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So, just for fun I decided to punch some numbers in order to figure out which runestone maximizes expected damage for the Hungering Arrow skill. Here are the results with a brief explanation of my method. Comments or corrections are welcome, but this should give a good idea about whether one runestone dominates over another for this particular skill:
First, let's go from base attack to active skill. Base attack will be normalized at damage 1 (that is, we'll assume 1 attack for 1 damage). This normalization doesn't matter, and could be set to any arbitrary number (your attack damage, i.e.), but is set to 1 for ease of calculation.
The result: a geometric series that I'll truncate at n=5 monsters on screen and in range of the arrow. Sums to 1.3x weapon damage. I'll keep n=5 from here on out. Lowering this number could potentially have large effects. Raising it will not.
So now from 1.3x base multiplier to each of the runes:
White: Add the crit possibility of 10% for this calculation for each possible hit. Assume explosions damage all 5 monsters. With this assumption I get a (decent) 1.66x multiplier. Of course, higher crit chance would raise this multiplier.
Red: Simply increase the base from 85% to 115%. Get 1.76x multiplier. Not bad.
Gold: Raise geometric sum from 35% to 46%. Easy and assumption-free. Get 1.81x. Getting better.
Indigo: Wow. 5 arrows. Even if we assume the 5 arrows have no additional chance to pierce, a quick expected value calculation of 0.85(1+(.35)(5)) gets us a 2.34x multiplier. Seems rather high. Am I missing something here?
Obsidian: Blah. A 24% increase is quickly gobbled up by the fast decrease of the piercing chance in the geometric series. A lowly 1.48x multiplier. Seems rather low, maybe I missed something here.
Anyway, seems the indigo wins out here by quite a bit. Anyone wanna try this for a different skill?
Edited by Mandlebarb on 2/15/2012 5:31 PM PST
So, for the, like, 3 of you actually interested, I did some numerical simulations on this as well. I made some Matlab code that simulated 10,000 attacks with each of the runstones. Here's what I came up with for average multiplier:
Base to skill: 1.292x
Edited by Mandlebarb on 2/16/2012 10:25 PM PST
It's also interesting to note that part of the reason why indigo is so high is because of the assumption that there are 4-5 other targets on screen for the arrows to hit. Its damage drops precipitously for one or two targets. Still, there seems to be a correct answer here: usually indigo, when there are hordes of monsters to fight. For the boss, switch to red.
And unfortunately, this is a pretty big if. I trust that Blizzard is doing similar calculations on even larger scales and balancing everything behind the scenes more precisely than I can extract from their tooltips. I just went from what I could glean about the Hungering Arrow skill from reading the descriptions. I don't pretend to know all of the little things and mechanics involved in churning out the final numbers. But, these should be close.
You need to truncate it because otherwise you're assuming an infinite amount of monsters to kill. I was using the simple formula for truncation, though for certain other color runes it gets a little more complicated. But, as you can see, stopping after the 5th order only gives about 0.8% change, so it really doesn't matter too much. Though now that I think about it, can the arrow jump back to the same target multiple times without needing a different target? Then we have a case for the full infinite series.
Also, my numerical simulation seems to support my methods generally. Though with that one weird exception that I haven't looked into.
Edited by Mandlebarb on 2/15/2012 5:24 PM PST
Some of the math may be getting over my head, I only understand basics on geometrics, (wolfram alpha is my crutch) could you provide a more verbose calculation on how you are arriving at the conclusions; can't seem to reproduce the alabaster 1.66x multiple. you mean 141.1% weapon?
as for the red, do we assume the fire can stack on a single target?
1.81 on the 46% pierce? 1 / (1-.46) = 1.85 * .85 = (157.4% weapon)
does the 5 split mean 5 hungering arrows that can no longer pierce, could all 5 spawns hit the first target if it is the only one? or is it a simple multi shot spray out the back >.<
obsidian, we can't determine if it is 24% multiplicative or +24% static weapon additions.
too many unknowns but eventually I'd like to see if it is possible to do a table on a collection of samples 1 - 5 monsters, with returns on the rune given n over infinity for averages.
Edited by Korba on 2/16/2012 9:54 AM PST
can't seem to reproduce the alabaster 1.66x multiple.
The alabaster has by far the most assumptions. I added a 10% crit chance to any hit in a series and assumed that, if successful, it would do (.39)(5) normalized damage, or 39% weapon damage to all 5 enemies in the radius of the explosion. It came up with about 1.6x, or 160% weapon damage. This could vary drastically by changing the aforementioned assumptions, however.
Yeah, I apparently messed up when punching in the numbers. My simulation had it at 1.53x, though, so close to what you calculated. Truncating at n=5 probably accounts for a little of that difference.
If all 5 could pierce, it would do significantly more damage. And it's already the highest. If all 5 could pierce AND retained the shatter effect, it would cascade and the expected value for damage would grow towards infinity. This would be hilarious, and broken.
No, we cannot. I assume a 24% compound increase, meaning first 124%, then 124% plus 24% of that 124%. It actually barely has an effect because of how low of a chance there is for more than a pierce or two.
considering how bola splash works, I get the feeling you should assume the explosion excludes primary targets. so just against 4(.39) to be conservative
Yeah, you're probably right about that. Again, though, it wouldn't have all that much of an effect. Changing the crit chance would incite a larger change. Would be nice if Blizz's tooltips were a little more detailed. For example, we also don't really know if shatter shot will fire 5 arrows in a cone, or 5 seeking arrows, or whether or not they will be able to re-target the original monster hit. That could change things quite a bit.
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