Does Increasing AC Work on a Curve?

[DrowPiratRobrts]

Note: I'm simply comparing AC to AC with this question. I'm not trying to address the value of AC vs Saving throws or anything like that.

Good, now that that's out of the way. I'm wondering if increasing your AC works on some sore of mathematical curve, such that you see diminishing returns when increasing a very high (>20) AC or very low (<10) AC versus increasing an average (~15) AC. Has anyone done the math to search this out? If not, what would the process entail? Below in my first comment are some of the thoughts that brought this question about.

[sophontteks]

You would want to create a line graph for each to hit value I think.
X would be the AC
Y would be the % chance of hitting vs. that AC.

Like +0 attack vs. 10 Ac is a 50% chance to hit.

I believe it would look like a bell curve. Eventually a high enough Ac will drop the hit chance to 5% where additional AC does nothing unless the enemy attack value increases.

EDIT: you would need to get the derivative of the above graph to determine the value of another AC. Thats when I think it starts to look like a bellcurve
EDIT had the X and the lines mixed up.

[DrowPiratRobrts]

Most ACs in the game fall around the 10-20 range. Because the game is balanced around this, it seems logical that increasing your AC by 1 or 2 while in that range should yield significant benefits. Going further down this line, let's imagine a Fighter with an AC of 16:

-A creature with no modifiers on their attack role has a 25% chance to hit our fighter (hit on 16-20). But almost every creature adds some sort of modifier to their attack, so let's say this creature gets a +3 to hit.
-Now our fighter gets hit 40% of the time (13-20). So every +1 to this enemy's attack adds a 5% chance to hit.

Here's the problem. That's only AC vs one specific enemy and their to-hit modifier. To find out the value/efficiency of AC it seems necessary to take the max, min, and mean of all the "to hit" bonuses that all monsters get. I don't want to do that, which is one reason I asked if someone already had. But we might be able to "fake it" by taking a "best guess" at how common enemies are with each "to hit bonus" from +1-+16 (or whatever the highest hit modifier is for a monster out there). Then it seems logical that we could determine what the most efficient AC is, and when it's better to spend gold, spells, etc. on more damage or healing rather than extra AC.

Making sense so far? Thoughts?

EDIT: It's important to note that we need to look at all monsters in the game since they all use different to-hit modifiers but we use the same AC vs them all. So our question is really, on average what AC will help me avoid attacks most efficiently vs all the monsters in the game?

[DrowPiratRobrts]

Like +0 attack vs. 10 Ac is a 50% chance to hit.


Just a quick note for anyone who continues doing math, 10 AC gives 55% to-hit since you're only safe on a roll of 1-9.

[sophontteks]

Each to hit just shifts the exact same line on the axis. Its not much work.

[DrowPiratRobrts]

Each to hit just shifts the exact same line on the axis. Its not much work.

Right, but that's only against one monster with a predetermined to-hit bonus. I'm talking about generic AC vs all potential monsters.

So for example if

-25% of monsters have a to-hit bonus of 3 and
-50% of monsters have a to-hit bonus of 6 and
-25% of monsters have a to-hit bonus of 8 then
-with an 18 AC then you would be hit 45% of the time across a campaign with an even smattering of monsters right? So maybe it is a linear progression, and not a curve after all...though I did just make those numbers up completely.

[DrowPiratRobrts]

https://i.imgur.com/nJaw5dZ.png

This might prove a helpful starting point?

Without accounting for the percentage of monsters at each CR, this indicates that the average to-hit bonus is 7.

[PhoenixPhyre]

The following stats come from the full set of MM + VGtM creatures (spreadsheet linked in my sig):

* Average ATK = 5.86 (over all monsters)
* Average ATK (CR <= 1) = 3.63 ; 198 creatures
* Average ATK (CR <= 5) = 4.52 ; 389 creatures
* Average ATK ( CR <= 11) = 5.12 ; 480 creatures
* Average ATK ( CR <= 16) = 5.44 ; 515 creatures
* Average ATK ( CR <= 20) = 5.60; 526 creatures
* Average ATK ( CR <= 30) = 5.86; 539 creatures

Checking it against a range of ACs from 10 to 28, the following trends appear:
* Every additional point of AC increases miss chance by 5 percentage points regardless of the ATK bonus and the AC (thus the absolute benefit is linear)
* There is a CR-dependent hard cap (at which point the average monster will only hit on a 20):
** CR <= 1: AC 23
** CR <= 5: AC 24
** CR <= 30: AC 25

Since the calculations are dominated by the lower-CR monsters (which is plausible due to bounded accuracy), the max AC under this model plateaus quite early.

