# Dreamgrove

## On Diminishing Returns

I originally wrote this article back in August 2018, and decided to shelve it to focus on other work. Please pardon the dated references to Uldir.

I’ve seen a lot of questions surface recently about diminishing returns — what they are, how they are (or aren’t) related to damage reduction, and what stats are (or aren’t) affected. Let’s talk about it.

Be warned: we’re doing a deep dive into the theory of diminishing returns and damage reduction. Here be dragons mathematics. You can skip to the conclusion for a TL;DR.

## What are diminishing returns?

“Diminishing returns” refers to a property of some stats where every point of a stat you add is worth less than the previous point. It’s sometimes unhelpfully abbreviated as DR (not to be confused with damage reduction, a concept which is also abbreviated as DR and is often discussed simultaneously with diminishing returns which can be extremely confusing); something I will try to avoid doing in this article.

When we talk about diminishing returns in World of Warcraft, we are typically referring to one of two different but related concepts by which the effects of our stats are diminished.

### Diminishing Returns from Marginal Utility

The first is the property of diminishing marginal utility, which is the idea that every point of a stat added to a total is less valuable relative to the previous point. For example: adding 1 rating when you have 100 rating to begin with is a larger marginal gain than adding 1 rating when you have 200 rating.

\begin{aligned} \frac{100 + 1}{100} = \frac{101}{100} &= 1.01 \ \text{(1\% gain)} \\[3ex] \frac{200 + 1}{200} = \frac{201}{200} &= 1.005 \ \text{(0.5\% gain)} \end{aligned}

Many people understand this intuitively as “stat X gets worse the more you have of it”. This is inherent in the way that WoW computes stat ratings, and all stats are affected equally by this phenomenon.

For the sake of simplicity I’ll be ignoring diminishing marginal utility in this article. Just know that it exists, and that whenever I say “stat X provides more/less/equal value the more you have of it”, it comes with an invisible disclaimer about diminishing marginal utility. I’ll revisit it in the conclusion.

### Diminishing Returns from Rating Penalty

A second use of the term diminishing returns refers to penalties applied to some stats to reduce their effectiveness from what is advertised on the tooltip. Stats such as Armor, Block, Stagger, and Dodge/Parry are all subject to a diminishing returns penalty as part of the formula that converts stat rating into percentage. This is the definition of “diminishing returns” that players use most often in WoW, especially in discussions involving tanking stats, and will be the primary focus of this article.

Let’s take the dodge/parry conversion, for example. For simplicity’s sake I will use terms relevant to Guardian, but know that anywhere that “Dodge” is mentioned, it can be replaced with “Parry” (and “Agility” with “Strength”) as the formula is the same for both.

Dodge chance gained from gear is subject to a diminishing returns penalty. The full dodge formula is longer (and not all dodge is penalized), but only this part of the equation is relevant to the current discussion.

$$effectiveDodge = \frac{dodgeFromGear}{dodgeFromGear * verticalStretch + horizontalShift}$$
• verticalStretch is 0.01 (for Demon Hunters it’s 0.02).
• horizontalShift is 1 / 0.97 for Druids (for other tanks, it’s 1 / 0.94)

What this results in is a function that limits the effectiveness of stacking stats that grant Dodge (namely, Crit rating), by diminishing the amount of effective Dodge you gain from the Dodge on your gear as you add more of it.

Armor is subject to a similar penalty, except it’s baked directly into the conversion from Armor rating to damage reduction (unlike Dodge, all Armor is penalized):

$$damageReduction = \frac{armorRating}{armorRating + K}$$

where K is a fixed constant that depends on the level your attacker and the type of content you are doing (for example, a Mythic Uldir boss has a K value of 9311.4).

In fact — any stat that is converted using a function with the shape $$f(x) = \frac{x}{Ax + C}$$ where A and C are constant, positive numbers — will be subject to diminishing returns.

Stats that use this type of formula and are therefore subject to diminishing returns include:

• Dodge/Parry
• Armor
• Block
• Stagger

Notably, Versatility is not subject to any diminishing returns penalties. We’ll get to that later.

A misconception I often hear is that because stats that reduce your damage intake like Armor are subject to diminishing returns, they therefore provide less value when you already have a lot of that stat. This is partially true, but it’s only half the story.

Consider for a moment what damage reduction actually implies. Suppose you currently have 0% damage reduction, and you take a hit for 100 damage. You reduce 0 of that incoming damage, and so you take the full 100 damage to the face. Now, suppose you gain 1% damage reduction, and you take another 100 damage hit. You are now reducing 1% of that damage, and taking 99 damage. Going from 0% to 1% damage reduction has decreased the amount of damage you take by 1%. This might seem painfully obvious, but bear with me for a moment.

Now suppose you currently have 98% damage reduction. A 100 damage hit would be reduced to 2 damage. If you then gained another 1%, that 100 damage hit is now reduced by 99%, and you take 1 damage. Going from taking 2 damage to 1 damage has decreased the amount of damage you take by 50%. You’re removing the same amount of damage, but the pool you’re removing from is much smaller.

As such, gaining 1% damage reduction is 50 times more valuable when you already have 98% damage reduction than it is when you have 0% damage reduction. Rather than diminishing returns, damage reduction has “increasing returns” — every percent of damage reduction you gain is more valuable than the last.

To see this more clearly, let’s express it in terms of effective health. Effective health (or EHP) refers to the amount of damage required to kill you, taking into account all of your damage reductions. This is very useful for judging exactly how much impact an increase in damage reduction will have on our survivability, as we can now define it in terms of how much additional damage we can take.

