Guild Wars 2

Math of Vitality and Toughness for the Elementalist

Suppose you have decided to invest in vitality and toughness. Let’s not discuss how much you really need for survivability. Rather, assuming that you do want more vitality and toughness, let’s discuss how the numbers work out and how you can decide which one is more helpful.

The following has of course been discussed in various forms, so only some of it is new, as far as I know. See this post for a nice, perhaps even the earliest discussion. See also the discussion here. I look at equipment options here. I hope someone finds it useful, at least making these notes helped to explain it to myself.

Claimer: I never make mistakes, and I never get downed in fractals ;-)

Setting up the problem

Let us consider a level 80 elementalist, one of those squishies that braves the world in light armor with little health to begin with. (No, this is not the problem I mean, although it is a problem :-) Let’s assume exotic armor. We only consider direct damage in relation to health and armor. We ignore condition damage, dodging, killing your foes quickly. We ignore healing for now, but see below.

For direct damage, it is not obvious how to compare builds with different amounts of vitality and toughness. However, there is a compelling suggestion, namely to consider the so-called effective health.

For example, suppose a single attack would do 15000 damage against a character with zero toughness. With 33% damage reduction from toughness, the actual damage dealt is 10000 for the same attack. The game only shows you the 10000 and subtracts that from your health. The idea of effective health is to undo the damage reduction calculation and define 15000 as your effective health. Given two stats, vitality and toughness, we compute one number, your effective health for zero toughness, which allows us to easily compare builds with different combinations of vitality and toughness.

Effective health – basic result

Further down I outline the math behind this. However, let me give the basic result first.

Warning – entering math zone :-)

For the elementalist at level 80 in exotic armor, effective health is given as a function of vitality and toughness by

eh(v,t) = (10805 + 10*v) * (1836 + t) / 1836

where v and t is the extra vitality and toughness added beyond the base value of 916.

The question is how to invest your stat points for optimal return. Well, first you have to define a budget! And your budget is finite, else it is trivial to make eh arbitrarily large. For example, v + t = 1000.

If we insert t = 1000 – v into the formula for eh, then we get a quadratic function eh(v), that is, a parabola. This is great, because we know how to find the maximum of a parabola.

There is one complication. It can happen that the math finds the maximum for some negative number t. This means that we should take out points from the base value of 916, which we can’t. In other words, there is an additional condition of the form t >= 0. If the maximum is obtained for some negative value of t, we can
set t=0 and only invest in v.

A figure that could be worth a thousand hitpoints

I love plots. Attached is a figure that shows effective health for different values of total points = 0, 250, 500, …, 1500.

All curves are pieces of parabolas.

The allowed range of values of v and t are shown in color. Outside this range the parabolas are continued as a grey, dashed line.

We show the maximum of each parabola with a dot. However, the maximum does not always fall into the allowed range. The actual maximum is then the upper edge of the colored bit of parabola.

The basic conclusion is that (without healing!)
- invest the first 750 points in vitality,
- divide any additional points equally between vitality and toughness.

For example, suppose you have 1000 points to spend. The plot tells you to put about 875 into vitality to get the maximum effective health. That is 750 in vitality to reach the balance, plus 125 in vitality and 125 in toughness.

Note that the parabola is rather wide near the maximum. This is good news because we may not have the flexibility in choosing our gear to hit the optimum precisely, but being off by 200 points in toughness or vitality only changes the effective health by about 1% (near the maximum).

However, if you move too far away from the maximum, then your investment may become rather inefficient. You can get higher effective health with less total points. 1000 points spent optimally give you more more effective health than v=0 and t=1500.

Effective health – long explanation

Actually, it is not that hard, but the entire post is definitely tl:dr considering the topic. Yeah, I went into overkill mode, and I sometimes hit frogs for 200k or so, too. Yeah, I love math.

Don’t say I didn’t warn you! :-)

Effective health

Effective health is defined by

eh = h * a / a0

where

health h = h0 + 10 * v
armor a = a0 + t

and v and t are the points in vitality and toughness that we add beyond the level 80 base value of 916,

vitality = 916 + v
toughness = 916 + t

For the level 80 elementalist in light armor, the base values of health and armor are

h0 = 10805
a0 = 1836

See the wiki here and here. Sometimes effective health is defined without the division by a0, which I don’t find quite as useful.