If, however, you expect to face dominantly threats of CR ~ Level, the breakpoints change drastically:



CR
ATK
Plateau


1
4
23


5
7
26


11
10
29


16
12
31


20
21
40



Under this assumption, being miss-capped is basically off the table without burning resources. Best case (+3 shield/+3 plate armor + shield) gives AC 31, which is hit-cap for a CR 16 creature.

The game expects you to be hit. Frequently. Note that those CR 20 creatures will only miss on a 1 unless your AC is 23+.

[Lunali]

It depends on how you look at it.

All else being equal, each point of AC added will reduce the average number of successful attacks against you by the same amount. So out of 100 attacks made against you, each point added will reduce the number of hits by 5.

Looking at it a different way, if your AC is such that 50 of those attacks would be successful, 1 point of AC reduces your hits taken by 10%. If only 25 of those attacks would be successful, 1 point will reduce your hits taken by 20%.

[PhoenixPhyre]

It depends on how you look at it.

All else being equal, each point of AC added will reduce the average number of successful attacks against you by the same amount. So out of 100 attacks made against you, each point added will reduce the number of hits by 5.

Looking at it a different way, if your AC is such that 50 of those attacks would be successful, 1 point of AC reduces your hits taken by 10%. If only 25 of those attacks would be successful, 1 point will reduce your hits taken by 20%.

Calculating the % increase in miss chance per AC as |Miss(AC + 1) - Miss(AC)|/Miss(AC) shows that the relationship is roughly inverse (higher AC, the less relative benefit you get from an additional AC until you hit the threshold of unhittability--getting over that threshold is huge, getting there is costly. The slope of this curve is very shallow against low ATK bonuses, but that's because the miss chance is already quite high even at low AC.

For some numbers:

Against an average CR 11 creature, going from an 11 (miss chance 5%) to a 12 (miss chance 10%) is a 100% improvement. Going from an 12 to a 13 is only a 50% further improvement, etc.

Anyone interested in this can PM me and I'll send them a copy of the spreadsheet.

[Lunali]

Against an average CR 11 creature, going from an 11 (miss chance 5%) to a 12 (miss chance 10%) is a 100% improvement. Going from an 12 to a 13 is only a 50% further improvement, etc.

It's a 100% improvement in miss chance, but only roughly a 5.3% reduction in expected damage taken. So it is a huge improvement in your odds of taking no damage, but not a very big reduction in your average damage taken.

[PhoenixPhyre]

It's a 100% improvement in miss chance, but only roughly a 5.3% reduction in expected damage taken. So it is a huge improvement in your odds of taking no damage, but not a very big reduction in your average damage taken.

Which is important for avoiding single big hits, but less so against a death of a thousand cuts.

Calculating the % change in expected damage taken as |MISS(AC) - MISS (AC + 1)|/(1-MISS(AC)) (since the DPR stays constant) gives the expected exponential increase in effectiveness with increasing AC (up to the unhittability threshold, after clearing the un-missable floor).

[DrowPiratRobrts]

Calculating the % increase in miss chance per AC as |Miss(AC + 1) - Miss(AC)|/Miss(AC) shows that the relationship is roughly inverse (higher AC, the less relative benefit you get from an additional AC until you hit the threshold of unhittability--getting over that threshold is huge, getting there is costly. The slope of this curve is very shallow against low ATK bonuses, but that's because the miss chance is already quite high even at low AC.

This is the idea I was looking for I think. I just wasn't sure what the mathematics process was that might justify it. So if I understand this correctly, it would seem that what we might call "AC Value" decreases with each point as you approach what you've called the "Threshold of Unhittability." It also stands to reason that the threshold will be determined by the monster with the greatest "to-hit" value in any given campaign. I'm going to do some work on this, but I'd like to see if we can arrive at a formula that gives a a sort of golden number for any given campaign where said number equals "the minimum ideal AC before you put resources (feats, spells, etc. elswhere)." Obviously getting past the "Threshold of Unhitability" is intriguing, but it's generally not realistic and probably never worth the investment do to so many high powered spells requiring saving throws rather than attack roles.

