[RPG] Mathematically, is a +2 bonus to AC better than attackers having disadvantage


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.

Best Answer

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.