In short, we compute effective health by

eh = (10805 + 10*v) * (1836 + t) / 1836

The key thing is that given v and t, we compute one number.

Effective health is your health increased by a factor

a/a0 = 1 + t/a0 >= 1.

Basically, we convert the extra armor due to toughness into extra health.

The claim is that effective health is a good measure of survivability.

Let me give not one, but three explanations for the meaning of effective health. Each gives an interesting angle on the problem!

Damage and Survival

The basic fact about life and death is that you go down when the damage d that is dealt to you is equal to your health h,

d = h,

or larger.

To proceed, we need the formula for how the damage you take depends on your armor and toughness. We can write this damage as

d(t) = A/a = A/(a0+t)
d(0) = A/a0

Here the constant A lumps together all those power and modifier factors you find on the wiki into one constant. Big A may be provided by Lupicus, little a is you. The larger the value of a, the smaller is the damage taken. This is nice, but we would need t=1832 to halve the damage, and you need special builds to even get to 1500 or so.

Damage to death

The damage you take before being downed is a function of toughness,

h = d(t).

The idea of effective health is to ask how large your health should be to survive the same attack (same attack factor A) without extra toughness,

eh = d(0).

We immediately get eh/h = d(0)/d(t) = a/a0, or

eh = h * a/a0

as advertised. The key thing is that the unspecified factor A has dropped out!

You can think of eh therefore as a measure of your one-hit survivability that is independent of the type of direct damage attack. It doesn’t matter whether it is one hit, a burst, or continuous direct damage, as long as there is no healing. Just imagine that either there is no time to heal, or that all your heals are on cooldown.

Time to live

Alternatively, we can ask about the time-to-live given some amount of incoming damage per second (dps). Our time-to-live is

ttl = h/dps

For example, with 15000 health and incoming dps of 1500/s, we live
15000 / (1500/s) = 10s.

Again, the dps is inversely proportional to the armor, now with a different constant B,

dps = B/a

so that

ttl = h * a / B

We can define effective health as the health we need for the same time-to-live for zero extra toughness t=0,

ttl = eh * a0 / B

Again, the unspecified constant drops out and we have

eh = h * a / a0.

Initially I found it surprising that your time-to-live depends linearly on your extra toughness,

ttl = h * (a0 + t) / B

Your time-to-live is proportional to your health (assuming constant incoming damage), and it is proportional to your armor a0+t.

Damage reduction

Yet another way to look at effective health is based on damage reduction, as was done here. If without toughness you take damage d(0) and with toughness d(t), then the relative damage reduction is

drd = (d(0) – d(t))/d(0) = 1 – d(t)/d(0) = 1 – a0/a

The relative damage taken is

dtk = 1 – drd = a0/a

For example suppose you build for extra toughness that is half your base armor, t = a0/2, which for the elementalist means t = 918. Then a = 3*a0/2 and

drd = 1/3, dtk = 2/3, or
drd = 33%, dtk = 67%.

In terms of your health we can now introduce an effective health equal to some damage before damage reduction. We reduce this damage to get the damage versus actual health, so we have

h = eh * dtk = eh * (1 – drd) = eh * a0/a

and again

eh = h * a/a0

Optimizing the vitality and toughness balance

Now that we have established, understood, and celebrated effective health, we want to get it efficiently. How do we maximize effective health as a function of vitality and toughness? We have

eh(v,t) = (10805 + 10*v) * (1836 + t) / 1836

Not allowed is changing h0 and a0, because then you wouldn’t be playing an elementalist!

Warning – entering an even worse math zone :-)

GW2 uses formulas with products of linear functions in several places, in particular for effective health, but also for effective power, so let me make some general comments first.

Essentially, we are asking how to maximize a function of two variables,

f(x,y) = x * y

But as stated this problem is not yet fully specified, because there are several extra conditions. We are dealing with a problem of constraint optimization.

