A stacking guide

General Discussion
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0. Intro

Hello. This is supposed to be a guide to one of the games most important mechanics overall. Since i see people including myself getting things wrong quite often i figured it was necessary. You might believe it’s unnecessary at first but there will be some impact for builds that are currently played. Let’s start with the basics that i think most people know, the guide will get more important as you continue to read.

1. What is stacking?

First of all the easy explanation to stacking is that different effects of your gear interact multiplicatively instead of additive. This effect can work for and against you: Having two items equipped INcreasing your damage by 100% gives you an increase of 300% to your damage (your doing 400% of the initial damage which is a 300% increase) instead of the additive 200%.
Having two items equipped DEcreasing your cooldowns by 10% each gives you a total of 19% CDR instead of the additive 20%.
Wether stacking works for or against you depends on the multiplicator on the item: If it’s less than 100%, stacking decreases the total effect and vice versa.

2. Why is stacking so important?

This is the part where i will start to do the actual maths. Remember to multiplicate/divide before you add/subtract.

Let x be the item modifier(e.g. legendary affix) as stated in the description. The actual multiplicator is (1+x) if it’s an increase and (1-x) if it’s a decrease.
For items having different values the total modifier you get is just a product of all the multiplicators.
Let’s assume you have n items equipped all with the same modifier value of x, it means the total modifier is
(1+x)^n-1 for an an increase and
1-(1-x)^n for a decrease.

As these are exponential functions we can immediately draw one very important conclusion: Having lots of items with the same increasing modifier equipped is way more important than having some items with huge modifiers. For decreasing modifiers it’s the other way around. This is why legendary gems, passives, elemental damage rolls and items like the mage fists etc. are so important even though they have low item modifier values.

Another, less obvious conclusion is about the distribution of the value on your items: Stacking is most impactful when all items equipped have the same x-value. So when there’s a choice between two sets of items (mathematical sets, not ingame sets) which add up to the same value, the first all having roughly the same value and the second having huge differences between the modifiers, remember the following rule: if it’s an increasing modifier, choose the first set. if it’s a decreasing modifier, choose the second set.

3. Which items stack? Which don’t?

Now here comes the first interesting part by using an actual examples:
First of all i want to point out that this is no offense to Rhykker, i do like and use his builds quite often. The example just fits perfectly to my point.

In this build guide Rhykker talks about how the Tal Rasha 4 piece bonus stacks. As many pointed out in the comments the Tal’s 4-piece gives resistance and not damage reduction but here is what a lot of people didn’t get: Even if it were damage reduction, would they stack as he explained in the video? The mechanic of Tal’s 4 piece is determined by the amound of stacks of a buff you get from the set.

Let’s use our formula by taking a look at the most extreme case of this exact mechanic: vyrs 6-piece bonus. Let’s assume you have 200 archon stacks. Additive this would give you 20.000% more damage. But if they stacked by using our formula we came up with roughly 1.61*10^58% damage increase.

Since that is obviously not the case (it would probably crash your game) we can draw one very important conclusion from this: Buffs you get from the same item or spell don’t stack with themselves, only different items or spells can stack.
Any further question you have about wether something stacks:
-The short answer is: yes.
-The long answer is: maybe you ask the wrong question. e.g. there’s a debate wether or not the new captains set does stack with cdr. Obviously the 2 piece bonus cdr does stack, but that’s not what you want to know. The question wether the captains set uses the sum of all CDR you have or the final calculated stack of cdr you actually get for it’s 3-piece-bonus is not a question about how stacking works but a question about the mechanic of the set. Since obviously every single item, set, spell, skill and so on uses individual programmed mechanics you will never ever get a general answer that applies to all items on a guide like this one. Search for the specific thing you want something to know about, e.g. in the diablo wiki and if you don’t get an answer try to test it ingame and then ask your specific question in the diablo forum.

