A friend of mine is looking to play a dual wield fighter (I did try to talk him out of it), and was curious about the tradeoff between attack and damage: i.e., which feats should he take, when should he power attack, etc. So I started thinking about the math.

Basic Premise: In d20, unless you can only miss on a 1, each +1 is an additional 5% of your base effectiveness, per roll.

This is pretty easy to calculate:

  • Each roll has a chance of success based on the number you need to roll on the die to succeed. If you succeed on an 11 or better, that’s half the possible rolls on the die, so you have a 50% chance of success. If you only succeed on a 16 or better, you have a 25% chance of success. And so on.
  • Because there are 20 possible results, each represents 5% (100%/20).
  • Each +1 to your roll makes you 5% more likely to succeed (e.g., you go from needing a 11+ to a 10+ to hit, going from 50% chance to 55% chance to succeed).
  • Thus, over time, you can add 5% of base effectiveness (e.g., 50% hits at your normal value vs. 55% hits at your normal value). If you do an average of 10 damage, a +1 is going to give you half a point of damage per attack.

Attack bonus is, thus, more effective over time the more damage you do, base.

 

Basic Premise: In d20, each +1 damage is an additional X damage, per roll, where X is equal to 1 times your chance of success.

This is a similar calculation:

  • As above, over time you can base your expected damage on the chance to hit: if you hit 50% of the time, you will do 50% of your base damage over time.
  • At 50%, each point of damage is, therefore, actually worth half a point of damage per attack.
  • Meanwhile, if you have a higher or lower chance to hit, the worth of the point of damage scales accordingly.

Damage bonus is, therefore, more effective over time the greater your chance to hit.

 

Obviously, these two premises stack very nicely: if you have a higher attack, you’ll deliver your base damage more often, and if you have a higher damage, it will benefit more from having a high attack.

But what if you have to choose? Greater Weapon Focus or Weapon Specialization? Activate Power Attack or stick with regular hit bonus?

In these situations, attack bonus becomes more valuable the higher your damage already is, and damage bonus becomes more valuable the higher your chance to hit already is. Obvious based on the premises above, right? But where is the breakpoint?

Base Damage 2 4 6 8 10 12 14 16 18 20
Value of +1 Attack 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Necessary Roll to Hit 2 4 6 8 10 12 14 16 18 20
Value of +1 Damage 0.95 0.85 0.75 0.65 0.55 0.45 0.35 0.25 0.15 0.05

Effectively, at around 10-11 points of damage and having to roll 10-11 to hit, a point of damage and a point of attack are roughly equivalent. Lower base damage is equivalent to a higher chance to hit when trading off.

What about 2 damage? Power Attack in Pathfinder gives you +2 damage for each -1 attack (with one-handed weapons).

Roll Needed to Hit
Damage 2 4 6 8 10 12 14 16 18 20
2 19 17 15 13 11 9 7 5 3 1
4 9.5 8.5 7.5 6.5 5.5 4.5 3.5 2.5 1.5 0.5
6 6.3 5.7 5 4.3 3.7 3 2.3 1.7 1 0.3
8 4.8 4.3 3.8 3.3 2.8 2.3 1.8 1.3 0.8 0.3
10 3.8 3.4 3 2.6 2.2 1.8 1.4 1 0.6 0.2
12 3.2 2.8 2.5 2.2 1.8 1.5 1.2 0.8 0.5 0.2
14 2.7 2.4 2.1 1.9 1.6 1.3 1 0.7 0.4 0.1
16 2.4 2.1 1.9 1.6 1.4 1.1 0.9 0.6 0.4 0.1
18 2.1 1.9 1.7 1.4 1.2 1 0.8 0.6 0.3 0.1
20 1.9 1.7 1.5 1.3 1.1 0.9 0.7 0.5 0.3 0.1

The chart above shows the ratio of the value of 2 Points of Damage to 1 Point of Attack. So if you’re doing 2 points of damage and hit on a 2+, 2 points of damage is 19 times as good as +1 attack. Meanwhile, if you already do 20 damage and need a 20 to hit, 2 points of damage is less than 10% as good as +1 attack. The breakpoints are in red: past that number, you shouldn’t Power Attack (for example). If your chance to hit is decent, though, you need huge amounts of base damage to make Power Attacking a bad idea.

The more you know!

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