Mathematically, is a +2 bonus to AC better than attackers having disadvantage?

Question

Mathematically speaking, I'm trying to determine if Bracers of Defense (AC+2) is better than a Cloak of Displacement, which causes all attackers to have disadvantage on attack rolls against you.

If it's pertinent, I'm a monk, my character level is 15, and my AC without the bracers is 18. I don't have any other sources that will consistently impose disadvantage on my opponent's attacks. I'm trying to choose between these two items and I'm trying to determine which one would ultimately result in less loss of HP.

It's been too long since my high school statistics classes, so I'm finding the advantage/disadvantage statistics threads too hard to follow and don't really take into account AC.

Answer 1

In this case, the Cloak is better

To explain this, let's first look at some stats.

When no bonuses are involved, the Chances of rolling a 20 normally is 5% (1/20), while the chance for rolling 18 or higher normally is 15%. So, from the start, it seems as if having a 20 AC is a better thing.

But not so fast. According to this source, who used a Monte Carlo simulation, and this function, the chances of rolling at least an 18 are around 2.2% when the attack is made with Disadvantage.

Also, it's important to think about how these effects will scale. At CL 15, you are already going to be facing enemies with strong bonuses to hit. A Wyvern (CR 6), for example, gets +7 to hit on all of its standard attacks, meaning that it has to roll a 13 or greater to hit a target with 20 AC and 11 or greater to hit a target with 18 AC.

Let's compare the stats between hitting at least 13 normally versus at least 11 on Disadvantage:

• ≥13 Normally: This is a 40% chance (5% chance to hit 13 + 5% for every value between 13 and 20, which ends up being 5% + 35% = 40%)
• ≥11 at Disadvantage: This roll has a 25% chance. It doesn't quite halve the chances that the Wyvern scores a hit versus the ≥13 standard, but it is quite a significant drop.

Already, you can see a drastic increase in your chances of not getting hit when the Wyvern is attacking on Disadvantage.

Another thing to consider is that some creatures will have situations that give them Advantage against you. With the Bracers, you would simply take the attack at Advantage, however, the Disadvantage imposed, you would negate Advantage for the enemy, meaning that an enemy rolling with Advantage would end-up rolling normally.

Let's use the Wyvern again, and say that it attacked you from Surprise, but otherwise had no bonuses other than the +7 to hit.

• ≥13 On-Advantage: Having Advantage shoots the likelihood of a hit for the Wyvern up to a Staggering 64%, and you better hope it's not using the Stinger
• ≥11 Normally: The likelihood of making an 11 Normally is a flat 50%. Toss a coin.

While neither of those options are good, per se, I would definitely prefer the enemy having only a 50% chance to hit over them getting a ~64% chance to hit.

And one more thing: a Natural 20 is considered a critical hit, and results in an automatic hit for almost twice the damage no matter AC. Imposing Disadvantage can help guard against that, because that reduces the likelihood of a critical from 5% standard to 0.2% on Disadvantage.

Overall, Increased AC is better at lower levels where enemies don't have as many bonuses, while Disadvantage is better at higher levels where enemies get better Bonuses.

Answer 2

Cloak is Better

Agreeing with the above answers outlining that the cloak will usually be better (unless the enemy has disadvantage for a different reason). Thought the below table on the probabilities would be helpful is showing how much better the cloak is. The table clearly shows that the benefit of the cloak over the bracers is higher for enemies with higher attack bonuses:

Enemy Attack Bonus	Roll to Hit (Bracers)	Roll to Hit (Cloak-Disadv.)	Odds to Hit (Bracers)	Odds to Hit (Cloak-Disadv.)	Cloak Benefit
+0	20	18	5.0%	2.3%	2.8%
+1	19	17	10.0%	4.0%	6.0%
+2	18	16	15.0%	6.3%	8.8%
+3	17	15	20.0%	9.0%	11.0%
+4	16	14	25.0%	12.3%	12.8%
+5	15	13	30.0%	16.0%	14.0%
+6	14	12	35.0%	20.3%	14.8%
+7	13	11	40.0%	25.0%	15.0%
+8	12	10	45.0%	30.3%	14.8%
+9	11	9	50.0%	36.0%	14.0%
+10	10	8	55.0%	42.3%	12.8%

Hope that this helps!

Answer 3

The Cloak is almost always better

Your AC and your opponents to hit bonus combine to determine a number that the oponent needs to roll on a d20 to hit you. This number can never be less than 2 (since a 1 always misses) and can never be greater than 20 (since a 20 always hits - indeed is a critical hit). We will call this number \$t\$.

For a start, irrespective of the target number, the cloak reduces the chance of a critical hit from 0.05 (1 in 20) to 0.0025 (1 in 400).

