Two-weapon Fighting

In response to a brief discussion of two-weapon fighting on Saturday (and sorry about the length, skip to the bottom for the conclusion):

An analysis of two-weapon fighting:

Let's consider a 5th level Fighter. We'll assume he's purchased the Ambidexterity and Two-Weapon Fighting feats. He's now considering his weapon options. Let's assume he's picked up enough bonuses from feats and the like to boost his attack bonue to +10 (for ease of calculation), and we'll assume he has a Strength bonus to damage of +2.

He's now considering two combat options. Option 1 is a Greatsword (2d6+3 damage, 19-20 critical), while option 2 is the longsword/shortsword pairing (-2 to hit with each, 1d8+2 and 1d6+1 damage, 19-20 critical).

Clearly, since everything else is equal, the key determinant on whether to go for two-weapon fighting or not is the average damage caused.

The average damage caused with option 1 is 10 points on a successful hit. With option 2, the longsword does 6.5 hit points, while the shortsword does 4.5. All values are doubled on a critical hit.

For AC 1 - 10, all attacks hit 95% of the time. Thereafter, the attacks of option 2 are 5% less likely to hit for every AC point up to AC 28, at which it reaches 5%. Thereafter, it remains stable. Additionally, option 2 threatens a critical hit 10% of the time, except at AC 28 and above, when the chance drops to 5%.

For AC 1 - 12, the attacks of option 1 are 95% likely to hit. This then decreases by 5% for each point of AC up to AC 30, at which point the chance of a hit is 5%. Thereafter, it remains 5%. Additionally, option 1 threatens a critical hit 10% of the time, except at AC 30 and above, when the chance drops to 5%.

For a given attack, then, the damage caused is:

(hit% * damage) + (crit% * hit% * damage) = (hit% * damage)(1 + crit%)

So, at AC 10:
Option 1: (0.95 * 10)(1 + 0.1) = 10.45 hit points
Option 2: (0.95 * 6.5)(1 + 0.1) + (0.95 * 4.5)(1 + 0.1) = 11.495

At AC 11:
Option 1: 10.45 hit points
Option 2: (0.90 * 6.5)(1 + 0.1) + (0.90 * 4.5)(1 + 0.1) = 10.89 hit points

At AC 12:
Option 1: 10.45 hit points
Option 2: (0.85 * 6.5)(1 + 0.1) + (0.85 * 4.5)(1 + 0.1) = 10.285 hit points

At AC 13:
Option 1: (0.90 * 10)(1 + 0.1) = 9.9 hit points
Option 2: (0.80 * 6.5)(1 + 0.1) + (0.80 * 4.5)(1 + 0.1) = 9.68 hit points

Thereafter, option 1 decreases by 0.55 hit points per point of AC, while option 2 drops by 0.605. In other words, the gap just continues to increase, until AC 27:

At AC 27:
Option 1: (0.2 * 10)(1 + 0.1) = 2.2 hit points
Option 2: (0.1 * 6.5)(1 + 0.1) + (0.1 * 4.5)(1 + 0.1) = 1.21 hit points

At AC 28:
Option 1: (0.15 * 10)(1 + 0.1) = 1.65 hit points
Option 2: (0.05 * 6.5)(1 + 0.05) + (0.05 * 4.5)(1 + 0.05) = 0.5775 hit points

At AC 29:
Option 1: (0.10 * 10)(1 + 0.1) = 1.1 hit points
Option 2: 0.5775 hit points

At AC 30 and above:
Option 1: (0.05 * 10)(1 + 0.05) = 0.525 hit points
Option 2: 0.5775 hit points

So, the end result is that Option 2 is better for AC 10 and 11, and for AC 30 and above. However, for everything in between, it's worse.

Now, there is an additional wrinkle thrown in once the Fighter gets to 6th level, in that he gains an additional attack. Moreover, the two-weapon fighter might consider purchasing the Improved Two-Weapon Fighting feat at this point, to get yet another additional attack. Between them, these issues make the calculations more complicated. The mathematics are left as an exercise for the reader, or until I next get bored.

The conclusion that I draw from this calculation is that even for the Ranger (who gets the two required feats for free), two-weapon fighting is a poor option, except when fighting extremely weak or extremely powerful creatures. This conclusion is reinforced when one considers the effect of feats such as Improved Critical or Weapon Specialisation, which apply only a single weapon, and would need to be bought twice by the two-weapon fighter. Additionally, it is obviously more cost-effective to equip a fighter with a single magical weapon than two, as would be required by the two-weapon fighter.

That said, there are a few cases in which two-weapon fighting is the way to go:

• Double weapons avoid Improved Critical issue that has just been mentioned. However, they still cost double to have enchanted. Additionally, most double weapons require yet another feat to use effectively.
• If the character has additional damage dice from any source (such as a Rogue's Sneak Attack), having more attacks is obviously of greater benefit than fewer, more powerful ones. In this regard, the Ranger/Rogue combination must be considered extremely powerful, to the extent that every Rogue should at least consider picking up a level of Ranger to complement his Sneak Attack.