Understanding Diminishing Returns

[Satorri]

Even a full year after its implementation, I'm still regularly seeing a general lack of understanding of what "Diminishing Returns" means when dealing with avoidance.

Two things should first be said.
1.) All the information I am about to present has been presented before by people smarter than myself. I am going to try to re-present it in such a way that I hope more people will be able to follow.

2.) Diminishing returns has been used in a nearly identical fashion on another primary survival tool in the game since inception. This was not introduced for the first time in WotLK, it was only then applied to avoidance. This is a tool that has been used to keep armor scaling appropriately from the beginning.

So, down to business. First, as a simple statement:

Diminishing returns does not mean you get less value out of the stat as you go, instead it means that you will receive an approximately linear return on investment as you gain more avoidance rating. This is the case because avoidance, like armor, builds towards 100%. At 100% damage reduction (i.e. armor) or 100% avoidance, the tank takes no damage at all. I will try to demonstrate below how, because of this, as the percentage approaches 100%, each equal step in scale becomes more valuable than the last.

Now for the math.
============================================

It does no good to simply say these things, and since we have the formulae, we might as well use real(ish) numbers, though obviously I'll be sandboxing the values a bit for illustration, not basing them on actual gear.

First, the armor curve for illustration of the point (as it is simpler). At level 80, the damage reduction by armor is calculated by the following equation:

DR% = Armor / (Armor + 16,635)So, first we'll set a few armor values for illustration, along with their corresponding damage reduction percentages:
24,000 = 59.06%
28,000 = 62.73%
32,000 = 65.80%
36,000 = 68.40%

So, the equal steps in armor become 3.67%, 3.07%, and 2.60% reduction increases. That is the act of the diminishing formula. Now let's consider these values against incoming damage. We'll keep it very simple and say we have a melee-based boss who hits once every 2.0 seconds for 60,000 damage per hit; that is 30,000 dps before any mitigation or survival tools. An old method of expressing a comparison was time to live, which is to say, if the tank has 30,000 health, 30,000 dps would be an expected lifespan of 1 second without mitigation. I'll use these methods to express for comparison. We'll set our tank at 45,000 health for this math.

Base/Unmitigated = 60,000 per hit, 30,000 dps, 1.5 sec to live
24k armor (59.06%) = 24,564 per hit, 12,282 dps, 3.66 sec to live
28k armor (62.73%) = 22,362 per hit, 11,181 dps, 4.02 sec to live
32k armor (65.80%) = 20,520 per hit, 10,260 dps, 4.39 sec to live
36k armor (68.40%) = 18,960 per hit, 9,480 dps, 4.75 sec to live

So while the damage per hit follows the decaying percentages as listed above (and dps will mirror these percentage gains), the time to live will improve by 0.36 seconds per step (forgive the slight margin by rounding to only 2 decimal places). Linear improvement from diminishing percent reduction.

So, if that is understood for diminishing returns applying a linear effect on functional survival, we can now look at avoidance.

The one way in which avoidance is slightly more complicated is that rather than being a single source contributing as with armor, avoidance has several sources (4 to be exact). Another, and somewhat more intriguing aspect, is that of the cap. Armor is designed to cap so that it will never give greater than a 75% reduction, that is a hard cap (i.e. the formula no longer applies, you will *not* gain more reduction regardless of investment). Avoidance however has three stats that each have the same affect (as far as survival, though parry has an extra non-survival mechanic), and each stat has its own cap. These caps are ~16% (miss), ~47% (parry), and ~88% (dodge). That means that, in theory, if you had enough combined stats (not attainable in game of course, but...) you could pass the 100% avoidance mark. Conveniently the curves are carefully crafted and the design is sound that you can never reach it, but the lack of a hard cap is still interesting.

On top of the complication of having 4 sources, the interaction of these contributions is also a little complicated to the 3 avoidance values. For the sake of simplicity, I'll only discuss dodge rating for the following rationale:

To keep things simple, we'll consider the tank's defense capped at 540, so miss will be constant, as will defense rating's effect on dodge and parry. To optimize avoidance ratings, it has been meticulously calculated (you can find it in this forum) that at a set defense value, until your dodge rating passes ~1.85 times your parry rating in totals, dodge rating will give you a superior amount of percent avoidance per point. For this reason we will set parry rating initially to start illustrating. We will also be neglecting the effect of agility, again, for simplicity.