The chance of a hit (including a critical hit) with the cloak is:

$$\begin{align} C&= ((21-t)0.05)^2\\ \end{align}$$

The chance of a hit (including a critical hit) with the bracers is:

$$\begin{align} B&= \begin{cases} (21-t -2)0.05 & t\lt19\\ 0.05 & t\ge 19 \end{cases}\\ \end{align}$$

Now I could solve this analytically, but I'm going to cheat and state that the cloak has a lower overall hit chance for \$t \ge 4\$.

For \$t=2\$ and \$t=3\$ the bracers have a lower overall chance of a hit (\$0.85\$ v \$0.9025\$ and \$0.80\$ v \$0.8075\$ respectively) but the cloak still reduces the chance of a critical hit (from \$0.05\$ v \$0.0025\$). If the damage on a hit is \$d\$ and the extra damage on a critical is \$c\$ a little algebra will show you that the cloak is better if \$c\gtrapprox 1.1053d \$ and \$c\gtrapprox 0.1579d \$ for 2 and 3 respectively. Now, I don't know of a way to make the base damage greater than the critical damage so the bracers are always better if your opponent needs a 2 but it is quite common for the extra damage from a critical to be greater than 16% of the base damage so the cloak retains the edge for 3 (probably).

Answer 4

Bracers are always there, whereas the Cloak can be rendered useless at times.

I would recommend the following reading material on how to understand the probability behind dis/advantage.

Probability for Gamers (Wall of Text crits you for 500dmg, yeah this guy is thorough).

That article does a great job breaking down the numbers but I will rehash a bit.

Bracers of Defense is a simple calculation an effective increase of 2 on the target die, base 10% difference statistically.

Disadvantage gets a bit muddier and, well it depends on the target number required on the unmodified roll.

Example: If your AC is 18 currently and you are facing a goblin with a +4 to hit you then he would need an unmodified roll of 14 on the die to hit you. This means that statistically disadvantage provides the equivalent of -5 or -4 depending on if you round up or down on the probability.

If you instead had a AC of 22 then that same goblin would require an unmodified roll of 18 to hit you which would be the equivalent, roughly of -3 or -2 depending on rounding.

The above examples are only to demonstrate the changes in approximate penalty to hit based on differing target numbers. They have nothing to do with the Bracers of Defense.

I do have to say that some could argue that the only true -5 modifier is at a target of 11 as the rest should be rounded down. But even then it would appear for the most part that the Cloak of Displacement might be the better choice at this time, but it could depend on what you were facing and the target numbers they were required to hit.

Important note: If you take damage while wearing the Cloak of Displacement its properties do not function until the start of your next turn. So you should factor that possibility in. Note it does not say hit by an attack so that means literally if you are damaged by a spell or trap or anything else you can't avoid you lose its benefits. Also, if you are incapacitated or unable to move. These are definitely things to consider as the bracers are a static +2 AC no caveats other than you cannot wear armor or use a shield.

If you take damage, the property ceases to function until the start of your next turn. This property is suppressed while you are incapacitated, restrained, or otherwise unable to move.

Other things to consider

You can only have disadvantage once, no stacking and if your enemy ever has something that grants them advantage then neither applies. What that means if they would only roll one die and your benefit is cancelled. Bracers of Defense can stack with other AC bonuses.

Example: If you have a Cloak of Displacement (that is currently active, i.e. you have not taken damage since your last turn) and let's assume your DM is using the flanking rules from the DMG, two goblins flanking you would normally have advantage which cancels your imposition of disadvantage so they roll standardly.

From Advantage/Disadvantage PHB 173

If circumstances cause a roll to have both advantage and disadvantage, you are considered to have neither of them, and you roll one d20. This is true even if multiple circumstances impose disadvantage and only one grants advantage or vice versa. In such a situation, you have neither advantage nor disadvantage.

I am sure others can be more eloquent with number magic but I am leaving the linked article as the main course and only narrowing the focus to the question at hand.

Answer 5

It really depends on what your fighting and what they need to roll to hit you.

First off let's cover what expected differences means. To calculate the difference in something and the modifier it provides you need to calculate the % chance of hitting you in each case. You then subtract the two chances to hit and divide by 5. For every 5% in the comparison, the change in item or ability provides effectively a +1 bonus. Why, because the hit dice is 20 sided. 100%/20 = 5%.

To see the differences in your two items let's look at two cases below. They really illustrate how it much depends on the creature trying to hit you.

For the first example let's say creature A has a +12 to hit. (For my lack of experience, high end creatures).

They need a 3 to hit with the cloak and a 5 to hit with bracers. Before you factor in the disadvantage for the cloak. This means they had a roughly 90% to hit you with the cloak and an 80% to strike you with the bracers.

However, the cloak imposes disadvantage which means they have to roll twice and take the worst. Thus this 90% chance to hit you decreases ever so slightly because for two dice, it is slightly more likely for them to roll lower. The odds shift to like 81% chance to hit you. 90-81 = 9 / 5 = (1-2). In terms of die rolls I typically floor percentages so its roughly a +1. As such in this case the DA is apply effectively a +1/+2 if you round up to your armor class.

