Is the Keen enchantment better than an enhancement bonus on a rapier?

Question

What weapon has a greater damage per attack: +1 Keen Rapier or +2 Rapier?

Here are the base numbers we can use:

STR 14
DEX 18 (uses Weapon Finesse for attack)
BAB +8
+1 Rapier Crit: 18/x2

Single Attack: +13
Single Damage: 1d6+3

Enemy AC: Since this depends on the target’s armor class, let’s evaluate for a range of enemy ACs. 15/20/25 should be enough to draw a conclusion.

Keen Weapons double the threat range, making the critical data for a rapier: 15/x2

Other considerations:

1. This doesn’t consider iterative attacks, calculating this is optional for me.
2. This is a warblade, and he gets +2 to confirm critical hits. I’m ok ignoring this calculation for now.
3. Without going into the wealth of strike maneuvers, obviously many of them deal a bunch of damage but don’t get multiplied on a critical hit.

So the question again: How does the increased threat range on a Keen rapier compare to an extra +1 enhancement enchantment?

Answer 1

With damage numbers that low (1d6+3 or 4), the multiplying effect from crits loses out to the flat +1 damage to all hits you get from the extra enhancement bonus (and that's not even considering targets immune to crits). Keen doesn't become useful until damage bonuses get high enough to balance out both that and the extra damage from the additional +1 to hit.

For example, while the +2 averages 8.1938 and the keen averages 8.0275 damage per attack against low-AC targets (at most 15, given the listed base attack and dex), this gap closes to 9.28625 vs 9.2625 with just 1 more strength bonus and swaps to 10.37875 vs 10.4975 if strength goes up to 18.

However, for ACs 27 and higher, the +2 is always strictly better than the keen due to the additional +1 to hit outweighing the extra crits. If we consider the +2 to confirm crits, the overall trend is the same (although the inflection doesn't happen until AC 29 and up).

While the calculations presented here are based on specific to-hit and damage values, the math is similar regardless (here "dmg mults" means multiples of your damage dealt per 20 attacks):

\$\begin{array}{|c|c|c|} \hline \textbf{AC - BaseToHit} & \textbf{+2 dmg mults} & \textbf{keen dmg mults} \\ \hline \text{<=3} & \text{21.85} & \text{24.7} \\ \text{4} & \text{21.85} & \text{23.4} \\ \text{5} & \text{20.7} & \text{22.1} \\ \text{6} & \text{19.55} & \text{20.8} \\ \text{7} & \text{18.4} & \text{19.5} \\ \text{8} & \text{17.25} & \text{18.2} \\ \text{9} & \text{16.1} & \text{16.9} \\ \text{10} & \text{14.95} & \text{15.6} \\ \text{11} & \text{13.8} & \text{14.3} \\ \text{12} & \text{12.65} & \text{13} \\ \text{13} & \text{11.5} & \text{11.7} \\ \text{14} & \text{10.35} & \text{10.4} \\ \text{15} & \text{9.2} & \text{9.1} \\ \text{16} & \text{8.05} & \text{7.8} \\ \text{17} & \text{6.9} & \text{6.25} \\ \text{18} & \text{5.75} & \text{4.8} \\ \text{19} & \text{4.6} & \text{3.45} \\ \text{20} & \text{3.45} & \text{2.2} \\ \text{21} & \text{2.2} & \text{1.05} \\ \text{>=22} & \text{1.05} & \text{1.05} \\ \hline \end{array} \$

As long as the value in the rightmost column is higher than that in the middle column, keen can win out for a sufficiently high strength bonus (or other bonus that gets multiplied by crits). Be aware that iterative attacks operate at a reduced to-hit bonus, moving you five rows lower on the chart (i.e. reducing keen's effectiveness vs. the extra enhancement bonus).