For our tank with 540 defense, and we'll set 300 parry rating, the tank will have a baseline of (approximating for simplicity) 6% miss and 9.4% parry, and we'll add a 5% parry from a Deflection talent. This gives us a baseline of 20.4% avoidance and we'll calculate dodge from there. Remember, parry rating will grant more percent avoidance when dodge rating exceeds 555. The base amount of dodge (not subject to or influencing diminishing returns) is roughly 3.4%.

200 dodge rating = 10.39%
300 dodge rating = 12.43%
400 dodge rating = 14.37%
500 dodge rating = 16.22%

As with armor, and as expected, the gains diminish with each step. The gains are 2.04%, 1.94%, 1.85% respectively. Again, if we compare with the same tools as armor we can see similar figures, but in a slightly different manner. As avoidance will negate whole hits instead of a portion of hits, we will not see the size of hits decrease but will still see the average dps and time to live change. Here are those changes:

Base/Unmitigated = 60,000 per hit, 30,000 dps, 1.5 sec to live
200 dodge (30.79%) = 20,763 dps, 2.17 sec to live
300 dodge (32.83%) = 20,151 dps, 2.23 sec to live
400 dodge (34.77%) = 19,569 dps, 2.30 sec to live
500 dodge (36.62%) = 19,014 dps, 2.37 sec to live

So, as with armor, while the percent gain diminishes, and the dps matches that change, the time to live extends in linear steps with the rating gained.

A common expressed (mis)conception is that stamina is better because it does not diminish with increases. The reason this is incorrect is because health is already a linear term in the equation. We'll do the same breakdown as above, but now fix the other values and change only health (remember above examples used 45k health). Here hit size and dps do not change, only time to live.

Base/Unmitigated = 60,000 per hit, 30,000 dps, 1.5 sec to live
30,000 health = 1.00 sec to live
35,000 health = 1.17 sec to live
40,000 health = 1.33 sec to live
45,000 health = 1.50 sec to live

Here you see a linear increase in health also represents a linear increase in time to live.

Reality gets a great deal more complicated. Armor and avoidance (generally) only work against physical attacks, and not all attacks can be avoided. The two major cited *not* misconceived reasons for health's superior value to avoidance are magic damage and the random nature of avoidance, where it is possible to not see its effects for multiple swings in a row (though that chance, and the typical of number of swings between avoided swings shrinks more rapidly than linearly with increasing avoidance). Furthermore it is hard to generalize about gains in avoidance through ratings as the value gained per increment varies for each character in each instance based on your current values of the four influencing stats (defense rating, dodge rating, parry rating, and agility). Another fun reality is the pulsatile nature of incoming damage. It comes in chunks more than many small constant ticks, or a bleeding away of the pool, because of that, it's hard to really appreciate the value of health as 1-2k health doesn't seem like a big deal compared to 20k hits. In reality it is, but only because of the complex interplay of heals, the complexity of what damage you take when, and the perks of having a deeper reservoir against uncertainty.

That said, bear in mind that regardless of that point, you will always get a linear return on that rating investment regardless of where on the curve you are.

============================================
Now for fun, let's look at some typical value ranges and the relative increase in time to live granted by each of these steps in stats!

We'll work from a baseline of 40k health, 28k armor (62.73%), 6% miss, 22% dodge (from 350 dodge rating), and 19% parry (from 200 rating). 47% total avoidance.

Base/Unmitigated = 60,000 per hit, 30,000 dps, 1.5 sec to live
Geared Baseline as above = 22,362 per hit, 5,926 dps, 6.75 sec to live

So, if we increase health in 2k increments:
Geared + 2k health (42k total) = 22,362/hit, 5,926 dps, 7.09 sec to live
Geared + 4k health (44k total) = 22,362/hit, 5,926 dps, 7.42 sec to live
Geared + 6k health (46k total) = 22,362/hit, 5,926 dps, 7.76 sec to live
Geared + 8k health (48k total) = 22,362/hit, 5,926 dps, 8.10 sec to live
Geared + 10k health (50k total) = 22,362/hit, 5,926 dps, 8.44 sec to live

Sufficient count to demonstrate that 2k health increments from this starting place will be a 0.34 sec increase in lifespan.