4. What affixes can stack?

This is the most important part of this guide since pretty much everyone i know gets this wrong: Every single one of the item affixes, being legendary blue or whatever, can and will stack in some way, even if it doesn’t seem like it at first. This has some pretty important impact; Let’s take the very popular LazyStorm build as an example.
As you might know LazyStorm is all about stacking as much crowd control (cc) chance as possible to proc the triple krysbins sentence bonus.
Now there is Crowd Control Resistance of mobs in Diablo 3. I won’t go all into detail (use the diablo wiki), but basically after you’ve procced some kind of cc like a stun to a mob, you can notice that the mob can’t be cc-ed for a couple of seconds. This is because every item and skill in the game that gives some kind of cc is only a percentage chance and even if the percentage is 100% (e.g. Command Skeletons - Freezing Grasp) this chance can and will be decreased by cc resistances. Every cc applied to a mob causes it to gain resistance to all ccs. This mechanic basically means that if you have different items with cc chance on it equipped you will increase the overall chance to cause any cc but you will decrease the chance for every single item to proc their specific cc. Now there are different values of the cc: The charm of bone spirit will last 10 seconds while the blind of blind faith will last around three seconds. If you had the choice you certainly would take the charm but by taking both the spirit and the helm you can’t choose. This leads to following calculations:

Let’s do it as an example with bone spirit and blind faith:

Bone spirit - Possession has 100% chance per hit to charm for 10 seconds. The mob will gain 10% cc resistance every second charmed up to 95% cc resistance. After that the resistance will drop 5% per second.
Blind Faith has 40% chance per hit to blind for 3 seconds. The mob will gain 10% cc resistance every second blinded up to 20%. After that the resistance will drop 5% per second.

By using the Gauss formula for the sum of all natural numbers up to a given value multiplied by the 5% per second we can calculate:
The average time to proc a charm after a charm ran out is 5.84 seconds.
The average time to proc a blind after a charm ran out is 17.84 seconds.
The average time to proc a charm after a blind ran out is 1.24 seconds.
The average time to proc a blind after a blind ran out is 3.73 seconds.
(Formula: (starting resistanc-proc chance)*x+5%*(x*(x+1)/2))=100%

The proc chance ratio is constant:
charm procs/blind procs=100%/40%=5/2
which means out of 7 procs there are 5 charms and 2 blinds.

Running only bone spirit means the average time between two procs
is 5.84 sec.
This means mobs are cc-ed 10/(10+5.84)=63.11% of the time.
Running only blind faith means the average time between two procs
is 3.73 sec.
This means mobs are cc-ed 3/(3+3.73)=44.58% of the time.
Running both means the average time between procs using simple statistic is
(5*5*5.84+2*5*17.84+5*2*1.24+2*2*3.73)/(7*7)=7.18 sec.
This means mobs are cc-ed
((2/7)*2+(5/7)*10)/((2/7)*2+(5/7)*10+7.18)=51,79% of the time.

So equipping blind faith decreases the chance to proc bone spirit charm resulting in a decreased cc uptime by 11.32%. (This does not account cooldown or attack speed though).
If you consider cc chances not as an increase but as a decrease of the chance that mobs will not get cc-ed, this works exactly like the kind of stacking we want to minimize.

For proc chance calculating while running both using the formula from part two for decreased stacking chance gets you pretty good results for the actual proc chance:
The average cc resistance of a monster before a proc is the average time formula from above with the proc chance of 0% and divided by the average time and scaled for the proc chance:
charm:
(5/7)*(95%-5%*(5.84+1)/2)+(2/7)*(20%-5%*(1.24+1)/2)=59.76%
blind:
(5/7)*(95%-5%*(17.84+1)/2)+(2/7)*(20%-5%*(3.73+1)/2)=36.55%
That means the overall proc rate of any cc using the decreased stacking formula from part two is
1-(1-(100%-59.75%))*(1-(40%-36.55%))=42.31% This is of course more procs than with a single cc, but it’s less than added both together.
Testing seems to confirm this. Now we can use statistics again:
charm proc chance is (5/7)*42.31%=30.22%
blind proc chance is (2/7)*42.31%=12.08%

As a result, Lazystorm should not use so much cc and instead stick to a single, valuable item like Brigg’s Wrath that has a knockback AoE. Since we don’t need blind faith anymore, we can easily swap Ring of the Royal Grandeur out for Brigg’s. Your follower can cc the rift guardian (who’s immune to knockback).

Now we’ve gotten all the formulas we need to calculate anything about cc. This is of course only using two cc sources. But it’s entirely possible to calculate more.

That kind of mechanics are applied to every possible offensive affix that first does not seem to stack. But it does. This is of course only one example for a stat stacking with itself, but there are a lot more stacking with other stats an/or itself. That is why everything in this game stacks with something and why it is so important to know.

