Damage Reduction Explained

Content of this article:
  1. Introduction
  2. Damage Reduction Math
  3. Armor
  4. Resistance
  5. Other Damage Reduction
  6. Block & Flat Damage Reduction

Intoduction

Damage Reduction (DR) in Diablo III is a stat that (big surprise here) reduces the damage that you take. You can gain damage reduction from Armor, various Resistances and a lot of different items and skills.

You can find your current damage reduction in the UI, albeit not in one place. Hovering over Armor and individual Resistances in defense section of details sheet will show you damage reduction provided by these stats. You may notice that in case of Resistances you have two numbers separated by a plus sign. Don't be fooled, these two numbers are not added together. The first one is your damage reduction against a particular element; the second is most your other damage reductions lumped together.

This is quite a convoluted way to display your damage reduction, and it also ignores some of the effects for no reason, but that's what we have to deal with.

This guide will cover all different sources of damage reduction and similar effects, as well as rules for their stacking, and different item and skill interactions.


Damage Reduction Math

The formula for damage taken with damage reduction present is extremely simple

    \[\textrm{Damage Taken} = \textrm{Unmitigated Damage} \cdot (1 - \textrm{Damage Reduction})\]

When you have multiple sources of damage reduction, they always stack multiplicatively, meaning if you have \textrm{DR}_1 provided by say Armor and \textrm{DR}_2 provided by some item, then you total damage reduction will be:

    \[\textrm{DR} = 1 - (1 - \textrm{DR}_{1}) \cdot (1 - \textrm{DR}_{2})\]

Or, for the arbitrary number of DR sources:

    \[\textrm{DR} = 1 - \prod_{i=1}^{n} (1 - \textrm{DR}_{i})\]

Lets say for example you have Aquila Cuirass providing you 50% damage reduction and use Vengeance Dark Heart, which also gives 50% DR. Your total DR, in accordance with above formula will be:

    \[1 - (1 - 50\%) \cdot (1 - 50\%) = 0.75 = 75\%\]

It may seem that there's some kind of diminishing returns here: your Aquila Cuirass provides you exactly 50% DR, while Vengeance only adds 25% on top of that. This is not true. Each source of damage reduction does exactly what it says on the tin: reduces damage that you take by stated amount. That doesn't change regardless of what other damage reduction you may have. Using the above example, if you were to be hit by an attack that does 100 damage, it will be reduced to 50 by Aquila Cuirass and to 25 if you also activate Vengeance. 25 is exactly half of 50 so, as expected, activating Vengeance cuts damage taken in half.

Another, more simple way to think of damage reduction, is to treat each source of it as a toughness multiplier. Reducing damage taken by 50% is the same as doubling your toughness. For arbitrary value of damage reduction that multiplier will be equal to:

    \[\textrm{M}_{\textrm{DR}} = \frac{1}{1 - \textrm{DR}}\]

So for 50% DR it will be 2, for 60% - 2.5 and 5 for 80%. Then you just have to multiply all these values together and you get your total toughness multiplier, which instantly gives you an idea of how tanky a build is. For instance, if we take LoD Grim Scythe Necromancer we get:

  • Legacy of Dreams: 52% damage reduction, 2.0833 multiplier
  • Aquila Cuirass: 50% DR, 2.0 multiplier
  • Razeth's Volition: 50% DR, 2.0 multiplier
  • Dayntee's Binding: 50% DR, 2.0 multiplier
  • Unity: effectively 50% DR, 2.0 multiplier
  • Bone Armor: 30% DR, 1.4285 multiplier

Multiplying all of these together we get a total multiplier of 47.61. That combined with massive Armor bonuses provided by Stand Alone and Stone Gauntlets produces an extremely durable build, capable of facetanking absolutely everything. In comparison we can take a look at LoD Skeletal Mage Necromancer "Rat", they have:

  • Legacy of Dreams: 2.0833
  • Razeth's Volition: 2.0 multiplier
  • Bone Armor: 1.4285 multiplier

for a total multiplier of 5.95. Comparing these two numbers we can instantly say that Grim Scythe Necro is at least 10 times tankier than a Rat. If we instead look at these builds' damage reduction values we'll get 83.2% for Rat and 97.9% for Grim Scythe. And while it's still easy to see that Rat is much squishier, it's not immediately obvious how much.