So, if we increase armor in 2.25k increments:
Geared + 2.25k armor (64.52%) = 21,288/hit, 5,641 dps, 7.09 sec to live
Geared + 4.5k armor (66.14%) = 20,316/hit, 5,384 dps, 7.43 sec to live
Geared + 6.75k armor (67.63%) = 19,422/hit, 5,147 dps, 7.77 sec to live
Geared + 9.0k armor (68.99%) = 18,606/hit, 4,931 dps, 8.11 sec to live
Geared + 11.25k armor (70.23%) = 17,862/hit, 4,733 dps, 8.45 sec to live

Sufficient steps to demonstrate 2.25k armor steps increase time to live by 0.34 sec (totally by accident of course!).

Finally, if we increase dodge rating by ~130 increments:
Geared + 2.5% dodge (49.5%) = 22,362/hit, 5,642 dps, 7.09 sec to live
Geared + 2.4% dodge (51.9%) = 22,362/hit, 5,384 dps, 7.43 sec to live
Geared + 2.1% dodge (54.0%) = 22,362/hit, 5,148 dps, 7.77 sec to live
Geared + 1.9% dodge (55.9%) = 22,362/hit, 4,932 dps, 8.11 sec to live
Geared + 1.8% dodge (57.7%) = 22,362/hit, 4,734 dps, 8.45 sec to live

Sufficient steps to show 130 dodge rating steps increase time to live by (...drumroll please...) 0.34 seconds.

Note, here: With avoidance the hits will connect for full value, so the value of avoidance will work out in the average in terms of overall damage reduction. In reality, shorter time spans, you will see periods of higher functional damage reduction with chains of avoidance, and periods of lower functional damage reduction with chains of hits. The final average will be as presented above.

[Satrina]

At level 80, the damage reduction by armor is calculated by the following equation:
The constant in the mitigation formula is based on the attacker's level, whoever that may be. For a boss attacking a level 80 player, the value 16635 is correct. For a player attacking a level 83 boss, the correct constant value is 15232.5


So, as with armor, while the percent gain diminishes, and the dps matches that change, the time to live extends in linear steps with the rating gained.

This is silly. You're talking about times to live modified by avoidance without showing how that would relate in equivalence to any other quantity to give it some sort of meaning. On one hand, it's true that over a statistically significant sample of attacks over time with healing such as a tank and spank boss fight you will see the average dps decreased by something approximating the avoidance amount. However, in terms of the reality of most fights where death comes in the form of two or three unavoided (and mitigated by armour) hits with no heal, it's a completely pointless measure. I don't see how that can clarify anything for anyone.

In any case, your math is wrong here. 540 defense (689 rating) goes to 5.6% dodge from defense, as we know. From there:

- 100 dodge rating becomes 100/45.25019 = 2.209935% dodge pre-DR.
- k/A = 0.956/(5.6+2.209935) = 0.122408
- 1/c + k/A = 0.011347 + 0.122408 = 0.133755
- inverting that gives 7.476346% diminished dodge
- Adding 3.4% base dodge gives 10.87635% dodge
- Total avoidance is then 31.2763%

Similarly, {100, 200, 300, 400, 500} rating -> {10.87635, 12.76703, 14.57110, 16.29438, 17.94217}

Using those dodge values and the approximated parry/miss, then using the "avoidance averaged" time to live, we get the following values (keeping the decimals since it's important):
+000 -> 2.121420s
+100 -> 2.182655s (+0.061234s)
+200 -> 2.244401s (+0.061747s)
+300 -> 2.306667s (+0.062266s)
+400 -> 2.369458s (+0.062791s)
+500 -> 2.432781s (+0.063323s)

We see that the increase in TTL is actually on increasing returns here. Note for comparison that when equating stamina to armour and substituting into the TTL formula (see below), the increase in TTL is exactly equal at each step whether you increased armour or health to calculate.