Most importantly, we are operating with a limited budget of stat points. It is trivial to make f(x,y) arbitrarily large if you can make x or y arbitrarily large. However, our resources are finite. Let us assume that we have a finite number C of points to spend,

x + y = C,

then, if we pick a value for x, we can put at most

y = C – x

points into y.

Given some C, our function turns into a simple quadratic function of x,

f(x) = x * (C – x) = C*x – x^2

This defines a parabola, and we know how to find the extremum of a parabola. Perhaps this tells you something:

f’(x) = C – 2x = 0 => x = C/2 and f’’(C/2) < 0: maximum at x=y=C/2.

Intuition

For some quick intuition I find it more helpful to recall that f(x,y)=x*y defines the area of a rectangle. And we all know that it is the square that gives you the biggest bang for the buck, that is, the largest area for the shortest circumference 2(x+y)=2C. For the square, x=y, and therefore x=y=C/2 gives you the largest area.

Back to the original problem

Let us write the effective health as

eh(v,t) = (10/1836) * (1080.5 + v) * (1836 + t)

where I have cleverly pulled out an overall factor. The value of the maximum depends on that factor, but not its location. I factored the 10 so that v and t enter with the same scaling. The law of the square tells us that eh grows fastest for some finite budget

v + t = totalpoints

if the two variable factors in eh(v,t) have equal size.

Imagine that you start with v = t = 0, and that you are investing points one after one into either v or t. Then you may want to first invest into v until 1080.5 + v = 1836, or until v reaches

vfirst = 755.5

Basically, given a rectangle with sides 1080.5 and 1836, we first lengthen the shorter
side to get a square. But once we have a square, we should grow each side equally,
adding 1 to both v and t for the optimal increase of eh(v,t).

This strategy takes into account another constraint of the optimization, namely

v >= 0 and t >= 0.

We are simply not allowed to lower the toughness below 916, which would mean t<0, to get the optimal square with shorter sides. In other words, yes, we know how to find the maximum of a parabola, but its location may fall outside the allowed range of our variables! In this case the maximum is obtained at the boundary of the allowed range.

Summary

In summary, we are looking for the maximum of

eh(v,t) = (10/1836) * (1080.5 + v) * (1836 + t)

under the constraints

v + t = totalpoints
v >= 0 and t >= 0.

If totalpoints <= 755.5, then set

v = totalpoints and
t = 0.

If totalpoints > 755.5, then set

v = 755.5 + (totalpoints – 755.5)/2
t = 0.0 + (totalpoints – 755.5)/2

In other words, add points to vitality until you have reached the “golden” balance.
Once there, invest 1:1 in v and t.

Healing and effective health

The most important omission so far is that we did not discuss healing. As far as I can tell it is rather difficult to model healing realistically, since this becomes a calculation where the damage coefficients do not drop out, and the timing of damage and healing matters. However, even without healing, our calculation is relevant, because it gives a reasonable estimate of how well we can survive damage that we are not able to heal. (Which usually happens to me before I die ;-)

With all the formulas in place, there are two simple models for healing that I want to mention.

Suppose you manage on average a certain amount of healing per second, hps. Our hps competes with incoming damage per second, dps. Inserted into the time-to-live formula we get

ttl = h / (dps – hps)

Clearly, this only makes sense for dps > hps, else we out-heal the damage and live forever. The point to make is that the time-to-live is inversely proportional to dps-hps. As hps approaches dps, we have a 1/x pole that blows up for x=0. In other words, when hps approaches dps our healing can have a very large effect on our survivability. On the other hand, if hps is much smaller than dps, the effect is very small.

Now suppose you manage to heal yourself for heal=7500 points before you go down. Basically, you get off one healing skill with a little extra. We can account for such a finite amount of healing by changing the health formula to

h = h0 + heal + 10*v = h1 + 10*v.

All the calculations for the vitality-toughness balance remain the same! Except that h0 has to be replaced by h1. I picked heal=7500 because this is equivalent to the offset v=750 that we need to balance our stats. So, if you get one such heal, your vitality and toughness are equally important. If you get in two such heals, toughness will be more important! Assuming that you have enough effective health to survive the damage spikes.

