Hungering Arrow ~130.8% Average Weapon Damage

[Bridgeburners]

I made a thread like this a while ago on the BNet forums, but I think it fits here.

I used to choose Evasive Fire over Hungering Arrow in the past, because I thought it did better damage per shot. However, the discipline guzzling backflip always drove me nuts, especially when you didn't need to back off and it just slows down your traversing speed, or if you get cornered and you spend discipline on nothing. This was before I decided to do the calculation, and realize that Hungering Arrow, on average, does a great deal more damage than Evasive Fire. In fact, its 130.769% average weapon damage beats Evasive Fire's 115% by a whole 15.769%. Now I always choose Hungering Arrow over Evasive Fire, and not only do I do more damage while generating Hatred, but I don't have that annoying backflip.

Hungering Arrow's impact damage is 85% WD, and it has a 35% chance of piercing. When it hits a single target, distant enough from others (this happens often), a pierced arrow will hit the same target. There is no limit to the number of times a single arrow can pierce. Every single impact has a 35% chance to pierce, no matter how many times it did so before.

I define "Average Weapon Damage" as the expectation value of a single HA's damage vs a target. To be clear, in the limit of infinite Hungering Arrows, the average damage each will have done would be exactly equal to the expectation value of a single Hungering Arrow, which I claim to be 130.769%.

For a single unleashed Hungering Arrow, we can calculate the expected damage of each hit. The first hit has a probability of 1 to land. The second pierced arrow has a probability of 0.35 of appearing, to hit the target. The third has a probability 0.35². Etc. The Nth arrow has a probability 0.35^(N-1) of landing (the Nth arrow corresponds to N-1 pierces).

The expected damage of each arrow is the damage it would deal multiplied by the probability of its appearance. (I refer, in this case, to all the possible pierced appearances of a single casted Hungering Arrow.) Thus, the expected damage done by the Nth arrow is [0.35^(N-1)]*85% weapon damage.

The expected damage of a single unleashed Hungering Arrow is the sum of the expected damage of each pierced shot. Thus

<HA Damage> = Sum{i=0..Infinity} [0.35^i]*85% Weapon damage

This is a well known geometric series. For any real value x such that |x| < 1, we have
Sum{i = 0..Infinity}x^i = 1/(1-x).

Thus, the expectation value of the damage of a single Hungering Arrow is

<HA Damage> = [1/(1 - 0.35)]*85% Weapon Damage = 130.769% Weapon Damage

One might argue that you should set an upper bound to the sum, since all enemies must die eventually. I argue that the upper bound need not be considered, because if an enemy dies to a sequence of Hungering Arrow pierces, you could perceive that as overkill. However, for argument's sake, we can look at the expected damage of HA assuming an upper bound. We do this by revisiting the well known geometric series. The following sum is true for all x in C except x = 1 (it's no longer constrained as it is in the unbounded sum):

Sum{i=0..N}x^i = [1 - x^(N+1)]/[1-x]

Note that this reduces to the previous sum quoted, if you set |x| < 1 and N -> Infinity

So assuming we only allow up to N pierces (i.e. N+1 hits) for a single Hungering Arrow, the expected damage is

<HA bounded damage> = [1 - 0.35^(N+1)]/[1 - 0.35] * 85% Weapon Damage

For example, if we limit to ten hits (N = 9), we get 130.766% WD, which is 0.003% WD different from the unbounded damage. This is an indication of how quickly the series converges; how heavily weighted the expectation value damage is by the first few hits.

Hungering Arrow is the best single target damage dealing hatred generator.

[Gelly]

Good post and solid calculations as far as I can tell. So there's going to be a trade-off between the defensive Evasive Fire or the offensive Hungering Arrow. There's still the runes though, so we'll have to see how that goes.

[Bridgeburners]

So the majority of rune benefits to HA are qualitative. However, the golden and obsidian benefits are purely a matter of increasing the expectation damage, and can be quantitatively analyzed.

Golden runed HA increases the piercing probability. This is a trivial analysis, as I've shown that for D damage and p chance to pierce, the expected damage is

<HA Damage> = D/(1-p).

For a rank 4 gold rune, with D = 85% weapon damage and p = 0.46, we get

<Gold HA Damage> = 157.41% weapon damage.

Obsidian runestone has a more complicated benefit. It adds 24% of its damage for each successive pierce, on top of the 85%. For example, the third pierced arrow (fourth hit) would do 85(1 + 3*0.24) weapon damage, or 146.2% weapon damage. Let's talk general terms. It has a base damage D, a bonus of f damage per successive pierce (f = 0.24 in the rank 4 case), and a probability p of piercing. Thus, the Nth pierced arrow will do damage D(1 + Nf). The expected hungering arrow damage in this case is

<Obsidian HA Damage> = Sum{n=0..Infinity} D(1 + nf)p^n
= Sum{n=0..Infinity}D*p^n + DSum{n=0..Infinity}f*n*p^n

We know the form of the first sum, that's just D/(1-p)
For the second sum, we need to know the following sum for |p| < 1:
Sum{i=0..Infinity}i*p^i

well that's simply p times the derivative of the sum

Sum{i=0..Infinity}i*p^i = p*(d/dp)Sum{i=0..Infinity}p^i = p*(d/dp)[1/(1-p)] = p/(1-p)²

Thus we can solve the obsidian average HA damage:

<Obsidian HA damage> = D/(1-p) + Dfp/(1-p)²

For a rank 4 obsidian rune, with D = 85% weapon damage, d = 24% weapon damage, and p = 0.35, we have
<Obsidian HA damage> = 147.67% weapon damage

Less than the gold rune benefit.