Here you see a linear increase in health also represents a linear increase in time to live.
That's right. The derivative of the time to live function loses the armour component completely and becomes a function of time only, and health is just a constant in there. However, we get the same effect for increasing armour because of the Effective Health effect. If you could somehow rationally factor avoidance into the time to live equation, it is a constant and is lost if you take dTime/dArmour. If you instead take dTime/dAvoidance you will get something, but unless you have some sort of valid equivalence between avoidance and armour/health, it's going to be meaningless. (this is not phrased right at all but I am sick and falling asleep, so it'll have to do)


We'll work from a baseline of 40k health, 28k armor (62.73%), 6% miss, 22% dodge (from 350 dodge rating), and 19% parry (from 200 rating). 47% total avoidance.

Base/Unmitigated = 60,000 per hit, 30,000 dps, 1.5 sec to live
Geared Baseline as above = 22,362 per hit, 5,926 dps, 6.75 sec to live

Using armour-stamina equivalence, we can calculate that the equivalence for 40k/28k is (16635+28000)/40000 = 1.115875. Using that we can calculate that 2231.75 armour will give you the same TTL gains as 2000 health (and will give you precisely the same TTL gain of 0.089439936 at each step to at least nine decimal places) How did you arrive at the 130 rating value to approximate the same TTL gains? (You didn't identify a formula for equating X rating to Y stamina or Z armour)


Sufficient steps to show 130 dodge rating steps increase time to live by (...drumroll please...) 0.34 seconds.

6% miss is wrong for 540 defense; you'll have 9.29% (assuming not a night elf). Similarly, 200 parry rating with base 10% undiminished goes to 18.57%, and the basic 350 dodge rating again with 10% undiminished does go to 22.04%.

Going and recalculating based on those and stepping dodge rating through {350, 480, 610, 740, 870, 1000} gives these TTLs using the avoidance average again:

+000 -> 7.140725
+130 -> 7.464939 (+0.324213)
+260 -> 7.799366 (+0.334427)
+390 -> 8.144497 (+0.345131)
+520 -> 8.500854 (+0.356357)
+650 -> 8.868994 (+0.368140)

Again we see increasing returns on TTL here. Even if you pick the "wrong" amount of rating to add to match the increase in TTL you got from the matching stamina and armour increases, the only thing you miss is the nice "coincidental match" of the TTL steps between armour, stamina, avoidance. Picking the "wrong" amount of rating would be impossible if you had an equivalence function to plug stamina or armour into. In any case, adding the same amount of rating, whatever it may be, should give you exactly the same increase in TTL at each step, which we don't see here.

(As noted, I am sick and fuzzy headed right now. Any mistakes in my math, even spreadsheet-assisted, are mine and apologies if I did mess up. Fall over time now.)

[uglybbtoo]

And at the end of the day what you have is proof that if you stack Health or armour you definitely live longer if you stack avoidance on AVERAGE you live longer.

However as the hits got bigger and fewer needed to kill you the question comes up will I live long enough to see the average which is what Satrina points out above.

Blizz's biggest flaw in the combat system at the moment is the hit size has got so large a percentage of a tanks HP it doesn't favour averages. You could have the same damage per time from multiple sources or less damage but faster hits and the averages play out with avoidance. I am not sure why the trend to fairly simply boss mechanics has crept in but certainly the bosses doing 2 types of attacks at the same time would have been a better mechanic than 1 horrid large hit

[Satorri]

Heh, thank you for Satrina for coralling the numbers into accuracy.


My only goal here was to demonstrate that say, dodge rating, didn't suddenly return no value, or greatly reduced value, when it reached a certain point (I've heard people relate that in multiple posts and I've heard "caps" at 20%, 22%, and 26%).

I was trying to use round numbers, rough approximations to make my point with figures without scaring people away with long trailing decimals. Forgive my engineering bent. =)

I'm not arguing the value of health vs armor vs avoidance vs effective health vs etc. I'm only trying to illustrate that, like each of our other mechanics, avoidance does not cap out, ever, in obtainable stats.