[PhoenixPhyre]

It's a smooth curve except at the thresholds, so no. There isn't a general optimum AC formula. You definitely need to know if you'll face

1) lots of low CR creatures (that is few solo bosses)
Or
2) a few high CR monsters.

That choice changes a lot of parameters in the optimization game, like most of the scaling curves.

[DrowPiratRobrts]

It's a smooth curve except at the thresholds, so no. There isn't a general optimum AC formula. You definitely need to know if you'll face

1) lots of low CR creatures (that is few solo bosses)
Or
2) a few high CR monsters.

That choice changes a lot of parameters in the optimization game, like most of the scaling curves.

Though it seems we could say this:

X=The Golden AC for your campaign. This is the most efficient AC in the average game with a wide spread of CR encounters at PC level and below.

B=The average "To-hit" modifier for every given monster up to the max CR monster you will encounter.

X=B+20

You did this above but I didn't fully realize it until just now. Here it is again below.


* There is a CR-dependent hard cap (at which point the average monster will only hit on a 20):
** CR <= 1: AC 23
** CR <= 5: AC 24
** CR <= 30: AC 25

So by-in-large for most games, it seems that an AC of 25 will keep you pretty safe (read as: As safe as your AC can even keep you). Though if you hit that magic number of 31 in a campaign focus more on monster CR=PC Level encounters then you'd get significant mileage out of the extra 6 points. It's also worth noting that in that each point of AC above 25 has a higher potential to save you a lot of HP in that sort of game.

[Lunali]

Though it seems we could say this:

X=The Golden AC for your campaign. This is the most efficient AC in the average game with a wide spread of CR encounters at PC level and below.

B=The average "To-hit" modifier for every given monster up to the max CR monster you will encounter.

X=B+20

But that isn't the golden AC number because you didn't account for the opportunity cost of raising your AC to that point. If using the ASI/feats/etc somewhere else results in shorter encounters, you may take less damage by going that route instead. There's also the possibility that investing resources in other areas might drastically change the encounters. (even ignoring the possibility of the DM intentionally changing them to adapt to AC)

There are far too many factors to make a general formula for optimal AC. A far more reasonable option is to weigh the expected benefits of a specific increase vs the alternatives for the same resource expenditure.

[DrowPiratRobrts]

But that isn't the golden AC number because you didn't account for the opportunity cost of raising your AC to that point. If using the ASI/feats/etc somewhere else results in shorter encounters, you may take less damage by going that route instead. There's also the possibility that investing resources in other areas might drastically change the encounters. (even ignoring the possibility of the DM intentionally changing them to adapt to AC)

There are far too many factors to make a general formula for optimal AC. A far more reasonable option is to weigh the expected benefits of a specific increase vs the alternatives for the same resource expenditure.

Right, maybe I didn't communicate this clearly though. Because I agree with everything you just said, I'm just thinking about "when to stop" based purely on AC. That is to say, when does AC stop paying off in and of itself? So if in your campaign going from 24 to 25 is more valuable (will cause more creatures to miss more often) than going from 25 to 26, it seems that it might be worth stopping at 25 max. Obviously this is build/party/DM/world/everything else dependent. But in a vacuum, it seems to me that there's are a few lines in the sand to be drawn.

EDIT: For instance a Wizard doesn't need to worry about this because their resources/ASIs/feats/etc will generally be focused entirely elsewhere.

[Daithi]

First, thank you to PhoenixPhyre. That was a great post with lots of relevant info.

Second, another thing that may need consideration for OP question is his character's hit points. If you are a glass canon that can get killed with one blow then AC may be much more important to you than say a barbarian/druid. Really fast guys like monks probably wouldn't benefit by trying to max their AC either if they aren't there to get hit in the first place. On the other hand, if you are expected to be in the middle of a melee and trading blows, then getting that AC up is probably a major priority even if you do have a lot of hit points.