D&D 3E/3.5 [3.5] Crit stacking?

[Mike Sullivan]

This part I understand, although you aren't taking AC into consideration obviously.

Well, I sort of am. I'm assuming that whatever AC the target is, you require exactly the same numbers to hit with a scythe as you would with a greatsword -- so it's the same factor on both sides of the equation, so I didn't bother to put it in in the first place.

Sorry, but you need to go through this part and explain. It sounds to me like you're pulling these numbers out of thin air, and your equation's answer also seems to just come out of thin air. Maybe I've been out of school for too long, but when you have a double variable, I wasn't aware you could solve one side first in only two steps. You're missing some in there somewhere. Go through each step of the equation. Also explain precisely where the entire top line comes from.

Goodness. Feeling demanding today?

Sure, whatever:

1.15 ( 5 + x) > 1.1 (7 + x)
Explanation: 5 is the average damage on 2d4, the base damage die for a falchion or scythe. x is what we're solving for -- the damage bonus that makes it worthwhile to use the falchion or scythe, in the expected-damage-scenario. 7 is the average damage on 2d6, the base damage die for a greatsword. 1.15 is the multiplier which represents the effects of a critical hit for an 18-20/x2 weapon or a 20/x4 weapon. 1.1 is the multiplier which represents the effects of a critical hit for a 19-20/x2 weapon. So, the inequality that we're solving for is what damage bonus x makes it true that the falchion or scythe's total expected damage per hit is higher than the greatsword's expected damage per hit.

5.75 + 1.15x > 7.7 + 1.1x
Explanation: I expanded the multiplication on both sides, using the, oh, hell, it's been like ten years since I took algebra, that rule that says: x(a + b) = xa + xb.

.05x > 1.95
Explanation: I subtracted 1.1x from both sides of the equation, and also subtracted 5.75 from both sides of the equation. If you want, you can expand this out into a bunch of steps:

5.75 + 1.15x > 7.7 + 1.1x
5.75 + 1.15x - 1.1x > 7.7 + 1.1x - 1.1x
5.75 + .05x > 7.7
5.75 + .05x - 5.75 > 7.7 - 5.75
.05x > 1.95

x > 39
Explanation: I multiplied both sides of the equation by 20 (or divided by .05, whichever you prefer.)

Thus, as we check out the original equation, we see that it holds true for all x such that x > 39. Since x must be an integer (you can't have fraction damage bonuses), that means that the lowest x for which it's true is x = 40. Thus, if you have a +40 damage bonus (or more), and do not have either keen or improved critical, assuming that your chances to hit are the same in both cases, it's advantageous to use a scythe or falchion instead of a greatsword against opponents susceptible to critical hits. If your damage bonus is +39, then they're exactly the same. If your damage bonus is +38 or less, it's more advantageous to use the greatsword.

 

[Anubis]

1.15 is the multiplier which represents the effects of a critical hit for an 18-20/x2 weapon or a 20/x4 weapon. 1.1 is the multiplier which represents the effects of a critical hit for a 19-20/x2 weapon.

Okay, one more question about this part. How did you come up with these numbers to represent the effects of a critical hit?

When I run the math, assuming a static AC, the average damage of say a greatsword is 7. The average damage of a critical hit is 14. Your chances of getting it, however, are less than 10%, as a 1 always misses.

This is also why I said you didn't take AC into account. You numbers assuming you always hit and always confirm. The maximum chance of a confirmed critical on a 19-20/x2 weapon, however, is 9.5%. This comes out with a factor of 1.095, not 1.1, so I am not clear where you got your numbers for this part. May sound like a small difference, but when the figures get bigger, this difference is noticable.

 

[Hypersmurf]

A weapon that grants a feat grants skill to the user. He actually has a skill he didn't have before. of course, this is beside the point, since there are no actual examples of such a weapon in any of WotC's published materials.

Sylvan Scimitar
This +3 scimitar, when used outdoors in a temperate climate, grants its wielder the use of the Cleave feat and deals +1d6 points of bonus damage.