Also, forgive the misuse of time to live, it's the only convenient metric I have for expressing the linear gains in survival. I tried to note the realistic way in which the different stats would be expressed, but hopefully the message survives though the math is criticized for precision (see: engineer).

[lyd]

i think you really need to make a distinction when you define the gains from diminishing returns as linear. gains are linear only in relation to the remaining variable percentage amount. the raw gains are anything but linear in the real sense, because you're taking a percentage of an ever-decreasing amount, which is an ever smaller number. it is in the very nature of a diminishing return effect that you put the same effort in, and get less out per unit of work. describing this as linear only works if you view it from the point of view of the %benefit received in relation to the ever-decreasing remaining chance to be hit (in the case of avoidance) or ever-decreasing amount of throughput damage (in the case of armor).

this concept is, of course, deductive from the math, but I think it needs to be spelled out for people who aren't familiar with the formulas (i'll focus on avoidance for simplicity's sake):

Diminishing Returns formula is set up such that you will always get a constant percentage benefit from your avoidance relative to your remaining chance to be hit. as you increase avoidance, you decrease your chance to be hit by a decreasing amount. meaning you get an ever- smaller raw percentage bonus from equal amounts of avoidance. this side of the equation is a regressive relationship, not a linear one.

i'll steal some of satrina's math to demonstrate:



- 100 dodge rating becomes 100/45.25019 = 2.209935% dodge pre-DR.
- k/A = 0.956/(5.6+2.209935) = 0.122408
- 1/c + k/A = 0.011347 + 0.122408 = 0.133755
- inverting that gives 7.476346% diminished dodge
- Adding 3.4% base dodge gives 10.87635% dodge
- Total avoidance is then 31.2763%

Similarly, {100, 200, 300, 400, 500} rating -> {10.87635, 12.76703, 14.57110, 16.29438, 17.94217}if you spell these gains out, you'll see both sides of the equation:

100 --> 10.87635
200 --> 12.76703 (+1.89068)
300 --> 14.57110 (+1.80407)
400 --> 16.29438 (+1.72328)
500 --> 17.94217 (+1.64779)

these gains show a regressive relationship. meaning as you increase X (parry rating) at a constant rate, Y (%chance to parry) increases at a decreasing rate of increase. that's the definition of a regressive relationship: increasing at a decreasing rate of increase.

now it's not wrong to describe this as linear if you really want to, but only if you are comparing it to the tank's remaining chance to be hit.

so for example:

let's say you have 50% avoidance. this means that if you're attacked 100 times, you should avoid 50 of them. adding an additional 1% avoidance increases this to 51% (yay!)

now let's say you're being attacked 100 times at 51% avoidance, and compare this conceptually to being attacked 100 times at 50% avoidance:

-as a percentage of the total attacks received (100), you have increased your avoidance by 1%.

-as a percentage of the attacks that will statistically hit you (50), you have increased your avoidance by 2%.

another way to understand this concept is to think of this from the opposite direction: let's say a tank had 99% avoidance. that means he has a 1% chance to be hit. if a second tank had 98% avoidance, you could say that the second tank has twice as much of a chance to be hit as the first one. in other words, by increasing your avoidance from 98% to 99%, you are effectively doubling the amount of attacks you avoid as a percentage of the amount of hits that can still hit you. to put it mathematically: the 98% tank has a chance to be hit equal to 200% of the 99% tank's chance to be hit. 2% = (200%*1%), or .02 = (2*.01)

another way you can analyze this concept is dividing a constant (x) by an ever-decreasing number (y):

X/Y
1/.05 = .2
1/.04 = .25
1/.03 = .333
1/.02 = .5

see how the quotient goes up at an increasing rate of increase? if raw avoidance gains were linear, this is the relationship that % avoidance would have with your remaining chance to be hit. your gains would be linear in relationship to your total chance to be hit, but they would be progressive in relation to your remaining chance to be hit. this is precisely the relationship blizzard wants to avoid.

