The math of Advantage and Disadvantage

Krachek said:

Simply remember.
Advantage or disadvantage have maximum impact when you are around 50% chance of success.
The closer you are from 0% or 100% the less it will impact you.

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Okay, I'm not beating on you, but this is... wrong. Understandably so, but wrong. Soapbox time!

Converting the difference in odds that occurs when using ad/disadvantage into a flat bonus leads to erroneous thinking. It's (for nerdy math reasons) just wrong to do this. It is, however, sometimes useful as a rough model to gauge effect. Like all models, though, it's still wrong, and will lead you into the thinking above where you assume the translation to the largest flat bonus represents the actual effect. It doesn't. Here's a good example of this:

Say you need to roll a 20 to succeed at a task. You have a 1/20 chance of success (or 5%). Cool. Now, let's say you have disadvantage on this roll. If you go with the relative flat bonus, the assumption is that at 20 the effect is very nearly a -1. But, if you need to roll a 20 to succeed and now have an additional -1 to that check added, then you cannot ever succeed. This, however, isn't true of actual disadvantage. Your chance of success just become much, much less.

And therein lies the point the makes your statement incorrect. Yes, the largest divergence occurs at 11, but not really the largest effect. Example, again. If you need an 11 to hit, you have a 10/20 chance of hitting. Suffer disadvantage, and that drops to 5/20. Your chance is halved. But, if you need a 20 to hit, that's a 1/20 chance. With disadvantage, your chance is 1/400, or twenty times less. The largest effect is actually at the ends, where it appears that the bonus translation is smallest. But, again, the bonus translation says that you can't hit a 20, ever, which means the impact of disadvantage is infinitely larger than the smaller success chance at 11.

To show this another way, let's assume you're attacked 5 times. Firstly, a case where your foe needs an 11 to hit you and secondly where your foe needs an 18 to hit you.

Case 1 (11 needed): The base chance that you are hit at least once in five attacks is 1 - p(all attacks miss). That's 1 - (10/20)^5 or 96.9%. If the enemy has disadvantage, the chance to be missed increases from 10/20 to 15/20, so disadvantage changes the chance you're hit at least once to 1-(15/20)^5 or 76.3%. That's a big change. But, let's look at case 2.

Case 2 (18 needed): Again, the base chance is 1-p(all attacks miss). For 18, that's 1-(18/20)^5 or 41.0% chance to be hit at least once. With disadvantage, though, that becomes 1-((1-3/20)^2)^5 or 10.8%. That's a Case 1 chance of 20%, but a case 2 change of 30% in the odds of disadvantage causing all 5 attacks to miss.

Again, converting the probability change due to ad/disadvantage is often a good rough gauge, but it's wrong, and relying on that shift (or even just the raw difference in probability) will lead you to believe that the biggest numbers mean the most impact. That's not exactly true, you just have to go a step further in exploring the impact to see it.

Actually, this error the OP made was in interpreting the % Diff. In his table (which is very nice and thorough), it seems he is simply subtracting the probabilities and assumes dividing the difference by 5% yields the effective +/- adjustment that advantage and disadvantage would give.

That was his error and oversimplification, but it really is a minor point as the result is close enough. The proper interpretation is in comparing the probability of success for a single roll without the advantaged probability.

For example, if the target number is 11, the normal probability is 50%. With advantage, it is 75%. So, what number for a single die roll would give you at least a 75% chance for success? A 6.

What does that mean? If you had the choice between rolling for an 11 with Advantage or a single roll for 11 with a +5 modifier (i.e. a 6), your chances for success would be the same.

Look at another example. The OP's table shows a target number of 8 has a 65% of success. With advantage, it is 87.75%. What target number has AT LEAST an 87.75% chance? A 3. So a single roll of 3 has at least as good a chance of success as an 8 with advantage. Or, another +5. Note: a single roll of 4 will not be as good as having advantage in this case, so the minimum modifier that would represent the benefit of advantage as +5 in this case.

At the other end of the spectrum, consider a target number of 18 with a 15% chance for success. With advantage, that increases to 27.75%. What number as a single roll has at least a 27.75% chance of success? A 15. A single roll with an additional +3 that targets 18 is slightly better than rolling for 18 with advantage.

Now, in all these examples, the division of the % Diff by 5% (the OP's approach) yields roughly the same numbers, especially when one considers rounding due to discrete results. But, understanding how these numbers should really be compared is a bit more complex.

Krachek said:

Simply remember.
Advantage or disadvantage have maximum impact when you are around 50% chance of success.
The closer you are from 0% or 100% the less it will impact you.