5. Some maths for fun

There is the natural exponential function hidden in the stacking calculating formula.
Let’s assume you have a total sum of +x% damage evenly distributed over n items.
The Formula for the final multiplicator is (1+x/n)^n-1
As the limit of n goes to infinity, the multiplicator goes to e^x-1.
That being said, the final multiplicator might converge to e^x-1 but e^x is a terrible estimate for the multiplicator since the average error also grows exponential the bigger x is.

6. Summary
-stacking is good for you at increasing stats (e.g. +%dmg, +%chd) and bad for you at decreasing stats (e.g. -%cooldown, +resistances(which decrease your damage taken), anything that has “chance” in it’s name)
-for the same increasing stat better have more less valuable items equipped than less more valuable items. For decreasing stats it’s the exact opposite.
-If you have the choice: for increasing stats choose items with roughly the same stat value. For decreasing choose items with big value differences.
-items, buffs, spells, skills and sets can’t stack with themselves.
-every affix can stack. even cc chances. We can only influence the amount.

I appreciate corrections and mistake spottings. Feel free to use, show or explain the guide as long as you don’t pretend you made it.

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I think there are some mistakes in this guide, or at least misleading informations. For example:

For elemental damage it doesn’t matter how the rolls are distributed among the items. If you have 10% + 20% + 10% Elemental damage on your gear or 12% + 14% + 14% or one item with 40% doesn’t make any difference: You will end up with 40% more damage for this element.

Elemental damage is so important, because it is a huge damage modifier for a single item. 20% more elemental damage on an item will push you by (at least) one GR-level. The same is true for +%Skill-Damage (in most cases). Even if Skill-Damage is in the “bad” additive / DIBS category, it is (in most cases) very strong, as there are only a few effects left in DIBS (exceptions like Wojahnni Assualter for LoN-DH should be considered as bugs, imho.) Therefore it’s “kinda multiplicative” as well. :wink:

This seems to mean that it is better to have more items with (low values of) +%chd than one item with (high value of) +%chd. Thats not true. All of your +%chd is added to a total amount of %chd. Again, it doesn’t matter how many rolls of chd you have. If the sum of all the values is the same, you will get the same damage modifier for critical hits.

And as a general comment:

That’s not what I understand as “stacking”. When I ask “does it stack”, I mostly want to know something about the mechanics of items like Strongarms (=> will there be more damage, if an enemy is knocked back by more than one player?) or the poison-gem (=> is there more damage when more than one player posions an enemy?). Whether this stacking mechanic is additive or multiplicative is then another question.

Or take Stricken: Players often speak about how quick they can add stacks to the RG. But the damage bonus by these stacks is added together, not multiplied.

I would rather say that “stacking” means exactly the opposite: If two effects (damage modifiers) “stack”, then both values are added first, and the sum works as one new multiplier in the total damage calculation.

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Source? Because in-game equipping shows exactly the doubled values (25% CDR/RCR for Diamond/Topaz, 46% extra life for Amethyst etc.). And you can’t even get more life from several ‘+% Life’ modifiers, because they are additive and actually suffer from diminishing returns.

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I agree that my point was misleading. When it comes to value distribution of items it does only matter if the stat stacks with itself. Of course things like elemental damage or chd can’t stack with themselves, but they stack with any other stats that increases the damage of a hit. Therefore having +10% elemantal dmg and +10% skill dmg is better than having +8% elemental dmg and +12% skill dmg.

You totally have a point here. It looks like i wanted to imply that, but of course it’s not true. Again, the point i wanted to make is that having many items with chd, elemental dmg, int, +%dmg and anything that boosts the dmg of a hit is better than having one single source with a single massive stat (like e.g. earthquake barb 6piece set bonus).

That is why blizzard has to buff sets so massively in order to push us just a few more GR-levels.

You are right. seems that somewhere in my testing i made a mistake. I’ll edit that out.

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I have never read someone using the phrase “chd stacks with elemental damage” or “skill damage stacks with mainstat” or “elemental damage stack with some set bonus”. That’s not what “stacking” means in D3.

Whenever the game guide speaks of “This effect stacks up to x times”, the bonus effect is added x-times, not multiplied x-times. For example the skill Soul Harvester. Which each stack, you get 3% more Int, up to a maximum of 15% more Int. That’s 3% * 5, not 3%^5. Or take Gogok, BotS, Taeguk … whenever we talk about “stacks” in D3, the different values on this stack are added together, not multiplied.

