d20 Math: AC, To Hit, and Diminishing Returns

[MadLordOfMilk]

I originally posted this on the WotC boards, but I thought it might get a bigger response here.

Goal of this post: To point out that AC (or other defenses, I'll just use AC to refer to them all for the sake of simplicity) is worth more the more of it you have, and that +to hit is worth less the more of it you have (they're non-linear!), and how to directly relate AC and +to hit bonuses to Damage Bonuses, Damage Reduction, &c

First, let me define a term: Effective Hit Points (EHP). Essentially, this means how much HP you "really" have. The base line is 100% - if you get hit by every attack 100% of the time, you have 100% of your actual HP, so your EHP is 100%. Now, if you only get hit 50% of the time, it's as if your HP doubled, giving you 200% EHP. If you get hit 25% of the time, it's like your HP doubled from getting hit 50%, giving you 2 * 200% = 400% EHP.

Essentially, this means that Hit % and EHP are inversely proportional:
Effective Hit Points = 1 / Hit Chance %


Yes, that's a curve. Essentially, it means a 5% gain in to hit doesn't quite mean a 5% increase in damage (it does in one sense and doesn't in another; see Nightson's reply). Cut chance to hit in half, EHP doubles. Double chance to hit, EHP is cut in half. Now, how does this correlate to D&D?



Min d20 Roll to Hit: The minimum d20 roll required to hit a target. 1 means 1-20 will hit, 2 means 2-20 will hit, 3 means 3-20 will hit, &c
Hit %: The % chance to hit given that d20 roll required (IE if they need a 20 to hit, that's only a 1/20 or 5% chance to hit)
Miss %: The chance to miss. 100% - Hit%
Target's EHP: The Effective HP of the target being attacked. EHP = 1 / Hit%
HP Gain from Last +1AC: How much the target's last +1 ac increased their EHP. For example, if a 1 still hits the target, their last +1ac did nothing, so it's a 0% hp gain. If it made the target need a 2 to hit instead of a 1, it's EHP(2) - EHP(1), or 105.2632% - 100.0% = 5.2632% HP gained. The math is just the change in EHP.
HP Reduced from Last +1 to Hit: How much the last +1 to hit the attacker gained "reduced" the target's EHP. Essentially, HP gain from AC, but diminishing in value. The reason that it looks the same in the chart is because essentially when you add +1 ac you go down the chart, but +1atk you go up the chart. So, going from needing a 20 to needing a 19 lowers EHP by 1000%.

ALTHOUGH +hit and +defense appear to both be increasing in value per point just at a glance of the table, they're not. The difference is +1AC moves you down one spot on the table, +1atk moves you up one spot. They're going in different directions. Thus, each point of defense bonus you get is worth more than the last, and each point of +atk is worth less.

Here's a graph in case you wanted a visual:


What you can do with this chart:
Say you go from needing a 12+ to hit a target to needing a 10+. That means you essentially "reduced" their HP by 222-182 = about 40%; this essentially means you gained ~40% damage against that target (relative to Base HP of 100%) from that +2 to hit. However, say you went from only needing a 6 to needing a 4 to hit, that's only about 133-117 = ~16% increase in damage (I say about because I'm ignoring the decimal places for those calculations). You can do the same thing with seeing the net benefit of AC bonuses. You can then compare these numbers to things like damage reduction and +dmg bonuses compared to AC and +hit.

Now, how do you relate the reduction in EHP to a relative increase in damage per hit? In a way, it's mildly confusing: going from needing a 20 to hit to needing a 19-20 is in one sense dealing "10x" as much damage, but in the typical sense most people would use (IE to apply it directly to damage rolls and whatnot) it's like doubling your damage (because you doubled your hit chance). Basically, it boils down to this:
-For a straight relative damage increase, just do Final Hit % / Old Hit % (for example, going from 5% to 10% hit chance means you do .10/.05=2x damage).
-For the change in EHP, use the final column of the table (the +atk bonus column)

 

[Nightson]

Well mostly yes. Specifically "5% to hit != 5% more damage." isn't quite right.

Imagine you have a ten damage attack.

Only hit on 20: Average damage is .5 (10*0.05)
Hits on 19 or 20: Average damage is 1 (10*0.1)

That .5 equals a 100% damage boost.

Misses on a 1: Average damage is 9.5 (10*0.95)
Never misses: Average damage is 10 (10*1)

That .5 approximately equals a 5% damage boost.

Each point of plus hit is an additional 5% of the average damage of the attack increase. It just matters less the more damage you're going.

 

[MadLordOfMilk]

Very good point! Hmm... yeah, I should probably re-word that, it is a bit misleading as-is. I meant it in the sense of relative increases in damage, I hadn't thought about it in the way you put it.

EDIT: How about that change? I changed the "5% to hit != 5% damage increase" thing and the final paragraph. The difficult thing is explaining the math behind it being both linear and non-linear, which turns out to be much harder to explain in a way that's clear and easy to read than I thought...

 

[blargney the second]

That's a neat way of looking at it! So wearing that shield in 4e makes a bigger difference than I thought. Hmm.
-blarg

 

[Stalker0]

That's a neat way of looking at it! So wearing that shield in 4e makes a bigger difference than I thought. Hmm.
-blarg

Yes and no. Against AC attacks, absolutely. Against aura damage or will attacks? Not a bit.

That's the thing you have to watch out for when looking at numbers like these is they don't apply all the time, and so are not 100% effective.

That said, it is a useful analysis. For example, most monsters probably hit a player's AC around 60% lets say. That means a heavy shield increases your hp by 34% if you are getting hit by nothing by AC attacks.

 

[Ycore Rixle]

Cool and useful. I had never thought of to hit increases vs. AC increases this way. Interesting implications for game design, for example, +1 to AC means a lot more at the end of the scale than at the middle. This says a little bit about grindspace: it is increasingly easier to get there as AC increases. That is, it's not a steady march to grindspace as AC goes up, it's an accelerating rush.

 

[Kraydak]

Scaling EHP gains to HP at auto-hit is silly. It is also why you are finding a ~40% DPS gain for a +2 to hit from needing a 12 rather than the more accurate ~20% DPS gain.

But yes, figuring out what number is important is, itself, important.
For DPS, every +1 to hit is a static +DPS. This does not mean a static percentile DPS gain, of course.
For target's time-to-die, you take 1/(number of dice sides that hit).
For risk-of-missing, you instead take 1/(number of dice sides that miss).

All three numbers above have their place. The numbers in your analysis don't have a place that I am aware of.

 

[Morrus]

That's fairly interesting!

I'd posit that the situation isn't that straightforward, though; your AC and other defences also prevent you from suffering status affects, some of which will have an influence on the amount of damage you're taking per round. In addition, some attacks do have effects on a miss.

In addition, if 20 is always a hit, defense increases beyond a certain point are wasted. It makes no difference if someone needs a 20, 21, 22, 35, or 456 to hit you - you're still going to be hit 5% of the time on a 20.

 

[pawsplay]

They're going in different directions. Thus, each point of defense bonus you get is worth more than the last, and each point of +atk is worth less.

They are both worth one move on the table, and hence, basically the same. The only way this analysis is very useful is if you receive either a +1 to hit or a +1 to AC and you want to know what you have really just gained. But assuming +hit and +AC are both in play, they are worth the same, simply an amount that is not predictable without knowing your opponent.

 

[Stalker0]

But assuming +hit and +AC are both in play, they are worth the same, simply an amount that is not predictable without knowing your opponent.

However in 4e it is pretty easy to know your opponent's approximate numbers for a challenge of your level.