d20 Math: AC, To Hit, and Diminishing Returns

[Kraydak]

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

It would be more accurate to say that 4e said "things get wonky when you get near the edges of a d20" and tried to pull everything towards needing a 10 (where small modifiers are stable, and even medium modifiers don't break things too much). That decision (which is why there is a flat 1/2 levels rather than 3e's BAB spread) was perhaps the single best decision in 4e's design (the flat xp-tables was perhaps 3e's, as killing THAC0 didn't change the underlying math). Unfortunately, much like 3e not getting class balance right, 4e didn't get the difficulty normalization right (and, frankly, got it wrong more spectacularly: how did +extra stat to d20 roll abilities get through playtesting?!).

But, basically, as long as your needed number is about 10, then you can have modifiers of about +/- 4 without things getting (too) wonky. Go past that, and each new modifier point is either worth hugely more or hugely less.

 

[Cadfan]

A challenge of your level doesn't mean a monster of your level. In my experience, about 8 points of variation on AC is plausible. This covers a significant portion of a d20's distribution.

 

[MadLordOfMilk]

All good points. And yeah, the key thing I think was said by Stalker0: "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."

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.

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

....*cough*

Also, your miss chance scales linearly, not with the equation you put. Unless of course you mean that when you have a 19/20 hit chance (95% chance to hit), it's actually a 100% miss chance, and that when you have a 1/20 hit chance, it's actually a 5.26% chance to miss. Because.... that's what you just said.

I'd touch on the DPS thing, but ok, I get what you mean, I'll give you the benefit of the doubt. Now, any other "silly" comments (to use your word), Kraydak?

 

[Kraydak]

...
I'd touch on the DPS thing, but ok, I get what you mean, I'll give you the benefit of the doubt. Now, any other "silly" comments (to use your word), Kraydak?

You are still scaling things to 100% hit rates. *That* is not useful. *That* is why you would expect a 40% DPS gain for going from hitting on 12s to hitting on 10s from your table. You in fact expect a 20% gain because you should be normalizing to the EHP at (either, depends on exactly what you are looking at) 12 or 10.

Or, to put it differently, the last two columns in your table are actively misleading. Going from 19s to 20s is a 100% EHP gain, NOT a 1000% EHP gain. Going from 2s to 20s IS the 1000% EHP gain you list (assuming 1s hit), but that involves a whopping +18 AC.

 

[MichaelSomething]

So how does this chart help me kill monsters faster? Does this mean I should focus on increasing my damage as much as increasing my hit?

 

[MadLordOfMilk]

@MichaelSomething: well, damage-wise, it does mean you need to keep increasing your damage along with your +hit. However, for the most part you shouldn't get too far into the extremes on either end of the table, given how 4e pushes for a general ~50% hit rate. Still try to max hit rate as much as possible of course (as it only gets lower as you level up), especially because powers have effects outside of damage. Given how it's kind of a losing battle the higher in level you get, it shouldn't really reach that "not nearly as useful" point way up there in high hit rates. However, don't make +hit your only source of damage boosts... just make sure to keep raising both.

For a relative damage gain, do (Final Hit% / Initial Hit%), so going from 5% to 20% hit rate is 20%/5% = 4x damage, whereas from 50% to 55% is 55%/50% = 1.1x damage, or a 10% increase. The values in the table are more for putting survivability at various hit rates into perspective, and they can certainly be used for calculating how LONG something will live, but for straight damage relation go with what I just mentioned.

You are still scaling things to 100% hit rates. *That* is not useful. *That* is why you would expect a 40% DPS gain for going from hitting on 12s to hitting on 10s from your table. You in fact expect a 20% gain because you should be normalizing to the EHP at (either, depends on exactly what you are looking at) 12 or 10.

Or, to put it differently, the last two columns in your table are actively misleading. Going from 19s to 20s is a 100% EHP gain, NOT a 1000% EHP gain. Going from 2s to 20s IS the 1000% EHP gain you list (assuming 1s hit), but that involves a whopping +18 AC.

You go from living 10x as long as someone who gets hit every time to 20x as long. It IS a 1000% net EHP gain, but a 100% relative gain. The former makes the point better IMHO, and lets one see exactly how much of a gap there is in survivability from 100% hit rate to 50% to whatever, but both are worth noting and useful.

When figuring out how much a character's relative gain is, of course you'd have to do 2000%/1000% = 2x survival duration. Unfortunately, there is only so much one can put on one table... I had to leave *some* calculations to people.

You're basically arguing about where the baseline is. Of course I could make 20 copies of the table with the baseline at each respective point... but it'd be the same scaling rate regardless. Scaling to getting hit 100% of the time is an easy-to-understand reference point, and IMHO makes understanding the concept of "effective hp" (or, in other words, how long you're going to live) much easier to relate to the actual numbers. YMMV.

Plus, all of this gets down to jumping between the relative gains and the literal values and linearity and non-linearity and... blech. It wasn't meant to be a comprehensive guide on every factor of damage