So why start using “stacking” with two different meanings?

Now you’re implying that some well-rolled legendary items with a good amount of chd, skill-damage, elemental-damage, mainstat and so on are better than a badly rolled class set. Again, that’s not true. (I do understand what you’re trying to say. But the “single massive stat” is often way better than the product of all the “regular stats”, because it’s - well- “massive”. :wink: )

That’s true, but it is mostly irrelevant, because you can never choose on a item between Skill and Elemental damage. However, the math behind this affects choices like “Mainstat or elemental damage on amulet”, or “skill damage or CHC on helms”, but it’s a little bit more complicated there, as the numbers have way different values in these cases.

Set bonuses do not push us “a few” GR levels. They teleport us into completely different GR-spheres. The sets are buffed so massively over time, because Blizzard had the philosophy “never nerf, always buff” when it comes to balancing. Through that, they started a power-creep-spiral, which spins faster and faster and lead us to completely absurd buffs.

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Well in your summary you made an essential mistake. The wording on your first point is so that it’s confusing.
You made it clear on your second point.

You get diminishing returns for stacking an “increasing stat” while the value for stacking a “decreasing stat” isn’t diminished at all.

for “decreasing stat” items/skills like RCR, CDR and DR that’s obviously the case.
however it’s easy to think the other way around especially since you need to take the armor/resistance interaction in consideration to this statement.

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Did you do your math correctly?

I am cornfused by your maths!

I can ask a group of people, “How are you doing?”

I am quite sure they would be cornfused If I asked them,
“How are you guys doings?”

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I knew my wording would confuse people since i’m not a native speaker. Me and my friends always used stacking in a double-meaning context. But i figured as long as it’s clear what i mean it doesn’t matter. Now maybe i gonna replace it with
“scaling”? What do you think?:wink:

The armor/reistance interaction is exactly one of the mechanics i talk about. resistances not only “scale” with themself giving diminishing returns but it also scales with armor in a 1:10 ratio.

The whole point of the guide was to have a single topic where players can share all their information about stat mechanics and correct each other since blizz is not that communicative when it comes to explaining game mechanics. So far i’ve learned that leorics crown doesn’t scale with itself which is great because i always believed the opposite. I’m currently working on a list including all Diablo3 stats and how and in which ratio they scale together.

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Just in case you didn’t know, apparently not everyone that speaks English is American or Canadian…
https://www.dictionary.com/e/math-vs-maths/

Both math and maths are short for the word mathematics. Math is the preferred term in the United States and Canada. Maths is the preferred term in the United Kingdom, Ireland, Australia, and other English-speaking places.

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You speak the obvious.
I wasn’t speaking about your mother tongue.
I wasn’t speaking about whether you speak English or some European language first.

I understand in India they use ‘maths’.

But in America it is ‘math’.

I assume you are not a Canadian, eh?

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That’s the point. Don’t assume that just because we’re on the US forums that everyone posting here is American. You were being condescending because someone used maths rather than math, perhaps without realising that for the vast majority of the English-speaking world, the term they use is maths, i.e. not the Americanised version.

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The part about CC effect immunity is a good nugget. Can we simplify the math into an easy to understand rule so decision making doesn’t require multistep calculations/excel?

Multiply the percent chance to proc by the effect duration, the bigger number is superior?
Only include another CC effect when the of value is less than X?

Is there a mathematician out there who can dumb it down?

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I am cornfused by this digression :sweat_smile::joy:

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Yes, I’m one outside the game but there is a lot of garbage and a few good bits of math in this thread, so sifting through stuff will take me a while. I just got here!

Edit: ok here are some initial comments:

The formula the OP states has no context on where it was derived from. While I could speculate where some parts of the formula come from, this is not the way math should be presented!

There isn’t anything in the game that stacks multiplicatively many times over to use the e^x approximation. Even Bane of the Stricken works by adding up all the stacks before multiplying by the modifier (e.g. 100 stacks of +1.2% is +120% which is a 2.20x damage multiplier).

Also, the “-1” in the formulas are wrong (if it was correct, then if x = 0 as in +0% damage, the multiplier in your formula would be 0 instead of 1).