Table: Intelligent Item Primary Abilities
d% Primary Ability
-- ---------------
01-04 Item can Intuit Direction (10 ranks)
05-08 Item can Sense Motive (10 ranks)
09-12 Wielder has free use of Combat Reflexes
13-16 Wielder has free use of Blind-Fight
17-20 Wielder has free use of Improved Initiative
21-24 Wielder has free use of Mobility
25-28 Wielder has free use of Sunder
29-32 Wielder has free use of Expertise...

And then there are the Eager, Grasping, Precise, and Proficient special abilities from the A&EG.

Battleaxe of the Bull, Breaking Blade, Disarming Flail, Spear of the Hunt, all from A&EG.

-Hyp.

 

[Storm Raven]

Sylvan Scimitar
This +3 scimitar, when used outdoors in a temperate climate, grants its wielder the use of the Cleave feat and deals +1d6 points of bonus damage.

You misunderstand. There is no instance in which a weapon printed by WotC gives the wielder Improved Critical and has the keen ability.

 

[Mike Sullivan]

Okay, one more question about this part. How did you come up with these numbers to represent the effects of a critical hit?

Okay, here's how it works:

What happens when you get a critical hit? You roll two d20's -- a specific one of them has to roll a hit, and the other one of them has to roll within the crit range.

As everyone knows, the odds of two independent events both coming out in a particular way are equal to the odds of one of those events times the odds of another of those events.

So, let's take a specific example: A longsword, say, without the keen trait or the improved critical feat -- in other words, a fully modified crit of 19-20/x2.

Now, let's say that the odds of hitting with his longsword is a variable we'll call m. What are the odds of critical-hitting with that longsword?

Well, it's the odds of rolling 19-20 on a d20, times the odds of hittimg (m). The odds of rolling 19-20 are 10%. So, the odds of critical hitting are .1m.

Now, we know that the odds of hitting at all are = m. We also know that a critical hit is, naturally, a hit as well. Thus, we can calculate the odds of rolling a normal hit as m - .1m, or .9m.

Thus, we can see that 10% of all hits are critical hits, and 90% are normal hits -- regardless of what AC we're attacking! (But see below). Naturally, both hits and critical hits are rarer with higher AC's, but the proportion between them is static.

Then, we note that a critical hit for this longsword does x2 damage -- so if the damage we do (before crits) is the variable d, then 90% of the time, we'll do d damage, and 10% of the time we'll do 2d damage. Thus, our expected damage for each hit is .9d + .1*2*d. That gives us an expected damage of 1.1d.

If you perform a similar calculation for every crit range, you'll find that this is true:

If your weapon has a 20/x2 crit, then to get your expected damage including crits, multiply your expected damage before crits by 1.05

If your weapon has a 19-20/x2 or a 20/x3 crit, then multiply by 1.1

If your weapon has an 18-20/x2 or a 20/x4 crit, multiply by 1.15

If your weapon has a 17-20/x2 or a 19-20/x3 crit, then multiply by 1.2

If your weapon has a 15-20/x2 or an 18-20/x3, or a 19-20/x4, then multiply by 1.3

If your weapon has a 12-20/x2 or an 18-20/x4, then multiply by 1.45


One problem with the abstraction above: it doesn't take into account the situation where your roll to hit is higher than your crit thresh -- so, if you have a 12-20/x2 critical range, but you require a 15+ to hit, you can't just naively multiply your damage by 1.45 like you ordinarily would. In that case, your crit threshold has been effectively reduced to 15-20, so you'd only multiply by 1.3.

This is also why I said you didn't take AC into account. You numbers assuming you always hit and always confirm.

No, they don't. They assume that the odds of hitting are the same between all the different weapons we're comparing, and they don't at all assume that you always confirm or threaten.

If it helps any, what I'm doing here is not controversial. This is about the 5th time I've posted these formulae on this board, and I've done similar stuff on RPG.net. Nobody has ever found any errors in the methodology, and a variety of other people have come to the same numbers from first principles -- and I'm hardly the first person to come up with these numbers. They are, to be totally blunt, correct.

