Wednesday, January 23, 2008

The Diminishing Returns is a Lie

Oh, and The Cake Is Too. But LOLcats speak only truth!

Anyhow, Karthis, author of Of Teeth and Claws has honored me with a great Recommendation in his blog. And while I feel excited and validated by this, thats far from the point here. In the comments for that post, the issue of diminishing returns was breached. Now, the theorycraft has already been done a million times over, including at least a half dozen times myself, but I like playing with numbers, so this gives me a good reason to do it again. And instead of addressing the theorycraft behind the issue in his comments, I realized that I now have the power to ramble all I want in my own blog. Fancy that!

First, I'd like to start with a simpler example of variable returns: Dodge. Both because there's one less layer of complexity to it, and because the exponential returns on dodge are fairly well accepted by all camps. I present the chart to your left. Nice and simple, only three columns. The first represents a character's dodge percentage, while the second represents the remainder, the character's percent chance to be hit. The third is a bit trickier, it is the relative value of additional dodge percentage based on your current dodge. You would expect that every 1% dodge you acquire is 1% fewer hits you are taking, and in the most simplistic way, you are right, and in fact if you have no dodge, that is the case.

But what if you already have, say, 50% dodge (I myself have somewhere around that when raid buffed)? Well, for every 100 swings a mob takes at me, 50 of them land. But if I get an upgrade and suddenly have 51% dodge, what happens? Well, obviously I now only take 49 hits for every 100 swings. I'm getting hit 1% less often right? Actually, no:

Yes, thats right, even though we only increased my dodge by 1%, I'm actually taking 2% fewer blows than before. And ultimately, its not the damage you DONT take that matters, its the damage you DO take. Taking this to the extreme for a second, if you have 99% dodge, and gain 1% more dodge, you actually have reduced your incoming blows by 100%!

Ok, now that we've laid some groundwork, on to the real issue: Armor mitigation. For starters, it will help if we know how the game converts armor to damage reduction:

This is the formula for Damage Reduction when being hit by a level 73 mob. Since raid bosses are level 73, and are of the greatest concern for us tanks, this is the one everybody uses. Note that if you plug in the accepted armor cap value of 35,880 into this equation, you get a DR of exactly 75%, as expected, while if you plug in 0 armor, you get 0 DR. Now to show you some calculations. . . this data set is a BIT larger than the last, I apologize. . .

A good bit more information here, but we have THREE different ways to look at armor, and I want to give all of them their due. For now, ignore the HP and DPSwing boxes, those will come in later. The columns are, in order, your armor, the Percent Damage Reduction (as per the above formula for level 73 mobs), your Percent Damage Taken (Remember, its not the damage that doesn't come in that matters, its the damage that does), the change in the damage reduction number per thousand armor, the percent reduction in damage taken per thousand armor, and then the last column, which takes those two values at the top and tells you how many hits you can survive given those parameters.

First off, lets look at the simplistic (read: wrong) view of armor, which is to simply look at column 4. Going from 0 to 1k armor increases your damage reduction, as it would be displayed in your character stats, by 7.7 percent. To contrast, going from 34k to 35k armor would only increase that number by a measly 0.55 percent. Pretty heavy diminishing returns, eh? But, as we noticed in the dodge calculation earlier, these raw mitigation numbers dont accurately tell us how much an upgrade will truly affect our survival; they are misleading.

So lets do the same thing we did for dodge: Calculate the ratio of damage you take with 2 incremental armor values, in this case, increments of 1000 armor. Using the same basic logic we did with dodge, you come up with the numbers shown in column 5. The first thing you'll probably notice is that, yes, there ARE STILL DIMINISHING RETURNS on armor when you look at it from a more practical angle. But, they aren't nearly as steep, and this shows that as you approach the armor cap, your armor increases are nearly 4x as valuable as the first approach would have suggested. Even gaining your last 1000 armor to hit the cap is over a 2% reduction in damage taken, and thats nothing to sneeze at!

Now, you might be thinking to yourself, "Surabear, people keep saying that there is no such thing as diminishing returns on armor, does this mean they're wrong?". No, it doesn't, it just means they subscribe to the third way of looking at armor: Time until death. Assuming a tank has 20k health, and is taking hits that, unmitigated, would deal 1673 damage, we get column 6, Swings to Death (I'm assuming 0% dodge for simplicity). Yes, the numbers were chosen specifically so that every 1,000 armor will allow the tank to survive one more strike, but any combination of health and damage will yield the same relative results: For every 1000 armor you have, your survival time increases the same amount. No diminishing returns at all. This means that armor linearly affects how long you can live without heals, no diminishing returns.

So, why do I look at armor in terms of mitigation, instead of in terms of survival time? Well, to put it simply, right now I have a little over 33k armor unbuffed, so would survive 45 hits in this scenario. In order to increase that to 46 hits, I can do one of two things: Find another thousand armor (182 pre-bear form) somewhere, or pick up approximately 450 health (30 stamina) At this level of gearing, both of these options are relatively equal in difficulty to obtain.

But here's the clincher: you only need to worry about the number of hits you can survive without heals if you're actually having a problem dying to spike damage. If you're NOT dying to spikes, then you're either dying to healers going out of mana, dying due to wipes, or not dying at all. In the first case, its your MITIGATION that will save you, because the more you mitigate (or avoid), the less your healers have to cast, and the more mana they conserve. In the second case, no amount of armor or stamina is going to save you, unless it affords you enough time to run past a reset point for the fight (good luck, my friend). And if you're not dying? Well then, all you're doing is making your healers' jobs easier, and once again extra mitigation is pretty much just letting them save some mana.

I hope that this has proven to be informative, even for those who have already been exposed to the Great Diminishing Returns Debate. If nothing else, I hope that those who used to be "Type 1 Theorycrafters" have come to realize the error of their ways and embraced either Type 2 or Type 3 (or both!). Knowing is half the battle!

EDIT: I went back and ran a few quick numbers, and came to a realization. If you convert the time until death numbers into a "% increased time to death per 1k armor", like I do for armor mitigation, you get the EXACT SAME PERCENTAGES for both. So if you instead look at the percentage of extra time that armor gives you, its exactly the same as the percentage of mitigation increase it gives. Kind of intuitive, but an interesting correlation nonetheless.


Leukos said...

Not sure if it is just me, but I can't see the dodge chart in the article. Great stuff otherwise. I am enjoying your blog so far, keep up the good work.

SuraBear said...

Ack, it must have gotten deleted when I was trying to redo my formatting (Inserting images adds extra spaces in between all of my paragraphs, I probably deleted it accidentally when rectifying that problem)

I'll get it back up there for y'all

Xanathos said...

diminishing returns is objective. Say every 1000 armor lets you survive 1 more hit. still moving from the 10 to 11 hits is more effective then moving from the 20 to 21 as the first lets you take 10% more hits and the later only 5%. But still the diminishing returns isnt nearly as bad as what it looks like at first glance.

SuraBear said...

Well, yes, diminishing returns is very subjective. But if we use that logic, so is Attack Power, Mana Pool, hit, crit, and most importantly Stamina (As most people who poo poo on armor's DR point to Stam as an example of something that doesn't diminish).

Given that angle, Dodge's exponential returns probably become the only non-diminishing stat in the game. Crazy semantics are why, even though I love the theorycrafting of game mechanics, when it comes to actually make gear choices, I go with my gut instead of my calculator.

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