In summary, when faced with lots of incoming damage that we cannot out-heal, we can still maximize our dps and effective health so that we kill before being killed. When there is not too much damage, we should consider dps-hps, but the result depends on several situational factors that are not easily discussed in general. However, a fixed amount of health through healing is simply equivalent to some additional vitality in the the vitality-toughness balance.

Exercises :-)

Repeat for other classes with different basic health and armor. See the nice table attached to that post.

Apply a similar logic to effective power, which is described by the function
g(x,y,z) = z * (1 + x*y).

As always Abraxax it is a pleasure to read maths well done

My head hurts. Please no more math.

So to sum up, if I read your graph correctly:

-You’ll want your toughness and vitality to be roughly equal for optimal health.
-If you have lower max health, you’ll want to have a bit more vitality.
-If you have higher max health you’ll want a bit more toughness.
-The breakoff point being at around 775 toughness/vitality.

Is that correct?

The base breakpoint is 755

I can’t do math, but you remembered me that I had to say this in some post at some point: I was doing a test with a friend (he has a warrior), trying to get the same level of armor and HP and seeing who lasted more.
Warrior still lasts more during fights than Elementalist, despite the numbers being pretty similar. Is there any kind of hidden rule, aside the super OP warrior healing signet?

Grevender.9235:

I can’t do math, but you remembered me that I had to say this in some post at some point: I was doing a test with a friend (he has a warrior), trying to get the same level of armor and HP and seeing who lasted more.
Warrior still lasts more during fights than Elementalist, despite the numbers being pretty similar. Is there any kind of hidden rule, aside the super OP warrior healing signet?

Do you mean against eachother? Against certain enemies? Were you both equally skilled? What about traits?

ThiBash.5634:

Do you mean against eachother? Against certain enemies? Were you both equally skilled? What about traits?

I don’t know if OP is ok with derailing the original thread xD
if you want, I can pm more info.
BTW, the gap in damage mitigation was so evident, that it lead me to think that behind the numbers may lie something else, unless the above mentioned warrior healing signet is THAT broken of being capable of single handedly making the difference.
I will do more testing, that’s granted.

@ Zelyhn:
Thanks :-)
(By the way, it’s nice that you keep your guide up-to-date, good reading.)

@ Grevender and ThiBash:
Yes, in principle you should be able to add vitality and toughness to the Elementalist to match the starting values of a Warrior.

The way effective health is computed, the Elementalist needs
18372 – 10805 = 7567 additional health or 757 vitality and
2127 – 1836 = 291 additional toughness
to get the effective health of the starting Warrior. That’s 1050 points!

Well, as you say, we then have to make sure everything else is equal, too! No extra boni from Signets or healing etc!

But if the other things are equal, then your upgraded Ele should take as little damage from direct hits as your basic Warrior. At least this is what the formulas for effective health say, i.e. there should not be any “hidden” boni for warrior.

@ ThiBash:
The basic conclusion is that (without healing!)
- invest the first 750 points in vitality,
- divide any additional points equally between vitality and toughness.

For example, suppose you have 1000 points to spend. The plot tells you to put about 875 into vitality to get the maximum effective health. That is 750 in vitality to reach the balance, plus 125 in vitality and 125 in toughness.

If you assume a single 7500 heal, then this already reaches the balance, so then
vitality and toughness are equally important. But healing can be very situational.

Edited the original post to make this a little clearer.

Speaking of signet, it´s OP.

Ele with perfect healing rotation(everything cast the moment they´re off cd), atleast 15in water and 30 in arcane can get slightly better healing per second than what warrior gets from his passive.

Condition damage. Toughness is worth zero.

Abraxax.7135:

@ ThiBash:
The basic conclusion is that (without healing!)
- invest the first 750 points in vitality,
- divide any additional points equally between vitality and toughness.

For example, suppose you have 1000 points to spend. The plot tells you to put about 875 into vitality to get the maximum effective health. That is 750 in vitality to reach the balance, plus 125 in vitality and 125 in toughness.

If you assume a single 7500 heal, then this already reaches the balance, so then
vitality and toughness are equally important. But healing can be very situational.