They don't necessarily tell the whole story -- no figure that's based on averages ever will -- but they're correct as far as they go.

The maximum chance of a confirmed critical on a 19-20/x2 weapon, however, is 9.5%.

Yes, and the odds of hitting for that same scenario are 95%. So, as you can see, 10% of all of the hits will be critical hits (10% of 95% being 9.5%).


EDIT: On a second reading of your post, I think I see the point of confusion. What I am saying is that (in the scenario above), 10% of all HITS are critical hits. That is not the same as saying that 10% of all ATTACKS are critical hits -- since no attack has a 100% chance to hit, as you correctly note, that means that there is always less than a 10% chance that a given attack is a critical hit. However, when we're talking about differences between weapons, we don't want to talk about ATTACKS, we want to talk hits -- because adding AC's into the picture complicates the scenario, to not particular advantage.

On the other hand, if we were discussing power attack, or two-weapon-fighting, or whatever, where the chance to hit is different in two seperate scenarios, then we do have to discuss attacks -- and when that's appropriate, I take it into account. For an example of the rather more complicated analyses that this engenders, take a look at some of the several 3.5 Power Attack threads that have gone on recently -- in at least two of them, I worked out the math extensively.

 

[Anubis]

Another thing I would like to point out about the greatsword and falchion is that they are both 2d4 damage weapons, whereas the greatsword is 2d6. That has a great effect on things.

For accuracy, we should compare the mercurial greatsword (2d6/x4 crit) and the greatsword (2d6/19-20/x2 crit).

By the way, Mike Sullivan, I forgot to thank you for explaining the formula. Like I said, my math is rusty at this point, but now I understand your numbers perfectly.

Here I will do the above. We will assume that the attacker attacks once at +10.

Mercurial Greatsword v. Greatsword

Against AC 11

Average damage is 7(0.95) because 1 always misses. The critical factor for the mercurial greatsword (x4) works out to be 0.05*3*0.950+1, or 1.1425 overall. The critical factor for the greatsword (19-20/x2) works out to be (0.10*1*0.95)+1, or 1.095 overall. (To get the base factor, you put one less than the actual critical multiplier in there because weapons always do x1 damage to start with.) Anyway, moving on . . .

1.1425(7+X) > 1.095(7+X)

7.9975+1.1425X > 7.665+1.095X

0.0475X > -0.3325

X > -7

Now looking at this, when damage is the same, the x4 multiplier wins hands-down over a 19-20/x2 multiplier. Now I must ask, what on earth is the point of these numbers? How do we see exactly what happens with the Keen and Improved Critical situations? I think perhaps we need to look at different numbers.

We'll use a keen rapier (fighter has improved critical and ki critical) and a normal keen rapier. This time, the purpose is to find out either how much more damage it gets to unbalance things OR show how much is actually lost in the revisions. Our fighters will have Str 18 and Dex 26, both have Weapon Finesse, Weapon Focus, Superior Weapon Focus, and Weapon Specialization, both weapons are also +5, and both will attack at +35/+30/+25/+20.

Keen Rapier (w/Improved Critical and Ki Critical) v. Keen Rapier

Against AC 37

For Fighter #1, the hit factors for the four hits, counting the critical factors (10-20/x2 each) are (0.55*1*0.95)+0.95, (0.55*1*0.7)+0.7, (0.55*1*0.45)+0.45, and (0.55*1*0.2)+0.2, or 1.4725/1.085/0.6975/0.31. Average damage is 14.5. That gives us the following average damage per round:

14.5(1.4725) + 14.5(1.085) + 14.5(0.6975) + 14.5(0.31) = 51.6925 damage per round

For Fighter #2, the hit factors for the four hits, counting the critical factors (15-20/x2 each) are (0.3*1*0.95)+0.95, (0.3*1*0.7)+0.7, (0.3*1*0.45)+0.45, and (0.3*1*0.2)+0.2, or 1.235/0.91/0.585/0.26. Average damage is 14.5. In fact, I just realized this formula can be shortened to [(0.3*1)(0.95+0.7+0.45+0.2)]+(0.95+0.7+0.45+0.2). That gives us the following average damage per round:

14.5(1.235) + 14.5(0.91) + 14.5(0.585) + 14.5(0.26) = 14.5(1.235+0.91+0.585+0.26) = 14.5(2.99) = 43.355 damage per round

So there you have it.

