"Ouch?!" - improved crit or +2 damage?

[MonsterMash]

I'd take the +2 damage in most circumstances.

 

[technophile]

One thing that is assumed here is that you confirm every threat. That is not the case, so it pushes the balance further towards +2 damage...

Orsal, I couldn't follow your math, so I'll present mine. It also ignoers what happens at extremes (1 only misses or 20 only hits).

AD = Average Damage with weapon (dX+str+magic+misc)
BD = Damage you're adding (+2 in this case)
TR = threat range (i.e. 2 for 19-20/ 3 for 18-20, 1 for 20)
ATR= Added Threat range (typcially a doubling from the previous)
CM = Critical Multiplier (x2,x3,x4)
NAD = NEW Average Damage

....

NAD = (AD+BD)*(1 + (TR+ATR)/20*(CM-1))

There is two conditions you want to compare:

I) BD=+2, ATR=0

and
I) ATR=ATR, BD=0
ATR=Increasing the trheat range. I'll leave this value as avraiable since I want to see the effects when I include keen/Imp Crit stacking. If not, you can simply reduce ATR=TR.
this leads to an equalityu condition that sets the break even point (the AD you must do to make the choise equal)

AD=20*BD*(1+TR/20*(CM-1))/(ATR*(CM-1))

or in this case in which BD=+2

AD=20*BD*(1+TR/20*(CM-1))/(ATR*(CM-1))

So Let's anlayze the following cases

ICR= Initialt Critical range
FCR= Final Critical range

  ICR	  FCR	 AD
20/x2	19/x2	  42.0
20/x3	19/x3	  22.0
20/x4	19/x4	  15.3
19-20/x2	17-20/x2	  22.0
18-20/x2	15-20/x2	  15.3
19-20/x3	18-20/x3	  24.0
19-20/x4	18-20/x4	  17.3
17-20/x2	15-20/x2	  24.0
15-20/x2	12-20/x2	  17.3

* are situation in which keen and Imp. Crit are allowed to stack.

So that the Average damage that you do is of course the deciding factor in the choice.

Of course the above analysis can be extrapolated to include a different Bonus damage, which will of course determine a differnet threshold. If you want to do an analysis for +1 BD v/s increased threat range just correct the treshold values of AD by 1/2.

This method also allows you to compare the threshold values in which adding an ergy damge (BD=3.5) is equal to increasing the threat range:

 ICR	 FCR	 AD
20/x2	19/x2	  73.5
20/x3	19/x3	  38.5
20/x4	19/x4	  26.8
19-20/x2	17-20/x2	  38.5
18-20/x2	15-20/x2	  26.8
19-20/x3	18-20/x3	  42.0*
19-20/x4	18-20/x4	  30.3*
17-20/x2	15-20/x2	  42.0*
15-20/x2	12-20/x2	  30.3*

* are situation in which keen and Imp. Crit are allowed to stack.

Which emphasizes that (barring energy resistance and mosntrous damage dealing ability) it's better to have a flaming axe than a keen axe in 3.5

 

[orsal]

One thing that is assumed here is that you confirm every threat. That is not the case, so it pushes the balance further towards +2 damage...

No, that's not assumed. If your threat range is 19-20, it means that 10% of attacks are threats... but also that 10% of hits are criticals. Since all these calculations are based on what happens after it's been determined that you hit, this is the correct probability to use.

Look at it this way: let's change the procedure a little. Suppose you're using a weapon with a 19-20 threat range, and against your current opponent you hit on 15 or higher. Instead of taking 19-20 as a threat and 15-20 on the subsequent roll as confirming, we'll take 15-20 as a threat and 19-20 to confirm it. The probability of a critical is exactly the same -- it's just as likely to roll 19+ followed by 15+ as vice versa. But using this procedure all hits are considered threats, and exactly 10% of them get confirmed.

(There is one slight complication: if the number you need to hit is higher than the number you need to threaten a critical, use the higher number for both rolls.)

 

[iwatt]

One thing that is assumed here is that you confirm every threat. That is not the case, so it pushes the balance further towards +2 damage...

If you include the chance to confirm a threat (PtC):

{ (20-TR)/20*AD + TR/20*(PtC*CM*AD+(1-PtC)*AD) }

a little math and

AD*{ (20-TR)/20 + TR/20*(PtC*CM+(1-PtC)) }

and then

AD*{ 1-TR/20 + TR/20 + TR/20*PtC(CM-1) }

=

AD*{ 1 + TR/20*PtC(CM-1) }


which of course when PtC =1 (always hits) collapses to my initial formula. Of course, this more complete formula still has one caveat: It's based on the premise that every threat is actually a hit, which probably holds tru most of the time for weapons in the 15-20 threat range, but breaks down when you enter th 12-20 range.

