Best average damage weapon

[Selion]

I'm bored, time to do some pointless calculations. We'll just calculate the average damage of various weapons, taking in account critical multipliers and the "improved critical" feat.
(Sorry for any grammatical mistakes, I'm not native english)

Weapons taken in account will be:

Scimitar 1d6 18-20 x2
Sword 1d8 19-20 x2
Axe 1d8 20 x3
Pick 1d6 20 x4

For the sake of simplicity we assume that a 15 will suffice to hit the target, we also compare damages with the same BAB, ability scores and magic enhancement bonuses


RESULTS:
Swords and axes have the same average damage.
Scimitars and picks have the same average damage.
Scimitars (and picks) are better than swords (and axes) when the average weapon damage is more than 23 (requiring a total bonus damage of +19)
When improved critical is taken in account, scimitars (and picks) are better than swords (and axes) when the average weapon damage is more than 13 (requiring a total bonus damage of +9)
Weapons are fairly balanced anyway, so don't be afraid to use a keen axe if you like it.




Let's start with the basic formula: without critical hits average damage per hit is pD, where p is the probability to hit and D is the average weapon damage (namely 3.5+bonus for scimitars and picks, 4.5+bonus for axes and swords)
Lets begin with swords and axes.
If we take in account critical hits in the damage formula, we have to add the probability of a critical hit and multiply it to the additional damage of a critical:
- a sword has additional damage D (the 2x adds another D to a normal hit), the probability is 2/20*p (p is required to confirm critical hits).
- an axe has additional damage 2D, the probability is 1/20*p
so the average damage is:
sword: pD+ (2/20) pD
axe: pD+ (1/20) 2 pD
You can easily see that swords and axes have exactly the same average damage, no matter the BAB or the strength modifier of your character.
Things wouldnt change with the improved critical feat.
Sword (improved critical): pD + (4/20) pD
Axe (improved critical): pD + (2/20) 2pD
For scimitars and picks, we notice the average basic damage is one less than swords and axes, so we use the formulas above with damage (D-1) instead of D, bonus damage in case of critical hits is D-1 for scimitars and 3(D-1) for pikes
we have
scimitar: p(D-1) + (3/20)p(D-1)
pick: p(D-1) + (1/20)p 3(D-1)
scimitar(improved critical): p(D-1) + (6/20)p(D-1)
pick(improved critical): p(D-1) + (2/20)p 3(D-1)

You'll notice scimitars and picks share the same average damage, but how they compare to swords and axes? We need to solve the following inequality:

average sword damage < average scimitar damage
namely:
pD+(2/20)pD < p(D-1) + (3/20)p(D-1) -> D>23

lets do the same with improved critical:
pD +(4/20)pD < p(D-1)+ (6/20)p(D-1) -> D>13

Differences are actually slight, let's stretch the numbers and consider 40 average damage (requiring a bonus damage of +36), let's say we have 50% to hit the target, a sword with improved critical has 24 average damage per hit, while a scimitar with improved critical has 25.35 average damage per hit.

[denthor]

Halberd two handed. 1 1/2 strength damaged 1d10x3 on a critical 20.

[John05]

Halberd two handed. 1 1/2 strength damaged 1d10x3 on a critical 20.

Even greatsword has that beat.

It's 1.5x strength damage and 2d6 x2 on crit 19-20.

Let's say you have a much higher atk than target's AC, so you only miss on 1.

That's average of about 7.35 damage for greatsword.

Meanwhile, the halberd only has 5.78. That makes sense though, because if we just assume a normal hit (no crit hit or misses), greatswords do 7 (2d6) on average while halberd does 5.5 (1d10).

If you look at auto-misses except on crit, then halberd does average of 5.5 and greatsword 7. Even if we auto-confirm crits, that's 16.5 on natural 20 for halberd and 14 on crit for 19-20 range on greatsword.

The only areas I see halberd winning out on are 1. lower cost and 2. two choices of damage type (piercing and slashing vs greatsword's slashing).

[Selion]

Even greatsword has that beat.

It's 1.5x strength damage and 2d6 x2 on crit 19-20.

Let's say you have a much higher atk than target's AC, so you only miss on 1.

That's average of about 7.35 damage for greatsword.

Meanwhile, the halberd only has 5.78. That makes sense though, because if we just assume a normal hit (no crit hit or misses), greatswords do 7 (2d6) on average while halberd does 5.5 (1d10).

If you look at auto-misses except on crit, then halberd does average of 5.5 and greatsword 7. Even if we auto-confirm crits, that's 16.5 on natural 20 for halberd and 14 on crit for 19-20 range on greatsword.

The only areas I see halberd winning out on are 1. lower cost and 2. two choices of damage type (piercing and slashing vs greatsword's slashing).

Actually according to the calculation the only difference between a 19-20 x2 and a x3 weapon is the basic damage. They would both be out-damaged by a 18-20 x2 weapon or a x4 weapon at high strength scores, but just slightly

[John05]

Actually according to the calculation the only difference between a 19-20 x2 and a x3 weapon is the basic damage. They would both be out-damaged by a 18-20 x2 weapon or a x4 weapon at high strength scores, but just slightly

Yes, for the majority of cases, only the basic damage is going to matter, because x3/20 is simply adding 0.05 (crit chance) * 200% (bonus damage) to 100 base damage, while x2/19-20 is adding 0.10 (crit chance) * 100% (bonus damage). They both amount to the same.

I'm being specific for edge cases. E.g. Constructs, or absurdly high AC foes. Those are functionally similar cases, where the 19-20 crit threat is better, because crits are guaranteed hits even if you don't confirm them and/or deal multiple of the damage. Ofc you're right that the difference is slight.

Frankly, I'll take the higher threat range, because there are some modifiers and effects that don't get multiplied. IIrc energy damage, negative level effects, and other effects on-hit. What is the x3 or x4 multiplier doing for me then? Also, although it's less cinematic, there's some inherent value in consistency for me.

[Khedrac]

I'm being specific for edge cases. E.g. Constructs, or absurdly high AC foes. Those are functionally similar cases, where the 19-20 crit threat is better, because crits are guaranteed hits even if you don't confirm them and/or deal multiple of the damage. Ofc you're right that the difference is slight.
That's a fairly common misconception, but it happens to be wrong. A critical threat is only a threat if it hits - only a natural 20 is a guaranteed hit.

[John05]

Increased Threat Range

Sometimes your threat range is greater than 20. That is, you can score a threat on a lower number. In such cases, a roll of lower than 20 is not an automatic hit. Any attack roll that doesn’t result in a hit is not a threat.

You're right, Khedrac! I missed that line. Thanks.

IIrc, I picked up that misconception from DDM tourneys. Have to remember DDM rules are obviously different from base d20 rules.