Is it true that Reckless Attack renders AC boosts less effective?

Question

I saw someone arguing recently that

A) Barbarians should always RA. and B) Assuming A) is true ... then it's a waste of time for a Barbarian to try to boost their AC.

I assume that the case for B) is supposed to be "since they've got advantage ... they're so likely to hit you that the AC boost isn't worth what effort / expense you put into gaining it."

i.e. that giving your opponents Advantage on attacks against you means that AC increases are less valuable.

Is this argument for point B) valid?

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Let us assume that point A) is in fact true. I'm not convinced it is ... but that's not what I'm interested in, for this question!

Answer 1

Enemies attacking with advantage lowers the value of AC boosts, especially in the mid-ranges. It does not remove the value entirely.

Zerohitpoints made a helpful graph of what the probabilities look like when rolling 2d20 and taking the highest.

Most enemies have a decent attack bonus, with even CR 1 foes having +3 or more to hit. At higher levels AC does not scale quite as well vs attack bonus, especially with special attacks or means for enemies to have advantage.

However if you go off the graph, there's only a 24% or so chance that an enemy with advantage will roll below 11. For a barbarian with 15 AC say, that means you are going to be hit around 85% of the time vs someone with +5 to hit, and around 80% vs someone with +3. A standard plot of 5%/number roll means you are hit around 50% of the time vs someone with +5 to hit, and only 35% vs someone with +3. Add to that (by having good medium armour and a +2 dex modifier, say, or wielding a shield on top) and you are dodging the vast majority of attacks at lower levels.

So yes, broadly - if you always use Reckless Attack, and your AC isn't particularly high to start with (10 dex, cheap medium armour, no magic items or buffs, no shield) then the odds of you getting hit increase significantly, to the point it may reach 90%+ vs higher to-hit enemies. At that point, increases to AC (from Rings of Protection or buffs or so on) may be less valuable than vs say, giving that AC boost to the paladin with a shield who enemies have to roll against without advantage.

However, even if you were always using reckless attack - if you stacked AC as much as possible you are still seeing a significant benefit, not THAT much lower than someone who isn't being attacked with advantage. The difference is mostly egregious at lower AC levels.

Answer 2

TLDR; I kinda screwed up. Yes, it is always true that AC points are less interesting if you're being hit with advantage.

I have posted an earlier answer that I want to correct. While the maths in it are correct, they are not actually taking into account what is most relevant for the game mechanics (I want to remind that I am not a dnd player). I will in this answer correct this with mostly the same maths, but a slightly different point of view on them which will alter the actual practical conclusion. Thanks to Matthieu M., Neil Slater, and Dulkan in the comments of the earlier answer for pointing out the part that I was missing about what actually matters in game

Odds of getting hit with and without advantage:

We will here use odds instead of probabilities to discuss chances of getting hit. That is to say, we will discuss "you have one chance in x to get hit". You could also think of it as "it will take your opponent x attempts to hit you". This x will be much more relevant to your game experience, and will show why AC is indeed less valuable if hit with advantage. From here on, I will refer to it as the required attempts to hit.

Here are the required attempts to hit a given AC with a +4 attack modifier, with and without adavantage:

See how the curves rise more and more sharply? That illustrates what commenters had been pointing out: the higher your AC, the more benefits you get from raising it.

We also see of course that it take significantly less attemtpts to get hit with advantage, especially for high ACs. This is what changes the conclusion.

What benefits do I get from raising my AC:

We now discuss how much gains you can expect from raising your AC by one more point. Of course, as we established above, that depends on your current AC: the higher it already is, the more benefits you will get out of one more point.

Here is a graph showing how many more attempts it will require for your enemies to hit you if you raise your AC by a single point, depending on your current AC:

Let me walk you through it: see for example that the value at 21 AC is almost 2 without advantage? That means that going from 21 to 22 AC will require your opponent to take 2 more attempts before they can actually hit you!

We notice two things: in both cases, as already discussed, you get much more benefits from investing in AC for higher AC values. And you can also simply see that the gains you get from investing in AC are always higher without advantage. So the value you get out of AC points is always reduced if you are being hit with advantage

N.B: In also checked the math in terms of relative gains but did not include it here so as to keep it rather short. It does not alter the conclusion. AC remains less attractive if hit with advantage

The math details:

The required attempts of being hit is simply the inverse of the probability of getting the hit. So without advantage, it is given by: $$P_{usual}=\frac{20}{21-AC+m}$$ with advantage (rolling the dice twice and keeping highest result, if i understood correctly): $$P_{adv}=\frac{400}{400-(AC-m-1)^2}$$

The gains can be obtained from derivation of those odds with respect to AC. The derivatives were calculated numerically by finite differences (this is more accurate than theoretically deriving it, since the AC values are discrete and can only increase by increments of 1)