The above rule about multiplicative stacking of damage reduction is only broken by effects that provide a stacking damage reduction buff, such as Zoey's Secret, Binding of the Lost and Aegis of Valor (4) Bonus. In these cases you could say that damage reduction stacks additively, but it's better to just think of it as a single source of damage reduction with magnitude dependent on number of stacks.


Armor

Armor provides damage reduction against all damage types equal to:

    \[\textrm{DR}_{\textrm{armor}} = 1 - \frac{3500}{3500 + \textrm{Armor}}\]

or in other words, Armor increases your toughness by a factor of:

    \[1 + \frac{\textrm{Armor}}{3500}\]

That number, 3500, is technically equal to 50 times the level of the attacker, but in all practical situations that level will be 70, so we just leave is as 3500, for simplicity's sake.

Acquiring Armor

There's two ways to increase your armor: you can get some flat armor or you can get a percentage Armor Bonus. There's only 5 sources of flat armor:

  • All armor pieces and shields have some base Armor
  • All gear except weapons can roll +Armor as a primary affix
  • Each point of Strength and Dexterity provides 1 Armor
  • Laws of Justice Immovable Object provides 7000 Armor when activated
  • Goldwrap provides Armor equal to your current gold streak

After that you can gain different percentile Armor Bonuses from skills, passives, and items. All these bonuses are added together and then your total Armor is equal to:

    \[\textrm{Armor} = \bigr ( \sum\textrm{Flat Armor} \bigl ) \cdot \bigr ( 1 + \sum\textrm{Armor Bonuses} \bigl )\]

There's a lot of different effects that provide Armor Bonus:

  • Paragon: 25% Armor
  • Bottomless Potion of the Tower: 20% Armor
  • Stone Gauntlets: 250% Armor
  • Istvan's Paired Blades: 30% Armor
  • Barbarian
    • Leap Iron Impact: 150% Armor, 450% with Might of the Earth (4) Bonus
    • War Cry: 20% Armor, 40% with Horde of the Ninety Savages (2) Bonus
    • War Cry Hardened Wrath: 60% Armor , 120% with Horde of the Ninety Savages (2) Bonus
    • Tough as Nails: 25% Armor
  • Crusader
    • Slash Guard: 25% Armor
    • Sweep Attack Inspiring Sweep: 20% Armor
    • Blessed Shield Divine Aegis: 20% Armor, 50% with Jekangbord
    • Akarat's Champion Prophet: 150% Armor
    • Divine Fortress: up to 31% Armor
  • Demon Hunter
    • Evasive Fire Hardened: 25% Armor
    • Fan of Knives Bladed Armor: 40% Armor
    • Perfectionist: 10% Armor
  • Monk
    • Mantra of Salvation Hard Target: 20% Armor
  • Necromancer
    • Blood Rush Potency: 100% Armor
    • Stand Alone: 100% Armor
  • Witch Doctor
    • Horrify Frightening Aspect: 50% Armor
    • Soul Harvest Languish: 50% Armor, 100% with Sacred Harvester
  • Wizard
    • Ice Armor Crystallize: 60% Armor
    • Energy Armor: 35% Armor
    • Archon: 150% Armor, a lot% Armor with Vyr's Amazing Arcana (4) Bonus
    • Glass Cannon: -10% Armor
    • Unwavering Will: 20% Armor
Estimating value of an armor bonus

While straight damage reduction always affects your toughness in the same predictable way, it's a bit different with Armor Bonuses. Their effectiveness depends on both your Flat Armor and your total other Armor Bonuses. While you can just plug the exact values into the above formula for Armor's Damage Reduction, it can be quite bothersome and instead the following approximate calculation can be used:

  1. Take your new Armor Bouns +X\%
  2. Divide it by your total Armor multiplier, you had before this bonus, equal to 1 plus all your Armor Bonuses
  3. If you are an Intelligence class with no significant Armor Bonuses, divide it by another 1.25
  4. Add 1 to the result. This is the approximate toughness multiplier you get from your new Armor Bonus

To illustrate, lets consider a rather common problem of choosing between Stone Gauntlets and some other item with 50% Damage Reduction. The latter always provides you a toughness multiplier of 2. With the former you need to do some calculations.