basically, when Satorri says this is a "linear" gain, he wants us to view it is as thus: in order to get a linear benefit from avoidance ratings as a percentage of your remaining chance to be hit, you must decrease the amount of raw avoidance you get per point of dodge or parry rating. an increase of 1% should be viewed, via the DR formula, as 1% relative to 100% of the remaining chance to be hit. to put it a little confusingly: if you have 50% avoidance, each percent gain in avoidance thereafter should be viewed as 1% relative to 100% of the remaining 50%. this is the view you must take in order to describe regressing avoidance returns as linear.

so to sum, there are two ways to arrange the system: you can either have raw avoidance ratings give a linear benefit or a regressive one:

a) if raw gains give a linear benefit, you get an ever-increasing amount of effectiveness in relation to your remaining chance to be hit.

b) if raw gains give a regressive benefit, you get an linear amount of effectiveness in relation to your remaining chance to be hit.

currently, we operate under (b) in WoW.



here's the TL;DR: whether or not gains from avoidance are linear or regressive depends on which side of the equation you're viewing. as a function of total chance to be hit, gains are regressive. as a function of remaining chance to be hit, gains are linear.

i'm sorry if I beat a dead horse here, but I really feel it's important to understand both sides of the process. the DR formula gives you a regressive benefit from increasing amounts of avoidance, which is linear as a function of your remaining chance to be hit. simply telling people that avoidance after DR gives a linear benefit is a little bit obfuscatory if you don't know how the function actually works.

[Satorri]

Bonus points to Lyd for finding a proper use for 'obfuscatory' in conversation!

[Satrina]

And then minus points to both of you for using words like "obfuscatory" and "pulsatile" in a discussion meant to make a complex subject easier to understand to more people.

I'm an engineer, too, Satorri. Don't get me wrong here, I understand what you're trying to do here and it's probably a good idea. What you need to realise is that people are going to literally walk away from such an article with the idea that 20 avoidance rating literally is equivalent to about 300 health or 350 armour, when no such relationship actually exists. (like, literally!)

Look at how long it took to mostly eradicate the concept of "there's always a 0.000001% chance to crit", rather than +140 defense being elimination of critical hits against a standard 5.6% crit mob. I take full blame for that little tidbit, as it was in the original Defense article that I was musing that one of the possible explanations might be that there's always a chance to be crit regardless of defense score. I could argue that at the time, we didn't know any better. Honestly, though, the evidence and parsing supported that no such thing existed and certain mobs just had an increased crit chance. By the time I revised it out, it was far too late. Even now, there are people who still believe that "once in a blue moon" (again, my regrettable words) thing.

My point is, if you're going to pull out any sort of numbers then precision is not optional. Even if you're an engineer who is used to saying "close enough" :)

[Satorri]

What you need to realise is that people are going to literally walk away from such an article with the idea that 20 avoidance rating literally is equivalent to about 300 health or 350 armour, when no such relationship actually exists.

And I'm really glad you point that out, because you're absolutely right. :)

[Satrina]

And you quoted my overuse of "literally", curses!

[Ion]

I'm an engineer, too, Satorri. Don't get me wrong here, I understand what you're trying to do here and it's probably a good idea. What you need to realise is that people are going to literally walk away from such an article with the idea that 20 avoidance rating literally is equivalent to about 300 health or 350 armour, when no such relationship actually exists. (like, literally!)

Honestly, it read EXACTLY that way. In fact, I expect if you don't go back and edit it, that is precisely the point people will take away from that post...most people won't read down further and get a clearer understanding they'll just look and say "oh, cool, ttl equivalencies...ezmode".

[Satorri]

Technically, Ion, the comparison was at the bottom of the page, but the point is a good one, people can and do just scan stuff and take away what they think it means, and that is a little too convenient. I've snipped that portion out.

[Satrina]

Speaking of precision, here's a poor example of it in words:


The constant in the mitigation formula is based on the attacker's level, whoever that may be. For a boss attacking a level 80 player, the value 16635 is correct. For a player attacking a level 83 boss, the correct constant value is 15232.5

What kind of idiot writes such crap? What that should say is this:

The constant in the mitigation formula will change based on the attacker's level. For a level 83 boss attacking any target, the value 16635 is correct. For a level 80 player attacking any target, the correct constant value is 15232.5.

