D&D 5E Attack bonus vs. damage bonus

[UngeheuerLich]

Because you have a formula, like savyomagy showed, that is dependend on 2 variables, so you can derive it and find out for which combination of those two variables you get zero. This is exactly when attack bonus + target number is 21. If you increase your attack bonus your target number decreases by 1, and so, if you are at 22 or above, you increase attack bonus. If you are below, you increase attack damage. This is always the best ratio you can get. If you are exactly at 21, it does not matter which you increase, because the damage gain is equal.

Notice, that it does not tell you anything about your actual damage, only which is the better way to increase your damage if you have the choice between +1 to attack or damage. You generally want target number as low as possible and damage as high as possible. And if you have the choice between getting 21 or getting a number that is different from that, but target number lower and attack damage higher, you chose the latter.
If your target number is very low however and your damage also low, you should maybe get a -5/+10 feat and look if you can get a bit more balanced stats. If your attack bonus and attack damage are well balanced, get +2 main stat instead.

 

[Saeviomagy]

Could you guys walk me through this? Why 22 (or 21, according to Saeviomagy)?

It's >= 22 OR > 21, which are identical (for integers anyway). If it's equal to 21, then it doesn't matter which one you do, so for the purposes of this metric we just chose to do it one way for simplicity.

As to why?

Every point of attack bonus you get increases your average damage by 1/20th of your current damage.

Every point of damage bonus increases your average damage by your chance to hit.

Choose which one of those is higher.

Notable consequences:
If you can do 20 average points of damage, always increase your chance to hit until you miss only on a 1.
If you can hit for 10 average damage, you're basically set for damage against most foes and should typically focus on attack bonus.
If you hit for 5 average damage, increase damage! There's not many foes that you'll need a 17 to hit.
The -5/+10 feats will mean that you should basically always focus on attack bonus if you want to use them.

 

[BoldItalic]

Could you guys walk me through this? Why 22 (or 21, according to Saeviomagy)?

The breakpoint is 21 (we agree on that) but at exactly 21 neither bonus is better than the other; there's a dead spot between 20 and 22. At 22 or above it is clearly one way (Hit bonus is better) whereas at 20 or below it is clearly the other way (Damage bonus is better).

To see how I arrived at the formula, let the roll to hit be H and the expected damage be D. The probability of hitting is (21-H)/20 so the DPR is Dx(21-H)/20. Now we want to compare giving a +1 bonus to hit, which gives a DPR of Dx(21-H+1)/20, with a +1 bonus to damage, which gives a DPR of (D+1)x(21-H)/20. To compare the two expressions we can first of all cancel the /20 and just compare the numerators. Then we can subtract Dx(21-H) from each, which doesn't change the comparison, leaving just D for the first expression and just 21-H for the second. So the question reduces to just D vs 21-H. If D > 21-H the attack bonus is better whereas if D< 21-H the damage bonus is better. I just re-arranged the terms to make D+H > 21 (which is D+H >= 22) vs D+H < 21 ( which is D+H <= 20 ) so that you could work out one number (D+H) and say is it more or less than 21?

(edit) Saeviomagy's explanation above is a lot simpler than mine

 

[UngeheuerLich]

A little addition: If (21-H)+D is a fixed value, the area of a rectangle D*(21-H) is higest, when it is a square.

And generally you want both 21-H (probability to hit) and D (Damage) as high as possible.

 

[BoldItalic]

I've worked it out allowing for critical hits.

If the damage bonus also applies singly to criticals (a straight +1 that doesn't get doubled) the formula is still correct.

If the damage bonus would be doubled on a crit then the breakpoint is 22 instead of 21. (An example of this would be switching from a d6 weapon to a d8 weapon).

 

[FrogReaver]

Damage * Chance to Hit = DPR

Let n = number of attacks
Let D = Base Damage
Let B = Damage Bonus
Let C = Base Chance to hit
Let A = Additional Chance to hit

Damage after increase = (D+nB)*(C+A) =

DC + DA + nBC + nBA

Now, Let x be the amount you could increase Damage Bonus by. Let y be the amount you can increase chance to hit by if you do not increase Damage Bonus.