Say you are a monk with just 25% Armor from Paragon, then from Stone Gauntlets you get +250% Armor. Divide that by 1.25, you get 200%. Add 1 to that and you get 3. So Stone Gauntlets will almost triple your toughness, handily beating Lefebvre's Soliloquy or a similar item.


Resistance

Resistance provides damage reduction against the damage of certain type. There's 7 types of damage in Diablo III: Arcane, Cold, Fire, Holy, Lightning, Physical, and Poison. There's no Holy Resistance, however, because no monsters deal Holy damage. In PvP, Arcane Resistance protects against Holy damage as well. The formula for Damage Reduction granted by resistance is:

    \[\textrm{DR}_{\textrm{res}} = 1 - \frac{350}{350 + \textrm{Resistance}}\]

You may notice, that this is exactly the same formula as the one used for Armor, except for the constant being 350 instead of 3500. And that's exactly the case; Armor and Resistances are the same, except there's only 1 Armor and 6 Resistances.

Acquiring Resistance

Again, everything is exactly the same as Armor, there's Flat Resistance that can come from:

  • Paragon: 250 All Resistance
  • Primary affixes on all gear except weapons: 130 All Resistance
  • Secondary affixes on all gear except weapons: 210 Resistance
  • Bottomless Potion of the Diamond: 100 All Resistance
  • Flawless Royal Diamond in armor sockets: 78 All Resistance
  • 1 All Resistance per 10 points of Intelligence
  • Laws of Justice: 490 All Resistance
  • Swampland Attunement: 3000 Poison, Cold, Fire and Physical Resistance
  • Harmony: 84 to all Resistances except one

And then there's Resistance Bonuses that provide +X% to Resistance and stack additively with each other:

  • Esoteric Alteration's secondary effect: 75% Non-Physical Resistance
  • War Cry Impunity: 20% All Resistance, 40% with Horde of the Ninety Savages (2) Bonus
  • Companion Boar Companion: 20% All Resistance
  • Perfectionist: 10% All Resistance
  • Mantra of Salvation: 40% All Resistance, 60 to 80% with Perseverance and/or Inna's Mantra (2) Bonus
  • Energy Armor Prismatic Armor: 25% All Resistance
  • Archon: 150% All Resistance, a lot% with Vyr's Amazing Arcana (4) Bonus
  • Glass Cannon: -10% All Resistance
  • Unwavering Will: 20% All Resistance
  • Tal Rasha's Elements (4) Bonus: 100% All Resistance

For whatever reason, All Resistance bonuses are very rare, compared to Armor. However, all the calculations and estimations say exactly the same.

Balancing Armor and All Resistance

Often you have to choose between flat bonus to Armor or All Resistance: choosing primary affix on gear, Diamonds vs Rubies in armor, Strength vs Intelligence augments. First, there's no magical Armor to All Resist ratio! You can often hear that you get optimal defense, when your Armor is equal to 10 times your All Resistance. This is only true when you can exchange Armor for All Resistance at 10:1 rate! (which only happens when choosing between Int or Str augments) In general case you need to see which option provides bigger relative increase to your (Armor + 3500) or (All Res + 350) respectively.

For example, let's say you are a Witch Doctor with 2000 All Resistance and 9000 Armor, choosing between +516 Armor or +130 All Resistance on a Shoulder piece. If you go with Armor, your respective stat (Armor + 3500) will increase from 12500 to 12500 + 516 \cdot 1.25 = 13145 (25% from paragon), that is 5.2% increase. On the other hand, your All Resistance+350 would change from 2350 to 2480, which is a 5.5% increase. Here we can see that these two options are almost equal, with All Resistance being a tiny bit better.

In most cases you can just follow a simple rule of always choosing All Resistance on Strength and Dexterity classes, and Armor on Intelligence classes. Even if above example demonstrates a situation where this rule fails, you will never be more than half a percent off, and at higher Paragon this rule becomes absolute.