(and even then, we can tighten that up by removing the words "boss" and "player", since it doesn't matter whether the attacker is an NPC or player at all.)

[Satrina]

Technically, Ion, the comparison was at the bottom of the page, but the point is a good one, people can and do just scan stuff and take away what they think it means, and that is a little too convenient. I've snipped that portion out.

You still have a fairly explicit equivalence between 2k health, 2.25k armour, and 130 rating in the last examples. As noted, the equivalence between 2k health and 2.25k armour is real, but you're drawing the reader into seeing an equivalance with 130 rating which I assume was retrofitted to approximate the same 0.34 second TTL step size. You're making those parenthetical comments which will draw the reader's attention to see an equivalence where there is not one.

("Oh, 2000 health increases TTL by 0.34 seconds, and 2250 armour increases it by 0.34 seconds, and 130 rating increases it by 0.34 seconds. 2000 health = 2250 armour = 130 rating, got it!")

[lyd]

("Oh, 2000 health increases TTL by 0.34 seconds, and 2250 armour increases it by 0.34 seconds, and 130 rating increases it by 0.34 seconds. 2000 health = 2250 armour = 130 rating, got it!")

i want to make sure I completely understand what you're saying: these ratios exist only for these exact quantities, and any change in these variables would result in a different equivalency ratio relative to the +/- change in each variable. additionally: as was pointed out above, there is a distinct difference between average TTL over time when talking about avoidance, and static TTL when talking about effective health. the TTL function ignores these residuals.

right?


P.S. obfuscatory is the most deliciously ironic word in the english language. although not as cruelly ironic as "lisp."

[Satrina]

You can always figure out the equivalence between health and armour with 1 health : (k+a)/h armour, where k is the constant from the mitigation function. As the formula implies, a unique armour-health equivalence exists for each triplet {armour, health, k}. That calculation comes from here: http://www.tankspot.com/forums/f63/41526-ac-stamina.html. This is an underlying foundation of Effective Health.

As noted in my initial post somewhere, for the 40k health/28k armour pair with a level 83 attacker (k=16635), that equivalence is 1.116. If you were to change the character to 45k/30k, the equivalence changes, in this case to 1.036

Given the character with 40k health and 28k armour, we know that we can increase that character's time to live by the same amount by adding 1 health or 1.12 armour. (Of course, assuming a purely physical DPS intake, with no damage that cannot be mitigated by armour. As soon as fireballs fly, this is not useful. Or, you need a compound description using TTL based on armour and TTL based on resistance. Fun!)

If by residuals you're talking about error in the method, that's the entire thing I am talking about. Using armour-health equivalence, and modelling on damage per second, there is no error in the first nine or so decimal places when comparing steps in TTL as you increase by fixed amounts of armour and health, and when comparing the stepsize when using the equivalenced values of armour and health (say, 100 health and 112 armour in the current example.)

(Of course, modelling on damage per second is already introducing error in terms of the actual in-game performance since damage is stepwise on a non-fixed swing timer, and time to live is really in swing-timer-sized chunks - but that's a whole other discussion! Here in theory land we care about showing the relationship between armour and health.)

When it comes to avoidance, the best you can do is massage numbers until you find an amount of rating that will approximate the TTL step for a given {a,h,k} triplet. If you could instead derive a formula for that, then a more comprehensive TTL formula that covers all three stats would be possible. Unless someone has been holding out, there is no avoidance-health-armour equivalence that will let you plug in numbers and have it all fall out.



Big words are cool, I like fancy words. When explaining concepts to the greater WoW audience you need to keep in mind that due to its wild popularity, your audience includes mechanics, people for whom English is not a primary language, young people, and so on across all all kinds and walks of life. You can't make assumptions about the vocabulary level of your readers. If people have to go looking up words to understand your article, you've failed (in my not-so-humble opinion, at least ;))

[lyd]

it appears that the vicissitudes of my pedantry have inundated this incidental colloquium with a patina of obtrusive nomenclature.

;)

[Satrina]

haha!