Thus we are comparing possibilities for (B, A) = (x, 0), (0, y)
*Also any other values could be compared here if desired

And for Damage Bonus to beat out to hit bonus:

DC + nxC > DC + Dy

which implies,

nxC > Dy for the damage bonus being considered to be better than the alternative to hit bonus being considered at any given time.
Dy < nxC
D < C*n*(x/y)

Thus, when the formula D < C*n*(x/y) is true then the additional damage x is better than the additional chance to hit y.

Example Calculation: n=2, x=3, y=5%, D=17, C=60%
17 < .6*2*3/.05
17 < 72, therefore it is better to add +3 damage when you have 17 damage and 60% chance to hit and 2 attacks than it is to add 5% chance to hit.

 

[Giant2005]

There is no right answer because whether or not more attack or more damage is better depends on your target's AC. If your target's AC is high, then more attack is better, if your target's AC is low, then more damage is better.

Having said that, I'm going to go ahead and declare that more attack is better anyway. Enemies that are powerful enough for maximizing your DPR to actually be a difference tend to have higher ACs. Effectively you will encounter two basic situations in game:

1. It doesn't matter whether attack or damage are higher - you are going to win pretty easily either way.

2. You want a higher attack.

 

[UngeheuerLich]

Yes. This formula was especially useful with 3.5 power attack. But it is still useful when to use the damage increasing feats and when not. It may also help to decide if divine favour or bless is the better choice.

Of course you neglect other variables like defensive abilities of your target, advantage, hp of your target, damage capabilities of your teammates.

Sometimes a low damage hit is what you need, no matter what damage you do, sometimes it would not matter if you hit for low damage or not at all (against heavy armor mastery, maybe against a fleeing target with 15 hp left and you only have one chance to hit.
You can't account for all variables and as long as your ratio of to hit and damage is close to t+d=21 it doesn't matter a lot what you increase. Other considerations have a bigger impact on your damage increase then.

Examples:
Against a goblin with 7 hp, you generally don't want to increase your damage even if you are way below 21, if your chances to deal 7 damages is high enough.
With a rapier and dex +4 you deal 1d8+4 damage, so you kill with a 75% chance. If you only use a dagger, suddenly your chance to kill drops to 50%. In that case maybe divine favour may be a good idea, because your chance to kill increases to 95% if you hit, while +1d4 to hit won't raise your chance to kill by a wide margin.

 

[BoldItalic]

How to apply the simple formula

A 1st-level elf rogue favours the longsword (it's an elf thing) but he has now acquired a greataxe. Which should he use against AC16 goblins? What about an AC17 goblin boss?

To calculate a baseline, we can disregard his Str bonus because it cancels out in the formula. The baseline roll to hit equals the opponent's AC (16 or 17 in this case). He is proficient in longsword but not in greataxe, so the longsword option represents +2 to hit (he's 1st level, so his proficiency bonus is +2). Against this, the greataxe option represents +2 damage over a baseline of 4½ (average damage is 6½ against the longsword's 4½).

Against ordinary goblins 16+4½ = 20½ < 21 so better damage is better - he should use the greataxe.
Against the boss goblin, 17+4½ = 21½ > 21 so a to-hit bonus is better - he should use the longsword.

Interesting.

It's marginal either way, but that's because I've cunningly chosen the example to straddle the breakpoint. If I'd compared kobolds to hobgoblins, it would have been more clear-cut. But the conclusion is the same - against AC17 or higher he should use the longsword whereas against AC16 or lower, he should forget about being an elf and wade in with the greataxe.

Also note that I haven't taken into account that goblins only have 7hp so there's a possibility of a one-shot. That complicates things; instead of looking at DPR, you have to start looking at kill rates.

Elf rogues with greataxes. Now there's a thought.

 

[JackOfAllTirades]

What's the formula look like if you're using one of the -5/+10 feats, and where is the break even point in that case?