Edited the original post to make this a little clearer.

I see. That wasn’t clear from the image. But regardless, you’re basically saying that toughness is virtually useless (beyond the first 100)?

Seriously can’t believe that someone did actually take the time do the math for this lol. Seriously?

Lettuce.2945:

Seriously can’t believe that someone did actually take the time do the math for this lol. Seriously?

Seriously, I can’t believe how useless your comments usually are. seriously.

Such an analysis only provides a narrow view of the overall problem, because it considers too few variables. In your analysis you forgot to take Protection into account. Boons define the elementalist. Some build can keep permanently protection up, giving you 0.33 damage reduction. Protection influences the effect and the optimal value of toughness. Ironically, protection reduces the marginal damage mitigation effect of toughness.

FirstBlood.7359:

Such an analysis only provides a narrow view of the overall problem, because it considers too few variables. In your analysis you forgot to take Protection into account. Boons define the elementalist. Some build can keep permanently protection up, giving you 0.33 damage reduction. Protection influences the effect and the optimal value of toughness. Ironically, protection reduces the marginal damage mitigation effect of toughness.

Not really. We can see how bad toughness is without massive amounts of vitality.

Lettuce.2945:

Seriously can’t believe that someone did actually take the time do the math for this lol. Seriously?

Well maybe his trying to prove that he knows how to use a calculator? 0_o

Guys, the maths behind these calculations is not that hard.
It looks complicated only because Abraxax is being meticulous and aims for perfect accuracy, so that people can understand what he is talking about.

No worries. These calculations are just that: calculations, theory. Don’t take them too seriously. I am the first to say: game play is everything!

Personally, I find it intriguing that some isolated parts of a game like GW2 can be calculated. Imagine you design a game. Somewhere some math happens. I find it fascinating that some “simple” formulas can make a rather balanced game. But great game play certainly doesn’t require you to know the math. Just my 2 cents.

@FirstBlood: I completely agree, there are many other variables.
Good point, we should also look at protection!

Protection

Protection is a really intriguing boon, together with Might my favorite for the Ele. The post was simply getting too long, but here are my thoughts. I’ll ignore other types of damage reduction like Frost Aura.

Here we go again, don’t read on if you don’t want the math :-)

Protection and Toughness

The formulas indicate that protection becomes more efficient with increasing toughness, and toughness is more efficient when protection is on. The reason is that in the end we have to multiply two numbers for the effect of protection and toughness, which enhance each other.

Protection reduces incoming damage by 33%, so we have to multiply by a factor 2/3,

dps = 2/3 * B/a

To obtain the time to live, we divide health by dps,

ttl = 3/2 * h * a/B

Reducing incoming damage per second by 33% means that we live 50% longer (that is we can take 50% more damage).

How does this relate to toughness? Basically, the combined effect of armor and protection is

b = 3/2 * (a0 + t)

As already mentioned above as an example, to get that factor 3/2 from toughness alone we would need t=a0/2=918 so that

a0 + t = 3/2 a0.

In other words, Protection is worth 918 points in toughness when you start out with t=0. This is huge! Just compare it with other ways to get toughness.

Suppose now that you do invest 918 points in toughness, and add protection on top of that:

b = 3/2 * 3/2 * a0 = 9/4 a0 = 2.25 a0

You would need

t = 5/4 a0 = 2290

to get such damage reduction from toughness alone.

That is, the more toughness you already have, the more effective protection becomes. And conversely, if protection is on, then adding toughness has a higher return than without protection. At least when looking at your time-to-live.

Protection and Effective Health

What is perhaps surprising is that the strategy for optimizing effective health does not change! (I hope I am not confused about this ;-)

The reason is that protection appears in time-to-live as an overall factor. We define effective health with respect to t=0:

ttl = 3/2 * h * a/B = 3/2 * eh * a0/B
=> eh = h*a/a0

and the factor 3/2 drops out together with B. So whether protection is on or not, the same rules for getting optimal returns for v and t apply. (The size of the maximum in ttl changes, but not its location.) That is, we do not have to change gear or traits depending on whether we get more or less protection.