*shakes fist*

[veneretio]

While I'm sure everyone has their heart in the right place here, this is a scary thread. I can safely say that if you really want people to understand Diminishing Returns then you need to find a way to say it in less than 4 paragraphs and without walls of math. Just math alone scares people. Walls of text scare people.

People don't need proof to understand something. I don't need to know how a bridge works to know that I can safely drive my car across it. Consider trying to explain DR plainly then linking to the sources of the math for those interested in it. By approaching the problem this way, I think you're going to get far more people able to wrap their hand around what the concept means to them as a tank.

Personally, I've never read much into the math behind this subject, maybe I should, but long ago I read that the 1st point of Armor you get is just as good as the 30,000th point and that Avoidance was being changed in a similar way. I think the overall point that really needs to resonate with the community from your post has to be that Armor and Avoidance don't become bad because you have too much of them. An upgrade is still an upgrade. I think you've definitely tried to drill this home, but I want you to consider something.

What if someone doesn't know what Linear means?

[Satrina]

One hundred percent true, Vene. That's why http://www.tankspot.com/forums/f63/40003-diminishing-returns-avoidance.html is laid out as it is. At the same time, I have always been pretty unhappy at the amount of reading in the first post even with minimal math. Indeed, I keep going back and make small revisions to clarify the writing in most of my articles. The avoidance DR one really does need a whiz-bang short form summary for the reduced math first post!

[lyd]

What if someone doesn't know what Linear means?

You raise an interesting question: what is the appropriate level of education to assume in your reader when discussing a complex topic?

I put a TL;DR at the end of my post precisely because of your concerns. I summed up the concept I was trying to communicate for people who might be afraid of the math.

"obfuscatory" was a poor choice of words for a broad audience on my part. but i don't know how else to describe something being "linear" without calling it linear, or simply referring to it as a constant rate of increase. I also very clearly defined "progressive" and "regressive" in my post. if I give someone a definition and they choose to ignore it, have I done wrong by them?

note: i am hardly a math person. I got to pre-calc in highschool and bombed in a huge way. I sucked at physics. In fact, anything more complicated than basic algebra is anathema to me (i know, more big words...). so i feel that if i can understand the math in my post, most other people probably can too. if that's not the case, i'm not sure how i could dumb it down enough to represent in a way that reflects the truth.

[Satrina]

It is a difficult choice that you do have to make, certainly. You have to choose an audience for your article and write to them. For example:

http://www.tankspot.com/forums/f63/41522-diminishing-returns-armour.html - Here I am trying to explain the concept to anyone who comes along, regardless of how proficient they are with math. The examples are simple and I have tried to present the formula in as simple a form as possible. That article has the fewest replies for the number of views it has, and none of them are actually looking for clarification. If you go back to when I initially wrote it in 2005, it is still one of the things I get almost no messages (PM or email) about. Was I successful? I think so.

However, you can note that I have failed my own test in that article - I use the word asymptote, even though it is a link to the definition of the word. More people will know what linear means than asymptote :) Quite possibly, people may not understand empirical, too! This is the sort of thing that I end up revisiting and clarifying later.

Anyway, compare that to http://www.tankspot.com/forums/f111/33109-diminishing-returns-math.html This article says exactly the same thing as the other, but it is aimed at an audience with a much higher level of knowledge. (And even so, I have edited it several times for clarity!)

I would suggest that if you are discussing regression in your article, you've already chosen your target :)

[lyd]

haha, i was going to hit you back for using "asymptomatic" after giving me guff for using "obfuscatory." no matter. I think the necessary points have been made.

[Satorri]

Vene, you are right of course, but the problem is you can say it very simply:

Avoidance chance (percentage) values gained per point of rating decreases as your rating increases. However, the effect avoidance has on your overall survival increases consistently with each point, so 10 avoidance rating has the same value if you have 20 rating already, or 300 rating.

But that's been said over and over and over, and people still keep coming back with "zomg avoidance sucks, it diminishes!!! That means once you have some it's worthless to get more!!"

Enter: my shot at the in between. =) For both good and ill, it's another shot in the dark, but maybe not a very accurate one.