My pervious threshold values are related to potential damage, if that is what interests you. If you want to make the choice to maximize damage output, you'd have to repeat the anlysis for different chances of confirming a threat.

so if we repaet the analyssi I did berfoer, but include the following PtC value

(AD+BD)*{ 1 + (TR+ATR)/20*PtC(CM-1) }

CASE I : (BD=BD, ATR=0)

(AD+BD)*{ 1 + TR/20*PtC(CM-1) }


CASE II : (BD=0, ATR=ATR)

AD*{ 1 + (TR+ATR)/20*PtC(CM-1) }

we then proceed to equate both cases, and simplify some terms:

AD*{ 1 + TR/20*PtC(CM-1) } + BD*{ 1 + TR/20*PtC(CM-1) }

=

AD*{ 1 + TR/20*PtC(CM-1) } + AD*ATR/20*PtC*(CM-1)


we obtain the new corrected formula

AD=20*BD*(1+TR/20*PtC*(CM-1))/(ATR*PtC*(CM-1))

 

[Patryn of Elvenshae]

So, here's a really simply Excel spreadsheet that can be used to compare options.

I've currently got it set up to compare a Two-handed Sword, a THS with Imp. Crit., and a THS with Weapon Spec.

Enjoy.

 

[Iron Sheep]

At least, if (X-2)Y(Z-1)=40 exactly, go with Weapon Specialization.

EDIT: After posting this I looked over the weapon table to see when you might get (X-2)Y(Z-1)>40. Basically... never. If you use a greatsword with 20 Strength, then X=12, Y=2, Z=2, so (X-2)Y(Z-1)=20. Not even close. If you qualify for Weapon Specialization, it's a lot better.

Actually, a scythe or falchion (each average damage 5, and Y(Z-1)=3) with sufficient strength, power attacking, and/or magical enhancement could make it interesting. You require more than a 15 average damage (so at least +11 to damage) to start making this worthwhile, which is going to be hard to achieve. But it is conceivable that Improved Critical could be better than Greater Weapon Specialization for some high-level fighters, and it would likely start to look quite appealing to large creatures with appropriately sized versions of these weapons.

EDIT: Of course, the true comparison should be the choice between a greatsword with Weapon Specialization against a scythe or falchion with Improved Crit (ie. +4 average damage vs. 15-20/x2 or 19-20/x4 critical). I think that the greatsword will be the clear winner based on orsal's analysis.

Corran

 

[comrade raoul]

Here's a simplified answer; assume for now that you're really just interested in situations where you're up agains things you can actually crit.

If you use a weapon with a 19-20 or x3 crit, take Improved Critical if you do more than 20 points of damage on average. Otherwise, take Weapon Specialization.

If you use a weapon with an 18-20 or x4 crit, take Improved Critical if you do more than 14 points of damage on average. Otherwise, take Weapon Specialization.

If you also want to take into account situations where you're up against crit-immune opponents, it gets a lot more complicated. Here, I'd only take Improved Critical if offered a really significant benefit over Weapon Specialization -- if you did an average of 30 (for a 19-20 or x3 weapon) or 20 (for an 18-20 or x4 weapon) points of damage.

In conclusion, unless you use a falchion or scythe and are high-level, take Weapon Specialization.

 

[FreeTheSlaves]

Here is some help if you really want to get gritty with the maths (Courtesy of 'Nail')

Layout is the following: A = P{D[1+Pc(Mc-1)] + Db}
For bonus critical dice I insert: +Bd[Pc(Mc-1)] after the Db & before the closing }
If the chance to hit is different to the chance to confirm the critical use: A = P{D[1+(Pt/P)Pc(M-1)]+Db}

where
A = average damage per attack
P = Probability to hit, as a fraction
D = average weapon damage plus Str, Magic, etc
Pc = Probability to Threaten, as a fraction
Mc= Critical Multiplier
Db = Bonus Damage dice that are not multiplied by a confirmed critical
Bd = Bonus die average damage that only occurs on confirmed criticals
E.g. 8th level fighter (ab+14/+9 vs AC20).

Vanilla Longsword
7.0125 = 0.75{8.5[1+0.1(2-1)]}
4.675 = 0.50{8.5[1+0.1(2-1)]}
11.6875

****

Just run the 2 sets of numbers for the same base character to see which is better; a rule of thumb is that the average damage needs to be 20+ with a martial weapon for improved critical start being better than the flat +2.

 

[clark411]

Narrrgh my brain!

 

[Endur]

Weapon Specialization is better. Except for extremely powerful characters.

All this math boils down to a simple calculation.

If your character already does tons of damage that can be improved by a crit multiplier(better than 20 or better than 40 per attack depending on which calculation you use), then improved critical hit is better.

If your character does not do tons of damage or faces lots of crit immune enemies, weapon specialization is better.

If you think about it, this makes sense. If your character already has lots of damage bonuses, an increased chance of critting increases the damage bonuses you already have. Otherwise, you need to increase your damage output.