Guild Wars 2

So, first let’s get this out of the way; toughness is linear. Now that you’ve quit reading I’ll go ahead and explain why anyway alongside what I call the Proportion Error and we’ll talk a little about fractions and how they work too. There’s also some logarithms in there.

Linear You Say?

Yeah.

Proof?

Yeah. Let’s start with a simple one about the nature of doubling numbers and what I call The Proportion Error. To do this first I’m going to use simple addition and subtraction but I want you to take note of the wording. Very important.

10 added to 10 equals 20.
10 doubled equals 20.

Boring stuff, right? The ratio is 2:1 for both, simple. Now let’s do it twice with the both the product and the sum of the previous.

10 added to 20 equals 30.
20 doubled equals 40.

So what’s the issue, simple math, right? The ratio changed. 20 doubled is still 2:1, or proportionately accurate while 20 + 10 is obviously not 2:1 and is in fact 1.5 or 3:2. Leaving that alone for a second The Proportion Error creates “false” diminishing returns even here adding but the reality is that when we add what we’re really measuring isn’t the turn over turn (30/20) but the turn over base (30/10).

And this matters because?

Power isn’t exponential. Let me explain with three fractions, 1/2, 2/2 & 3/2. Now the first thing we see is that I’ve just added 1 to the numerator, and (X+1) is totally linear, so there’s nothing to be looked at, but that’s the incorrect way to read this (surprise) because when talking about effectiveness 1/2 is half as effective as 2/2 but 3/2 isn’t doubly effective. “Duh”, but it’s not as simple as “duh”. This is the same ratio error from before, the effectiveness of the numerator dropped by 50% just by adding 1 without changing the denominator!

Isn’t that diminishing returns? No. No it is not. As you continue on your merry way you will find that doubling your effectiveness requires more and more though, for instance 2/2 -> 4/2 is two steps but 4/2 -> 8/2 is four steps. What this means is that your effectiveness can be (and is) an exponential graph for a linear equation. 1, 2, 4, 8, 16, 32, 64 … these numbers look familiar?

They’re the powers of two. But we know better, we know that even if we measure effectiveness in exponential format it’s still a linear equation.

And let me guess…

Congratulations, you’re right, the same is absolutely true if you change the denominator. Division is a linear graph that can be (and often is) graphed in logarithmic compression. 1/2 has half the effectiveness of 1/4 which is half of 1/8 and so forth and so on and you’ll see the same powers of 2 play out both ways. Now this is important because if it wasn’t true we’d have major problems.

You can write all logarithms as exponents which means that you can write any division problem as a multiplication problem and you can swap the numerator and denominator to get the natural reciprocals. 4/1 for instance is the reciprocal to 1/4.

… And?

Oh you… This means that it doesn’t matter where you put your numbers in a linear equation. Top or bottom, front or back, the answer will eventually come out to the same proportionately. You can put toughness on the top or bottom but so long as one element is linear the other will be too, for instance with 3 power and 1 toughness both statements are true:

“I will do triple damage.”, “I can only block one third of this attack!”

Statement one is actually coming from the aggressor, the one with 3 power. (written: 3/1)

Statement two is coming from the defender, the one with 1 toughness. (written: 1/3)

The exist in harmony, isn’t that nice?

Okay, okay, but where does the error in thinking come from?

How we speak.

Meaning?

First we swap between static values and effectiveness. “If I add 1 to power on top then it’s a line!” is measuring static value, not effectiveness, which we saw when we looked at 1/2, 2/2 & 3/2. 3/2 is not as effective a move from 2/2 as 1/2 was. We however stare at the +1 (base) and the slope of .5 instead of talking about the percentages.

Which means we…

Absolutely! We do the direct opposite when talking about toughness. We talk about the percentages and effectiveness and ignore the base often making the mistake of constantly changing it. You see, when comparing 1/2 to 2/2 and 3/2 we’re comparing it to the base of 1 but if we compared it to it’s previous iteration (remember the 20+10? this is where that comes back) we get that illusion of diminishing returns. If we look at turn over turn it all works out in the end perfectly too.

1+1 = 2 which is 1 doubled.
2+1 = 3 which is a .66 increase.
3+1 = 4 which is a .75 increase.
4+1= 5 which is a .8 increase.

Or:

1/2
2/3
3/4
4/5

and if you subtract them in order from greatest to least you’re going to get what looks like … diminishing returns.

Also note you get their reciprocals when taking them at base:

1/2 is still .5
1/3 is .33
1/4 is .25
1/5 is .2

Basically they “confirm” one another.

But the big thing here is to note that we constantly change the denominator and do not work from the same base which is (hue hue) what we do when we talk about power. You want to triple your damage? Then triple your power! Plus one for everyone! But if you took it turn over turn … that never pans out (and I mean never by the way, ever).

