D&D: Better Bonus: To-Hit or Damage?

I crunched some numbers* using D&D mechanics.

Damage bonus is better... unless the target is REALLY hard to hit (you need a 16 or above).

If both opponents need a 10 to hit one another, a player with a +1 to hit versus a player with +1 damage will lose 55.8% more often, all other things being equal. If they each need a 15 to hit, however, this goes down to a 7% advantage to have the extra damage. If your target is so hard to hit you need a 16, however, you will start to lose with the extra damage: better to get the to-hit bonus at that point (7% better, in fact).

Not that I'm in to min-maxing**, mind you... but in a heavy-mechanics game like D&D, it seems to be better for everyone if this kind of information is out in the open.

* Actually: this is brute-force die-rolling, not real statistics: I fought 4000 battles for each result, ignoring tactics. Basically, two guys standing toe-to-toe and rolling to-hit, then damage, one attack per turn. (
And, yes, I know two attacks would be more accurate if they were really toe-to-toe: oh well.) I gave each of them 100 HP and a base damage of 1d10. Neither of them healed at all. This was done with ruby code: if you want the source, ask.

** On the other hand, as a game designer, I think min-maxing is an essential practice to understand the mechanics.


Jeremy Rice said...

I decided to check whether this held true as the bonus went up: +2 atk vs +2 dmg. Actually, the effect intensifies. With a 10-to-hit, the damage bonus won 115% more often!

At 15-to-hit, the damage bonus fared about the same as before: 7.4% better.

At 16-to-hit, the to-hit bonus won 10% more often.

Same results, just a little more dramatic.

Victor said...

it should be fairly straightforward to calculate actually. The number that matters is your expected damage per round, which is the average damage (5.5 for 1d10) * chance of hitting.

A +1 to damage will give you about 18% boost in expected damage. For to-hit bonus, it will of course depend on the hit chance. If I remember the new D&D rules correctly, you simply need to roll 1d20 at above the AC value, right? So for, say, AC of 10, a +1 to hit would amount to an approximately 10% increase in expected damage.

For any given damage die, there would be a corresponding break-even AC value above which to-hit bonus is better, and below which to-damage bonus is better. The equation to derive it should be about:

(d+b)*(21-a)/20 = d*(21 + b - a)/20

which simplifies to:

a = 21 - d

d is the average damage, a is the break-even AC value, b is the bonus you are calculating for. As you can see, the break-even AC point doesn't depend on the magnitude of the bonus you are considering, but does depend on the average damage roll. The higher the damage, the lower the AC threshold for to-hit advantage.

Jeremy Rice said...

That's very cool. Thanks for doing the legwork!

In actuality, to-hit in 4E is 1d20 + 1/2 your level + proficiency bonus (each weapon has one, usually +2 and +3 in a few cases) + attribute modifier (usually Str or Dex) + other modifiers, which could be class-features (fighters get +1), magical weapons, power bonuses, and situational modifiers.

Interestingly, 4E did away with Fort/Reflex/Will as Saving Throws and made them another kind of "AC"-- a target that you try and hit with the above type of roll. ...which is to say, you're not always rolling vesus AC--even spells are a "to-hit" kind of thing, nw. It's rather interesting on paper; I look forward to testing it in action.