51.6925 v. 43.355

With Improved Critical and Ki Critical, a keen rapier does 19.23% more damage per round. This obviously mostly disproves my theory about high threat ranges causing balance issues, but with one more step, it could do something else as well. Let's go with a greatsword now, something that is a power weapon. Same AC. Our fighter will have Str 26, Weapon Focus, Superior Weapon Focus, Weapon Specialization, a +5 keen greatsword, Improved Critical, and Ki Critical and will attack at +35/+30/+25/+20.

This time I will not go through the formula as it's up there twice already. Feel free to calculate if you wish, but I'll just give the basics here. The hit factors for the four hits, counting the critical factors (13-20/x2 each) are [(0.4*1)(0.95+0.7+0.45+0.2)]+(0.95+0.7+0.45+0.2), or 3.22 total. Average damage is 26. That gives us the following average damage per round:

26(3.22) = 83.72 damage per round

Let's do the same thing with a fighter that is not a weapon master and does not have a keen weapon, but does still have Improved Critical. This one will attack , of course, at +34/+29/+24/+19 due to not having Superior Weapon Focus.

The hit factors for the four hits, counting the critical factors (17-20/x2 each) are [(0.2*1)(0.90+0.65+0.4+0.15)]+(0.90+0.65+0.4+0.15), or 2.52 total. Average damage is still 26. That gives us the following average damage per round:

26(2.52) = 65.52

This is why so many people are supposedly complaining. They claim that without the stacking, "finesse fighters" can't begin to compete with heavy hitters like greatsword wielders. What they fail to tell you, and what is the biggest part of this, is that even at base stats for the heavy hitters WITH all the stacking for the finesse fighters, they STILL don't come close. At the same time, the damage factor of the stacking only goes up by 19%, which obviously isn't as big a difference as we all thought. This should show that finesse fighters aren't meant to dish out as much damage as heavy hitters, and that really is the way it should be. Consider that the finesse figher will have a higher AC, a higher hit chance, and other stuff as well, whereas the heavy hitter is good at simply dishing out damage. This remains the same regardless of whether criticals stack or not.

This also does show the Andy wasn't lying when he said it was taken out to make crits more special, which is STILL a good enough excuse. Why have crits if they happen all the time?

 

[Plane Sailing]

I was assuming a fully-upgraded scythe and greatsword for the 3.0 case

Doh! I thought it was something simple I was missing.

Thanks!

 

[Staffan]

Another thing I would like to point out about the greatsword and falchion is that they are both 2d4 damage weapons, whereas the greatsword is 2d6. That has a great effect on things.

I think you mean scythe and falchion. And that's the whole point - the scythe and falchion have a lower base damage but better crits, but are supposedly balanced against the greatsword (one could make the same comparison with the rapier/scimitar/heavy pick vs. longsword/battleaxe, or light pick vs. handaxe/shortsword). The question we're asking is, where is the balance point between the two?

For accuracy, we should compare the mercurial greatsword (2d6/x4 crit) and the greatsword (2d6/19-20/x2 crit).

No, we shouldn't. The mercurial greatsword is an exotic weapon whereas the regular greatsword is a martial weapon. Exotic weapons are supposed to be better than martial weapons.

That, and mercurial weapons are silly anyway .