So what are diminishing returns?

Diminishing returns in economics is the notion that when one unit of labor is added to a system eventually that system no longer benefits from that unit of labor and eventually it becomes overlabored. Now this differs greatly from GW2 in many ways as there is only one stat (precision) that has real diminishing returns in it because diminishing returns, unlike logarithmic compression, will always have a down-turn in relation to production presuming no hard cap.

For those who don’t know what this means it actually means that eventually having too much toughness will cause you to take more damage. This never happens. There has to be some form of loss or real opportunity cost involved in order to show diminishing returns, for instance, if there were diminishing returns in toughness it would be against vitality, not power, because the “unit of production” would be Defense. It has to be in the same classification to work.

In basic terms if you can double it consistently then it’s a line. 2X and X/2 are the same line as well. Don’t fall for the hype bro! Compress and Prosper!

Anything else?

If you are going to post a graph of x/y+1 and say “’nuff said” you need to look in the upper left hand corner. If there’s an exponential curve in that corner (there is) when you decide to say “TOUGHNESS AIN’T LINEAR” just remember that the tiny curve up there is whispering “…then neither is power…”

http://image.tutorvista.com/content/feed/tvcs/graph_4.JPG

So I chose to use this stuff, or at least am thinking about it, but honestly what is it?

First let’s go over the types of Stats, as there are only three, primary, secondary and supportive.

Primary stats are Power and Vitality. The reason they are primary stats is because while they are effected by other stats they themselves do not effect other stats. They are raw and untamed!

Secondary stats are Ferocity and Healing Power. The reason they are secondary stats is specifically because they indirectly augment the primary stats. They are useless if the primary stats are too low or in the case of health, depleted.

Supportive stats are Precision and Toughness. The reason they are supportive is because they are the glue, they by themselves are actually pretty neat but they’re not made to work alone. You could think of them as the matchmakers, ferocity and power are nothing without precision, you can have all the ferocity in the world but with a 4% critical hit rate it’s worthless and Fury helps make that a bit easier but it’s often temporary. The same is true of toughness, Protection is a huge boost to it but it’s temporary and what toughness does is while it works on its own its best used with healing power and vitality since toughness only improves both.

Yeah, yeah…

Well that brings us to our first point. Toughness improves your healing power and vitality. The big thing here is that this is not reciprocated at all; vitality doesn’t help with toughness or healing power, it just increases the health pool, and healing power by itself can sort of buffer vitality but in this game the coefficients keep it in check and from really allowing that full heal every five seconds.

So about that healing power?

Because this isn’t the math section I’ll leave it out. But basically what toughness does is it reduces the damage you take making every point you heal worth that much more. It eases the burden of healing you which can help eek out those last seconds until you can use your main heal again if you’re in a sortie you’re uh … not exactly “winning”. This is closer to the relationship between Ferocity and Power, without adequate toughness your natural heals won’t be able to keep up (presuming you’re not a warrior… >_>)

And vitality?

Just about the same thing. Your vitality improves as your toughness improves (which is why you’ll never have a point where 1 point in Vit is worth more as a “distinctive truth”) sort of like precision improves power. There’s no magic point in which you have the best ratio or anything and you can definitely gauge that however you want but that is what it does. I will tell you that toughness is not a substitute for vitality even when talking about direct damage so don’t try to use it for one; you want both if you’re going to play defensive because again much like precision it’s so much better when paired up.

So Nomads?

What? No! Don’t use Nomad’s. But do feel free if you’re power to mix maybe Valk. with Knights if you’re on the defensive so you get that vitality and toughness as well as the precision and ferocity. If there were a Precision Major, Toughness/Power minor armor I would suggest that one too. Not many options in this vein really but with conditions you can and should just run Dire if it’s your real source of damage unless you need condition duration but even then a balance between those two should sort you out.

Anything else?

Yeah, actually. Your toughness behaves differently in PvE than PvP. What I mean by this is that it has more value in PvP than PvE and there’s a reason. That reason is that enemies in PvE attack about once every 2-3 seconds normally so they have to hit that hard to keep players on their toes. The sheer magnitude of the strike might be daunting while other players have the same stat caps and balance choices to make as you do making them less deadly.

I am not saying you can’t tank champions, you can, and handily might I add with little effort, but just adding this as a note for those who may not know or may not have thought about it. And one more thing:

Don’t think of toughness as “Taking One More Hit!”

Bad idea. That isn’t how this game is played. If you are not synergizing your toughness with vitality or healing power (or both) don’t take it. It is a supportive stat and it is not designed to save you by itself. You’ll be just as dead as the zerker kid and with less to show for it if you’re not thinking synergistically.

You said Precision had diminishing returns…

Oh ho! I got busted. Yes. It’s an S-curve graph with diminishing returns.

I demand an explanation!

Another thread maybe.