This 1:10 ratio is a common misconception. To avoid confusion and over-complicating things, think of armor and resistances as providing a toughness multiplier [TM] instead of a damage reduction [DR] (they are related by TM = 1/(1-DR). Thus at level 70 (so the equations are simpler):

DR_armor = (Armor)/(Armor + 3500)
TM_armor = 1 + Armor/3500

DR_resist = (Resist)/(Resist + 350)
TM_resist = 1 + Resist/350

From these equations, it’s clear where the 1:10 ratio misconception comes from! If you could freely choose between adding armor OR resistances by adding Strength/Dexterity or Intelligence, then you would want a 1:10 ratio. However, each class can only add Armor OR Resistances through paragon! And furthermore, the +armor and +resistance affixes do not appear in the same quantity. Pants for example can get +750ish armor or +130 resistance.

In these situations, you want to optimize your toughness by “affix slot” not by “armor or resistance point”! Thus it will not quite be the 1:10 ratio! And in actual practice, you want all resist anywhere you can if you play a Str or Dex class and armor anywhere you can if you play an Int class since you get endless amounts of the other through paragon!

Finally, as you can see from the TM equations, there doesn’t reach a point when armor or resistance becomes irrelevant, the toughness boosts are almost linear!

There is one relevant item: Holy Point Shot.

That’s a fancy analogy! Where’s your proof that GRs are spheres though? Could they be cubes, tetrahedra, or even flat planes (#FlatEarth)!

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Actually, GRs are more like a compact manifold of a non-deterministic turing-machine operated by a Hilbert-turtle, but spheres are easier to calculate. :crazy_face:

And in most cases it is sufficient to model GRs as spheres - unless (of course) you use items like Holy Point Shot, which I totally forgot in my previous posting. :grin:

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Hi mate, im a physicist in real life. Alway glad to have some people with MINT background around. Outside of university i use mathematics from a practical point of view and i figured most people who really wanted to understand the equations would already know where they come from. Unfortunately this doesn’t seem to be the case since there are some errors in your reply. Im always open for mistake spottings and corrections but that being said calling the guide “a lot of garbage“ is just rude. Even though i’m a physicist i have feelings. So i will explain every step of my thoughts so even a mathematician can understand it;-) Unfortunately i don’t have much time today and writing equations from a smartphone is always a pain so i will do it when im sitting in front of my pc again.

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The “a lot of garbage” comment was referring to the entire thread, maybe I should have said fluff instead since there was a lot of unnecessary or redundant information. I apologize if my initial comment offended you.

There are also errors in several places, I’ll just highlight your summary.

Stacking useful stats is always helpful, but yes you need to consider alternate choices for certain slots since there are competing choices. Like stacking damage, stacking resistances increases your toughness roughly linearly (it’s actually affine linear) as seen from the toughness multiplier equation I posted in my previous post.

I see what you’re getting at here, as in your example of two 10% CDR rolls being weaker than one 20% CDR roll. However, for some things like elemental damage, the distribution doesn’t matter, 17%+18% = 15%+20%.

This doesn’t make sense, you should always aim to maximize a given roll for every useful affix. Why would I choose a 5% CDR roll with an 8% CDR roll when I could get two 8% rolls? Your logic would only be valid if the sum of the rolls was forced to add up to 13%.

Lastly, while I appreciate the effort spent into making this analysis, it is simply too dense for most players to read. Also, there are already existing guides which overlap with a lot of what you’re posting. One such guide is from the old forum:

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Lol i completely forgot about this topic two years ago. Has been a while since i logged into the us forum, usually im on the eu. Not my proudest actions here.

Well, after reading it again it looks like i have been rude too:

Don’t know, seemed to’ve had a bad day or smth like that. It’s never an appropriate way to respond and i apologize for that, even if it’s like two years late. That’s also the reason why i revived this topic, can’t let my statements stand like that. So yeah, apology accepted and given back, hope we’re good.

Yeah there are way better theorycrafters than me out there that i wasn’t aware of at the time. It’s good to have those people, sometimes d3 can be a bit vague and there’s definitely some use for more in-depth explanations sometimes. Just sad that the link doesn’t work anymore, but thanks :smiley:

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The necromancer has arrived and is bringing back the dead. :ghost:

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“It’s alive! ALIVE!”

Happy it got brought back from the dead though, fun read through.

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