And you really don't need to involve hit rolls when calculating average damage per hit for different weapons, because you're already assuming that you hit (it would be different if you were calculating the effects of power attack, since that changes hit probability). If you have h% chance to hit, and t% chance to roll in the threat range, t% of all successful hits will be crits. Putting it in real terms: let's say you're fighting with a longsword (threat 19-20 - that's 10% chance) and need 11+ to hit (50% chance). Using my calculations above one in every ten hits (note: not attacks) will be a crit. Let's check if that's true:

Out of 100 attacks, 10 will be natural 19 or 20, and thus qualify as a threat in addition to be a hit. These then need to roll 11+ on a confirmation roll to actually crit. Thus, we have 10 hits, 5 of which are crits.
Then, we have those attacks that rolled 11 to 18 in the first place. That's 40 hits.
Finally, there are all those attacks that rolled 1-10. That's 50.
So, we have 100 attacks, 50 hits and 50 misses. Of the 50 hits, 5 were crits. 5/50 = 10% - just like I said at the beginning.

 

[Anubis]

I think you mean scythe and falchion. And that's the whole point - the scythe and falchion have a lower base damage but better crits, but are supposedly balanced against the greatsword (one could make the same comparison with the rapier/scimitar/heavy pick vs. longsword/battleaxe, or light pick vs. handaxe/shortsword). The question we're asking is, where is the balance point between the two?

No, we shouldn't. The mercurial greatsword is an exotic weapon whereas the regular greatsword is a martial weapon. Exotic weapons are supposed to be better than martial weapons.

That, and mercurial weapons are silly anyway .

And you really don't need to involve hit rolls when calculating average damage per hit for different weapons, because you're already assuming that you hit (it would be different if you were calculating the effects of power attack, since that changes hit probability). If you have h% chance to hit, and t% chance to roll in the threat range, t% of all successful hits will be crits. Putting it in real terms: let's say you're fighting with a longsword (threat 19-20 - that's 10% chance) and need 11+ to hit (50% chance). Using my calculations above one in every ten hits (note: not attacks) will be a crit. Let's check if that's true:

Out of 100 attacks, 10 will be natural 19 or 20, and thus qualify as a threat in addition to be a hit. These then need to roll 11+ on a confirmation roll to actually crit. Thus, we have 10 hits, 5 of which are crits.
Then, we have those attacks that rolled 11 to 18 in the first place. That's 40 hits.
Finally, there are all those attacks that rolled 1-10. That's 50.
So, we have 100 attacks, 50 hits and 50 misses. Of the 50 hits, 5 were crits. 5/50 = 10% - just like I said at the beginning.

The problem is that you're leaving out the most important factor, that of hitting in the first place. That has a great affect on average damage in any situation, and it most certainly affects how well two weapons compare to each other. It's like comparing keen to flaming (Sword & Fist shows three examples). If you only calculate average damage per hit, you're assuming everything hits. This simply isn't the case, and it will have a profound affect on statistics.

 

[Technik4]

In point of fact, in terms of increased damage dealing capability for a rapier, you are better off taking Weapon Specialization and an energy enhancement than Improved Critical and keen. If you actually looked at the math as opposed to ranting, you'd have figured this out.

I'm not rushing and I'm not ranting. You are totally correct that a 4th level Fighter would be better off taking Weapon Spec than Imp Crit. But wait, I was talking about characters with the potential to take Imp Crit (BAB +8). So would an 8th level fighter who managed to not have Weapon Spec be better of with it over Imp Crit? Maybe. But are fighters the only ones who use rapiers? What about a rogue with BAB 8+? Isnt it really the best feat he can take? I'm more than slightly baffled you are arguing this point, but if you feel you have a leg to stand on then go for it.

As far as Energy Bonus +1d6 vs Keen? I've never seen a player who has the option of what enhancement they get not chase Keen. Perhaps statistically it is better in most situations to have the +1d6 Energy, but I can think of a lot of situations where that damage is more useless than doubling your crit range (and vice versa).

You still haven't mentioned what else you can do with a rapier besides going for crits. If I want Weapon Spec and Energy Damage then why didn't I pick longsword? Because I wanted finesse? But finesse hoses my off-hand...and it still doesnt really count as another style since if Im a dex-based weapon finesse rapier wielder than *All* Ive got going for me is crits.

Technik "neither ranting, nor rushing" 4