Click to expand...

What's "maximum impact"? If I need a 20 then advantage almost doubles my chance to hit.

If I need a 1 to hit then advantage reduces my chance to miss by 95%. Almost guaranteed success seems maximally impactful.

If the enemy needs a 20 to hit then disadvantage reduces damage by a factor of 20. Seems impactful to me. But here you are saying reducing incoming damage by 95% isn't impactful.

Pragmatic case.
You got a character that enemy hit on a 20. 1/20.
You got a character that enemy hit on a 11. 10/20.
You can protect one character by giving ennemy disadvantage on attack roll against him.
Which one will benefit more the protection?

On a fight where each got attack 20 time, the character with big AC will benefit on average once from being protected.
He was already hard to hit, he will become almost invulnerable.

The other character will be protected 5 times. Without protection he will be hit 10 times on average.
That is what I call a big effect.

Yaarel said:

We can all agree, the usefulness of advantage depends on how often a player needs to roll a natural 18 or higher.

In my games, this is often enough.

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If you really have a statistically significant number of rolls at 18+, I believe we can then say that that campaign is outside the design space of bounded accuracy, and that the results don't apply to a general discussion of the mechanic.

It is less about the frequency of needing a natural 18 or higher, it is about when these situations come up, the advantage vanishes. Oppositely, advantage is only strong when the result of the roll has less consequence.

Advantage is a great mechanic to represent being reliable at average things. However, it is of diminishing value to achieve anything difficult.

Blue said:

If you really have a statistically significant number of rolls at 18+, I believe we can then say that that campaign is outside the design space of bounded accuracy, and that the results don't apply to a general discussion of the mechanic.

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Our campaign style mixes up the levels of the encounters. So for example, sometimes players can enjoy showing off while cakewalking a challenge and sometimes the way to overcome a challenge is to hide and go around it. Players have to think about the scenario and what tactics might work best. We enjoy this approach, and it feels more ‘realistic’. The world too is a mix of high power and low power.

However, when it comes to skill checks, I assume most playstyles need to make 18+ natural rolls. For example, a wizard with Strength +0 still needs to make difficult nonproficient Athletic climb checks. A party is only as sneaky as its clumsiest member. And so on.

For a rough comparison of how much advantage is worth.




When you need the bonus so as to roll high, the advantage mechanic rips like a paper tiger. Advantage can feel like a mirage.

guachi said:

What's "maximum impact"? If I need a 20 then advantage almost doubles my chance to hit.

If I need a 1 to hit then advantage reduces my chance to miss by 95%. Almost guaranteed success seems maximally impactful.

If the enemy needs a 20 to hit then disadvantage reduces damage by a factor of 20. Seems impactful to me. But here you are saying reducing incoming damage by 95% isn't impactful.

Click to expand...

If you're in a fight where you need to roll 20's to win you'll almost certainly lose, advantage or not. Advantage or disadvantage would have very little impact on the overall outcome. It's the same way if enemies need a 20 to hit you, you'll only take a small amount of damage in combat so a 95% reduction of that if you can impose disadvantage, isn't impactful when the damage was never a problem in the first place.

Krachek said:

Pragmatic case.
You got a character that enemy hit on a 20. 1/20.
You got a character that enemy hit on a 11. 10/20.
You can protect one character by giving ennemy disadvantage on attack roll against him.
Which one will benefit more the protection?

On a fight where each got attack 20 time, the character with big AC will benefit on average once from being protected.
He was already hard to hit, he will become almost invulnerable.

The other character will be protected 5 times. Without protection he will be hit 10 times on average.
That is what I call a big effect.

Click to expand...

The largest effect is the smallest reduction in times hit? At 11, disadvantage will reduce the number of hits by a factor of 2. At 20, disadvantage reduces the number of hits by a factor of 20. "Almost invulnerable" seems like a big effect to me.

Yaarel said:

For a rough comparison of how much advantage is worth.

Code:

Natural   Advantage
d20       ≈ Flat
Roll      Bonus
Needed

10        +4
11        +5
12        +4
13        +4
14        +4
15        +4

16        +3
17        +3
[B]
18       +2
19       +1
20       +0
[/B]

When you need the bonus so as to roll high, the advantage mechanic rips like a paper tiger. Advantage can feel like a mirage.

Click to expand...

Right, because almost doubling your chances when you need a 20 is a bad, bad thing. And, again, converting a normal distribution to a flat bonus is wrong -- you're confusing yourself when you do this.