Effective health can be calculated as a function of damage reduction:

$$effectiveHealth = \frac{health}{1 - \%damageReduction}$$

For example: if you have 100 health and 50% damage reduction, your effective health is $$\frac{100}{1 - 0.5} = 200$$ , meaning that it would take 200 damage (before mitigation) to kill you.

As you might imagine, going from 98% damage reduction to 99% damage reduction (in other words, reducing the damage you take by 50%) doubles your effective health.

\begin{array}{cc} \text{98\% Reduction} & \text{99\% Reduction} \\[3ex] \begin{aligned} EHP &= \frac{health}{1 - 0.98} \\[2ex] &= \frac{1}{0.02} \\[2ex] &= 50 \end{aligned} & \begin{aligned} EHP &= \frac{health}{1 - 0.99} \\[2ex] &= \frac{1}{0.01} \\[2ex] &= 100 \end{aligned} \end{array}

To make the comparison more clear, I’m assuming that health = 1.

As you can see, at higher levels of base damage reduction, adding additional damage reduction has a significantly higher impact than adding the same amount at a lower level of base damage reduction.

So, why does this matter?

Recall that the Armor to damage reduction formula is $$damageReduction = \frac{armor}{armor + K}$$ , which inherently suffers diminishing returns when expressed as damage reduction. Let’s try and express it as effective health by substituting the Armor formula into the EHP formula.

\begin{aligned} effectiveHealth &= \frac{1}{1 - damageReduction} \\[2ex] &= \frac{1}{1 - \big(\frac{armor}{armor + K}\big)} \\[2ex] &= \frac{1}{\big(\frac{armor + K}{armor + K}\big) - \big(\frac{armor}{armor + K}\big)} \\[2ex] &= \frac{1}{\big(\frac{K}{armor + K}\big)} \\[3ex] &= \frac{armor + K}{K} \end{aligned}

With a bit of arithmetic, we arrive at a slightly unintuitive (and maybe surprising) result. Since K is constant, effective health has a linear relationship with Armor. That is, every point of Armor increases effective health by the same amount, regardless of how much Armor you already have!

The same is true for Dodge/Parry, Block, and Stagger — any stat that provides damage reduction and is subject to a diminishing returns penalty will grant linear returns to effective health, and therefore to survivability.

## What about Versatility?

An interesting exception to this rule is Versatility. As I mentioned earlier, Versatility’s rating conversion does not have a diminishing returns penalty. Every point of Versatility rating provides the same amount of damage reduction regardless of how much Versatility you already have. If you’ve been following along, you already know what this implies.

Let’s walk through the math. The formula for converting Versatility rating to percent damage reduction is $$damageReduction = \frac{versRating}{ratingPerPercent * 2 * 100}$$ , where ratingPerPercent is 85 at level 120.

\text{let P} = ratingPerPercent * 2 * 100 \\[3ex] \begin{aligned} effectiveHealth &= \frac{1}{1 - damageReduction} \\[2ex] &= \frac{1}{1 - \big(\frac{versRating }{P}\big)} \\[2ex] &= \frac{1}{\big(\frac{P}{P}\big) - \big(\frac{versRating}{P}\big)} \\[2ex] &= \frac{1}{\big(\frac{P - versRating}{P}\big)} \\[2ex] &= \frac{P}{P - versRating} \\[2ex] \end{aligned}

As you can see, Versatility actually has increasing returns with respect to effective health. This is what we mean when we say “Versatility gets better the more you have of it”, and why it makes such a strong stat for survivability.

## Conclusion

What are we to make of all this? Well, let’s review what we know.

1. Stats like Block, Stagger, and Armor, have diminishing returns penalties applied to their conversion from rating into damage reduction. Dodge/Parry also has a diminishing returns penalty to its conversion from rating into Dodge/Parry chance.
2. Damage reduction increases in value the more you have of it. This is because when you have more damage reduction, adding an additional 1% of reduction is reducing a larger fraction of the remaining damage.
3. Expressing damage reduction gains in terms of effective health demonstrates that the diminishing returns penalties applied to some stats cancel out the increasing returns of damage reduction. This results in a linear relation between those stats and survivability.
4. Versatility is exceptional in that it is not subject to diminishing returns. Each point of Versatility you gain will be more valuable than the last, in terms of damage reduction.
5. All stats are affected by diminishing marginal utility, which is distinct from the diminishing returns penalty.

So when someone claims that “Armor has diminishing returns”, they are correct; Armor is subject to a diminishing returns penalty. However: in terms of effective health gain, every point of Armor grants the same amount of effective health as the previous, and the next. In this way, the diminishing returns penalty cancels out the increasing returns of damage reduction, resulting in a linear return. In practical terms, an increase in Armor is always worth the same amount, regardless of how much Armor you already have. The same is true of Dodge/Parry, Block, and Stagger.

It is also true that Armor loses value the more you have of it, but only in the context of diminishing marginal utility. Every point of Armor you gain is a smaller relative increase than the previous, with respect to how much Armor you already have. The same is true of not only the damage reduction stats but of every stat in the game. We tend to ignore this factor when comparing the values of different damage reduction stats, because it affects them all equally and whether a stat has a diminishing returns penalty is far more relevant. Being pedantic about whether a stat actually has linear returns because of diminishing marginal utility serves only to confuse those who are unfamiliar with the topic.

Hopefully this has helped clarify some of the less intuitive parts of diminishing returns, and enables you to make more informed decisions about defensive stats in the future.