EDIT: Actually, we don't know if the bonus for obsidian is additive or multiplicative. My above calculation assumed it's additive. If it's multiplicative (i.e. +24% of the damage done by the last arrow) then the calculation is much simpler:

<Obsidian HA damage> = Sum{n=0..Infinity} D (1+f)^n p^n = D/(1 - (1+f)p)

Thus, if it's additive, it does 147.67% weapon damage, and if it's multiplicative, it does 150.18% weapon damage, both below the gold rune benefit.

[snurrfint]

With only 7% more dmg for golden and rank 4 runestones there is a significant chance that obsidian will do more dmg with lvl 7 rune stones.

However, If indigo rune splits in to 5 piercing arrows I think we have a winner in the dmg department. Then we have 85+0.35*130*5 dmg = 315 % wd on average.

[Bridgeburners]

With only 7% more dmg for golden and rank 4 runestones there is a significant chance that obsidian will do more dmg with lvl 7 rune stones.

On the contrary. Note the expression for average damage as a function of "d", where d is the obsidian compound bonus:

<Obsidian HA damage> = D/(1-p) + d*p/(1-p)²

The bonus increases linearly as a function of d. Now notice how it benefits with p:

<Gold HA Damage> = D/(1-p)

It grows as a function of 1/(1-p) which is a rapidly accelerating function of p. In order to level the playing field for the two rune benefits, the increase in p with runestone rank better have a diminishing return. Otherwise, if both d and p increase linearly with their respective rune rank, gold will end up blowing obsidian well out of the water.

However, If indigo rune splits in to 5 piercing arrows I think we have a winner in the dmg department. Then we have 85+0.35*130*5 dmg = 315 % wd on average.

True. However, I doubt that's the case. I think they'll simply turn into straight firing arrows. But we'll see. I'd hate for such a horrible imbalance to be present simply because of a math failure of the developers.

[snurrfint]

True. However, I doubt that's the case. I think they'll simply turn into straight firing arrows. But we'll see. I'd hate for such a horrible imbalance to be present simply because of a math failure of the developers.

Yeah, I doubt that too. Then it's 85+0.35*85*5 = 233% if every arrow hits. Its probably more reasonable to think that 2.5 arrows hit at average. Then its 160% wd. Thats pretty ok aswell.

[Bridgeburners]

New Hungering Arrow does 176.923% average weapon damage. With the rune effect that increases its pierce chance, its damage is boosted to a whopping mother ****ing 255.556% average weapon damage!

This convinces me that the skill development team doesn't know their geometric series'. The rune effected Hungering Arrow does more average single target damage than Pierce or Rapid Fire, as well as most other class' resource costing single target spells.

[Elfik]

New Hungering Arrow does 176.923% average weapon damage. With the rune effect that increases its pierce chance, its damage is boosted to a whopping mother ****ing 255.556% average weapon damage!

This convinces me that the skill development team doesn't know their geometric series'. The rune effected Hungering Arrow does more average single target damage than Pierce or Rapid Fire, as well as most other class' resource costing single target spells.

Yeah it just doesn't make sense some of the skill balancing Blizzard comes up with.

For example: http://diablonut.incgamers.com/skill/wizard/arcane-orb

Look at "tap the source" from arcane orb skill. It reduces the casting cost to 23 AP, meaning one can cast it 50% more often (35/23 = 1.52). On the other hand "Obliteration" increases damage from 250% to 325%, only a 30% increase! Factor in overkill and "tap the source" is so much better. Of course, "arcane nova" increasing the area of effect by 300% is arguably better than both. I just hope these were hastily done.

[Buu]

Yeah it just doesn't make sense some of the skill balancing Blizzard comes up with.

For example: http://diablonut.incgamers.com/skill/wizard/arcane-orb

Look at "tap the source" from arcane orb skill. It reduces the casting cost to 23 AP, meaning one can cast it 50% more often (35/23 = 1.52). On the other hand "Obliteration" increases damage from 250% to 325%, only a 30% increase! Factor in overkill and "tap the source" is so much better. Of course, "arcane nova" increasing the area of effect by 300% is arguably better than both. I just hope these were hastily done.

One thing that must be pointed out is that monsters have defenses.

100 damage at half the cost vs 130 damage at normal cost.

The first option does seem more appealing due to twice the number of attacks that can be made.

but if monsters have say 80 def, that turns it into a matter of 20 damage vs 50 damage. Now the second option is more appealing because even though it's hitting 1/2 as often, it's still out damaging the first option.

And yes, this is a very simplified example but i feel it still drives the point home, that just looking at attack damage isn't the best way to judge these stats.

[Elfik]

One thing that must be pointed out is that monsters have defenses.

100 damage at half the cost vs 130 damage at normal cost.

The first option does seem more appealing due to twice the number of attacks that can be made.

but if monsters have say 80 def, that turns it into a matter of 20 damage vs 50 damage. Now the second option is more appealing because even though it's hitting 1/2 as often, it's still out damaging the first option.

And yes, this is a very simplified example but i feel it still drives the point home, that just looking at attack damage isn't the best way to judge these stats.

Okay thanks for clarifying that. I didn't know defense worked that way (thought it worked by percent damage reduction). However, if defense is such a big factor in inferno, then aoe attacks would start to lose effectiveness relative to high damage single target, and fast weapons would lose out to slower weapons with equal DPS. If the extreme case you mentioned above occurs, I think the least of my worries would be which rune I'm using. I kind of doubt defense will be much of a factor when deciding on skills and runes, and if it is, that's kind of disappointing.