Damage Output of Improved Critical vs. Weapon Specialization

Question

Let's assume the following:

• Rapier: 1d6 Critical Hit Range 18-20/×2
• Attack: +10
• Damage: 3d6+2 (extra dice from other feats, I believe only 1d6+2 gets applied on the crit.)
• Opponent AC: 20
• Battle Ardor: +2 to confirm a crit (warblade class ability)

What feat will give a higher average damage output per turn: Improved Critical (double the critical threat range) or Weapon Specialization (+2 damage)?

How does adjusting the chance to hit affect the outcome (attack vs. AC)?

How does adjusting the base damage affect the outcome?

Answer 1

Let:

$$\begin{align} m&\text{: chance of a miss} \\ h&\text{: chance of a hit} \\ c&\text{: chance of a critical hit} \\ D_m=0&\text{: damage on a miss} \\ D_h&\text{: damage on a hit} \\ D_c&\text{: damage on a critical hit} \\ D = mD_m+hD_h+cD_c&\text{: damage} \\ H \ge 2&\text{: the lowest number on the d20 which is a hit} \\ T \ge H&\text{: the lowest number on the d20 which is a threat} \\ C &\text{: the lowest number on the d20 which confirms a critical} \\ \end{align}$$

Then:

$$\begin{align} m&= {H-1\over 20} \\ h&= {1-m-c} \\ c&= {21-T\over 20}\times{21-C\over 20} \\ \end{align} $$

In general \$H=AC-\text{Attack mod}\$, however, this cannot be less than 2 or greater than 20. Normally, \$C=H\$ but because of Battle Ardor, in this case \$H=AC-\text{Attack mod}-2\$, again not less than 2 or greater than 20. We therefore need to consider a range of \$AC-\text{Attack mod}\$ from 2 to 22.

This spreadsheet crunches the numbers. For all of them Weapon Specialization is better, doing 4.3 more mean damage when you need a 2 to hit dropping to 0.2 when you need a 20.

Answer 2

Damage from Weapon Specialization

Miss Chance: 0.45

Non-Crit Chance: 0.4525

Crit Chance: 0.0975 (Crit threshold × To Confirm Chance, which is same as hit chance +2 (0.55 + 0.10 = 0.65))

Average Damage Expected = (Non-Crit Chance × Non-Crit Damage (14.5)) + (Crit Chance × Crit Damage (29)) = 9.39~ Damage

Damage from Improved Critical

Miss Chance: 0.45

Non-Crit Chance: 0.355

Crit Chance: 0.195

Average Damage Expected = (0.355 × 12.5) + (0.195 × 25) = 9.31~ Damage

Conclusion: Weapon Specialization is the Better Option for this specific case It's close enough that you should measure damage at slightly later levels.

The higher the base damage, the more damage Improved Critical will deal relative to Weapon Specialization. This is due to Weapon Specialization being an increasingly small proportion of your base damage, as compared to a critical doubling your damage.

The higher the enemy AC/the lower your attack bonus, the larger percentage of your damage will come from crits. Improved critical will deal higher damage the higher the AC of the opponent, eventually surpassing Weapon Specialization. This trend holds true until it peaks at the number you need to roll being your critical threat range, at which point it reverses for higher AC's. If you need a natural 15 to hit an enemy and a crit range of 15-20, about half of your hits will be criticals. If you need a natural 20, only 5% of your hits will be criticals.

Try a Scabbard of Keen Edges This wondrous item can cast Keen Edge three times a day on your weapon, effectively duplicating Improved Critical. Since this critical multiplier doesn't stack with Improved Critical, you might as well take Weapon Specialization instead. This is a particular choice given that you're a Warblade, likely looking to take Slashing Flurry. If any of these things don't apply to your game, I'd recommend updating the question to talk about what does go in your game.

Answer 3

Questions comparing damage calculations like this can often be solved with basic algebra, systems of inequalities specifically.

Your normal expected damage per hit is

• \$x\;-\$ %chance to hit (you know, AC-attack bonus). \$x\geq.5\$ (because of nat 20s) and \$x\leq.95\$ (because of nat 1s).
• \$y\;-\$ average damage. \$y\geq1\$.

$$d = xy$$

With Weapon Specialization, that is

$$d = x (y+2)$$

With Improved Critical, if \$z\$ is your crit range, then you do normal damage plus a \$z\$ chance of normal damage again (barring non-doubling stuff like SA and other feat shenanigans)

$$d = x (y + zxy)$$

So if you want to know when IC is better than WS, you are solving the inequality

\begin{align} x (y + zxy) &> x (y + 2) \\ y + zxy &> y + 2 \\ zxy &> 2 \end{align}

So if your chance to hit times your crit range times your damage is greater than 2, then IC is better (this stands to reason, right — you get +2 from WS, so when do you get more than +2 from IC? When you hit, and when you confirm, and the resulting expected damage is larger than 2).

In your example you have a 55% chance to hit, and a 15% crit chance, and an average damage of 14.5 (though you didn't provide information as to where those other dice come from, it's likely they're not actually doubled on a crit because that's pretty rare unless it's making the weapon bigger or something). But just to take it at face value,

\$.55 \times .15 \times 14.5 = 1.196\$, so Weapon Specialization is better.

After your edits, the same formula works — the confirm chance is \$x+.1\$ because of your +2 to confirm crits, and you can just ignore the non-doubling part of the damage because it's a flat addition to either scenario. So you're looking at \$z(x+.1)y>2\$ where \$z\$ is still .15, \$x\$ is still .55, and \$y\$ is now 5.5. So WS is now even more way better in your example.

With this formula you can now trivially run other scenarios. What if you're going to get some power next level that lets you autoconfirm crits? Well, then you just need \$zy>2\$ because the confirm chance becomes 1, right. But with that damage (\$.15 \times 5.5\$), IC is still worse than WS even if you auto-confirm. Let's say you take Improved Critical and get that .15 threat range up to .30 (15-20). Nope, \$.3 \times 5.5\$ is still less than the +2 you get from WS.

Any time you want to see a hit/damage scenario 1 vs. scenario 2, set up a system of inequalities, reduce it to the fundamental variables, and reduce, and you'll end up with a simple formula that will tell you under what circumstances one is better than the other without Anydice or spreadsheets or whatnot.