Other Damage Reduction

Instead of boosting your Armor or Resistances, many items and effects just outroght provide some Damage Reduction. It can come in many shapes and sizes: it can be general catch-all DR, it can be only Physical or Non-Physical, it can only work vs melee or ranged attacks, or only vs elite enemies. Regardless of all that, there's always a single unifying rule: all these sources and different kinds of DR always stack multiplicatively. In essence, this rule makes every instance of DR independent. 50% Damage Reduction will always double your toughness, no matter what.

Sometimes instead of being a toughness buff damage reduction comes in a form of debuff that you need to apply to enemies (Resolve, Threatening Shout, Numbing Traps). Still it works exactly the same and stacks multiplicatively with other debuffs and DR buffs.

Some effects instead of reducing damage that you take, redirect part of it somewhere else (Unity, Call of the Ancients Together As One). But as long as you don't care about this "somewhere else"(e.g. using Enchanting Favor), these effects are functionally identical to DR and behave the same way.

Captain Crimson's Trimmings (3) Bonus deserves a separate mention. It provides DR equal to your current Resource Cost Reduction (RCR) value. Since RCR stacks multiplicatively, using exact same rules as DR, you can just consider every item or skill giving you RCR to be also a separate source of Damage Reduction. Equipping Yang's Recurve or activating Laws of Valor Unstoppable Force will always double your toughness, regardless of all other DR or RCR that you may have.

One exception to the multiplicative rule is the stacking DR buffs. There each stack provides some DR and they stack additively but only with themselves. One exception to this exception is Roland's Legacy (6) Bonus, which gives a buff with 5 stacks, 15% DR on each, but they stack multiplicatively and only provide 55.6% DR instead of 75%.


Block & Flat Damage Reduction

Block in Diablo III introduces a unique mechanic: Flat Damage Reduction. When you successfully block an attack it's damage is reduced by a flat value. There's two components to block: Block Chance and Block Amount.

Block Chance is pretty straightforward and represents a chance for each attack that hits you to be blocked. There's number of ways to gain Block Chance:

  • Level 70 Shield: 20% Block Chance
  • Primary affix on a shield: 11% Block Chance
  • Justice Lantern: 16% Block Chance
  • The Helm of Rule: 11% Block Chance
  • Blood Brother: 20% Block Chance
  • Punish: 15% Block Chance
  • Hold Your Ground: 30% Block Chance

Block Chance from all these sources stacks additively. Provoke Hit Me provides 50% multiplicative increase to your Block Chance. It is possible to stack Block Chance all the way to 100%, the in-game tooltip claiming that it's capped at 75% is wrong.

Block Amount only comes from your Shield and only depends on its level and Ancient status. The only way to increase your Block Amount is using Hallowed Bulwark (although crusader passive Renewal provides a pseudo Block Amount via healing). Block Amount is subtracted from the attack's damage after all the (percentile-based) damage reductions are applied (Coven's Criterion being sole exception). This order of operations produces an interesting phenomenon in block-heavy builds: toughness hyperscaling. As you stack more and more damage reduction, your effective toughness rises faster than it would in a no-block build, and eventually reaches infinity.

For example lets consider a scenario when you have a 100% Block Chance, 25000 Block Amount and an average attack from a mob hits you for 100k (before block). Lets say your starting toughness is 100%. After Block Amount is subtracted, you are hit for 75k damage. Effectively, block gives you 25% damage reduction and thus your effective toughness is 133%. Now lets add another 50% Damage Reduction item to our build. The toughness obviously becomes 200%(2x increase) and the same attack now hits us for 50k. After blocking, that gets reduced to 25k and our effective toughness is now 400%(3x increase). Adding yet another 50% DR item, will increase the toughness to 400%, but now block reduces the damage of an attack to 0, and our effective toughness is infinite.

The above example is oversimplified, because in reality there will be stronger and weaker attacks and your effective toughness will be high vs the latter but low vs the former. Also some ground effects and DoTs are outright unblockable. But still this hyperscaling pattern applies to high Block Chance builds, often making them very dependent on Paragon to reach the necessary mitigation threshold.

Post by Northwar