But careful, we have to be clear about the question we ask. The optimization does not change if we assume that we get protection for free. It is a different question whether it is more efficient to gear/trait in order to get protection instead of toughness or vitality. This is complicated. One stat point buys us the same amount of toughness or vitality, but how to compare with protection? So you give up t and v from runes, trait, food, or gear in order to get ArcanaV, boon duration, etc. for protection up-time.

Basically, we would want to generalize effective health to a function of v, t, and protection. For that we would need the conversion rate, otherwise we cannot define our finite budget like v+t+something*protection=totalpoints. It would be nice to have some game-play results. I didn’t do this systematically, perhaps someone can share his experience.

My rule of thumb is as follows. Since protection is easily worth 1000 to 2000 points in toughness, get some. This is situational (do you run with a Guardian? how easily does your playstyle allow protection? …).

However, what ever resources you have left to invest in vitality and toughness, go by the law of balance. As an Elementalist, first get 7500 health or healing, then add vitality and toughness in about the same proportion.

Abraxax.7135:

However, what ever resources you have left to invest in vitality and toughness, go by the law of balance. As an Elementalist, first get 7500 health or healing, then add vitality and toughness in about the same proportion.

I believe a nude elementalist at 80 starts with 10300 health

Fuzzion.2504:

Abraxax.7135:

However, what ever resources you have left to invest in vitality and toughness, go by the law of balance. As an Elementalist, first get 7500 health or healing, then add vitality and toughness in about the same proportion.

I believe a nude elementalist at 80 starts with 10300 health

He means on top of starting values.

The only math required for the Elementalist is actually:

damage output = 3000 / number of autoattacks to kill you

Antiriad.7160:

The only math required for the Elementalist is actually:

damage output = 3000 / number of autoattacks to kill you

I thought it was:
TTL = worse then everyone else because this is the good at nothing profession.

On a serious note, thanks for all the calculations, esp the Protections ones. I currently run a 100% protection build and its good to see whakittenactually provides and that I should try to up my toughness. Though I play sPvP so uping toughness isn’t all that easy.

So basically we need to be bumped up to the next health value of 15k.

^I’d be fine with the low health if we had damage to make up for it. Anet has no clue about what glass cannon is. They gave us the glass, but forgot the cannon part. Our abysmal health, armor, and mobility would only be acceptable if they doubled or tripled our damage output on all skills. They either need to bump up the damage or buff our defensive stats to more reasonable levels. Being a glass slingshot is not acceptable. Give us the cannon part, or remove the glass part.

Nikkinella.8254:

Being a glass slingshot is not acceptable. Give us the cannon part, or remove the glass part.

This thread is great. But, I just wanted to say “glass slingshot” made me LOL; gunna shamelessly steal that phrase.

Theres a dev who’s in one of the opposing teams in the matchup i’m currently in. He’s playing an ele. My guild which is mostly warriors now has made it a mission to follow and kill him as much as possible. Its our own personal guild bounty. Let’s see if he still thinks eles are “ok for the most part” when we’re through with him.

Did he troll you all by dancing over your bodies with obsidian flesh and chant, “ele is balanced.”?

We only saw him a couple times and he always runs away back into a tower. Yeah any class is ok if you hide in a zerg or tower. I will get him eventually, it is my mission now.

Nikkinella.8254:

We only saw him a couple times and he always runs away back into a tower. Yeah any class is ok if you hide in a zerg or tower. I will get him eventually, it is my mission now.

So basically, you want to convince a dev that eles aren’t balanced because they cannot take down an entire guild of warriors all by themselves?

Do you see the logic flaw in your plan?

ThiBash.5634:

Nikkinella.8254:

We only saw him a couple times and he always runs away back into a tower. Yeah any class is ok if you hide in a zerg or tower. I will get him eventually, it is my mission now.

So basically, you want to convince a dev that eles aren’t balanced because they cannot take down an entire guild of warriors all by themselves?

Do you see the logic flaw in your plan?

we weren’t all going to jump him at once. take turns killing him 1 on 1.

Ah, ok. :-) I was having visions of a horde of warriors rushing towards that single elementalist like Gru’s minions to a banana…