) - (D(S(x)+5.6) + P(S(x)+5.6) + Md(5.6))

Where Md(x) takes a raw miss percentage and converts it into the diminished amount.

The formula you want to look at is

Md(x + c) - Md(c) = Dd(x,c) - Dd(0,c) - (Dd(x + c) - Dd(c) + Pd(x + c) - Pd(c))

with Dd(x,c) being the dodge chance given by x dodge rating and c defense rating and Dd(x) being dodge chance from defense rating. You end up with a quadratic polynomal which should be solvable for the miss cap needed. I tried to do so (M_Def.zip, with somewhat different nomenclature), but i think i screwed up somewhere since the result doesn't look right. I would appreciate it if someone could check my algebra.

[Satrina]

And don't you actually have the 6.12% reduced chance to be crit, as evidenced by PvP data and fighting against mobs that have increased chance to crit?

PvP is a different game than PvE in terms of mechanics. You can't point to something in PvP and use it as an example for PvE, and vice-versa.

As for increased crit chance mobs, I don't think anyone has ever bothered to test whether there's a hard cap on crit reduction or not. It would take a long time and a whole lot of data points.

[quadturtle]

Thanks Oghula, I'm traveling for quite a bit of this weekend so I'll take that with me to look at on the train.

Quinten I'd noticed for dodge and parry that if you plot the graph of the difference between the raw avoidance and the diminished avoidance you can see that the dr formula actually increases the amount of avoidance you get when the rating is quite low. I don't have the details with me just now but I think for dodge its around 175 rating that they provide the same return and then after that the dr formula will diminish the amount of dodge you get from rating.

Perhaps this crossover point for miss is at 200 rating, given the equation I was using this would make sense.

[Kazeyonoma]

Yeah, I was wrong in those posts Shorty, after re-reading Satrina's work, and talking to quad, I've seen that it's impossible for it not to diminish towards the cap when it is in the formula itself. Sorry for the misinformation.

[Blueduck3285]

With not factoring in the Block values, you could then apply these figures to DK's and pallies could you not?

Being a DK and not having the benefit of a Shield, Id be interested to know where my avoidance stacking bottoms out.

The avoidance of the Raw stat DR's faster than the avoidance gained from Def is handled different by DR? If that is the case, then how does the avoidance gained from Def factor into the DR from the raw stats and the overall "soft cap" your looking for?