On the other hand let's say what your fighting only has a +4 to hit. The creature needs a 13 to hit you with the bracers and a 11 to hit you with, assuming they only rolled once. That's a 40% to hit you with the bracers and a 50% to hit you with the cloak. Factor in disadvantage again and the odds he rolls below an 11 are much greater so the impact is much more. In fact his new chance to hit you is about a 25%. 50%-25%/5 = +5. So it provides a +5 in this case, whereas the bracers only applied a +2. Essentially it's the same as flipping two coins and computing the likelihood that tails doesn't show up. There are four combinations the coins could land but only double heads will work. (25%).

TLDR; and as general rule of thumb. Only in the cases where the creature needs a 3 or less to hit you or the creature needs a 19 or more to hit you, is the cloak worse. Other than those edge cases, the effects of having disadvantage on the cloak are better than a raw +2.

Comments have added that bracers can stack with other items. Fundamentally this doesn't change the answer at all. It shifts the answer's scope. It just shifts the creature's target number to hit you.

Answer 6

^{I think this question is technically a dupe of this question about average outcome, but it's hard to read, so I'll answer again here.}

Disadvantage is worth more than a bonus of 2 (usually)

On average, disadvantage (or advantage) is worth about a five-point difference on the die roll. Five is more than two, so the Disadvantage on attacks against you wins.

Because you asked about attacks, there is an added bonus... you're basically uncrittable when enemies have disadvantage. 1/400 is a lot less than 1/20. It'll be a rare day when something auto-hits you wearing the cloak.

It is worth noting that for creatures with very high bonuses to hit, bracers may be better. The basic rule is if the attacker can hit on a roll of two or three, then the +2 AC is better.

Below, are the chances of rolling any specific die number or better.

roll   %     %dadv
1     100   100
2     95    90.25
3     90    81
4     85    72.25
5     80    64
6     75    56.25
7     70    49
8     65    42.25
9     60    36
10    55    30.25
11    50    25
12    45    20.25
13    40    16
14    35    12.25
15    30    9
16    25    6.25
17    20    4
18    15    2.25
19    10    1
20    5     0.25

+2 AC imparts a penalty of ten percentage points (i.e. - 50% to roll that number becomes equivalent to 40%). By comparing the two, we can see that a hit-roll of three is 90% likely regularly; 80% likely with +2 AC; 81% likely with disadvantage on the attack.

For your character of 18AC, that means the creature would need a +15 to hit or better for the +2 to be better (y'know, like an Ancient Copper Dragon).

For a zombie of AC 8, +5 to hit or better. You get the idea.

Answer 7

+2 AC depends greatly on what the enemy needs to roll to hit you.

Obviously, if your AC is already so good that they already need a 20 to hit, then another +2 won't do zilch while enemies having Disadvantage will mean you get hit 1/400 times instead of 1/20 times. (if it is you who have Advantage then this chances nothing to your chances of being hit, but the example here is to talk abut AC, so I am using the Advantage/Disadvantage mechanic from a defensive stance).

Best case scenario, the enemy needs an 18 to hit you. With the +2 AC suddenly it needs a 20 instead. In other words: it will hit you a full three times less often. That is a whopping improvement!

However, that kind of situation will occur only when the party is facing a huge horde of crummy weak monsters aka those with a really low attack bonus. i.e. puny mobs that fireballs will take care of really quickly anyway. The horde has to be REALLY huge for this effect too make a real difference. in D&D, there simply aren't any uber-colossal monsters with tiny attack bonuses yet that hit for humongous damage.

Where it is really important is when facing the big strong ones that strike really hard, not when fighting tons of weak soap bubbles monsters.

Let's say the monster hits you on 9+ on the dice. Over all of the d20 results that is 12 hits and 8 misses. We could say it hits you 50% times more often than in a "fifty-fifty" situation. With the +2 AC means it now hits you on 11+ instead: 10 hits and 10 misses, basically a very typical "fifty-fifty" situation. So, the +2 AC means you got a "+50%" improvement of the hits vs misses ratio.

Meanwhile with Disadvantage: Each dice hits 12 times and misses 8 times. So the odds of BOTH missing goes from 8/20 (single roll without the Disadvantage) to 64/400 (two d20s rolled for with the Disadvantage). In other words: From 160/400 to 64/400, so 160/64 = a 150% improvement!

Personally,I generally consider Advantage to count as if it was a +5, except in special situations where enemies have a super hard time hitting you ergo where you'd probably win easily anyway.

Advantage is a huge bonus.

An exception is if you already have a source of putting the enemies at Disadvantage, in which case the Cloak becomes Redundant.

Answer 8

One thing to consider is that the cloak is less flexible than the AC. There are a lot of ways that you can reliably impose disadvantage which become useless when using the cloak.

For example:

• Lying prone
• Taking the Dodge action
• Using Patient Defense
• Hiding

In all of these situations, the AC is strictly better.

A class that often finds themselves forcing attackers to have disadvantage may find the cloak to be essentially useless, and the AC to be far superior.