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

[Driddle]

What factors should be taken into consideration when choosing between two feats (or equivalent abilities) granting either +2 damage or improved critical? What basic mathematical formula do you follow in picking one over the other?

 

[Viktyr Gehrig]

The higher your critical threat range is, the better +2 damage is. The higher your critical multiplier is, the better an increased threat range is.

I saw someone run the numbers once, but they showed that 19-20/x3 is considerably better than either 18-20/x2 or 20/x4.

 

[Crothian]

My complex formula goes something like this: does the character have 4 levels of fighter?

 

[orsal]

The higher your critical threat range is, the better +2 damage is. The higher your critical multiplier is, the better an increased threat range is.

I saw someone run the numbers once, but they showed that 19-20/x3 is considerably better than either 18-20/x2 or 20/x4.

19-20/x3 is the equivalent of 2 extra damage roll, 2 times out of 20, meaning 4 extra damage rolls every 20 hits. 18-20/x2 and 20/x4 each give 3 extra damage rolls every 20 hits.

Suppose your average damage is X, you have Y numbers in your threat range (e.g. Y=2 if your threat range is 19-20) and your critical multiplier is Z. Then you will get on average Y(Z-1) extra damage rolls from criticals every 20 hits, for a total of 20+Y(Z-1) damage rolls, doing on average X[20+Y(Z-1)] HP damage. You can either replace X with X+2 or replace Y with 2Y.

Weapon Specialization: (X+2)[20+Y(Z-1)]=20X+XY(Z-1)+40+2Y(Z-1)
Improved Critical: X[20+2Y(Z-1)]=20X+2XY(Z-1)
Now solve this inequality:
20X+XY(Z-1)+40+2Y(Z-1) < 20X+2XY(Z-1)
40+2Y(Z-1) < XY(Z-1)
40 < (X-2)Y(Z-1)

So calculate (X-2)Y(Z-1). If the number is more than 40, go with Improved Critical; if less than 40, Weapon Specialization.

This does ignore a few details. If you are fighting a high-AC opponent, so that you will only hit on a 20, the higher threat range doesn't do a darn bit of good. If you're fighting a bunch of 1-HD monsters, the extra +2 is likely to be enough to kill them, while the critical would be wasted, so you'd rather get the extra +2 with every hit rather than slgihtly more frequent criticals. So on the whole, Weapon Specialization comes out slightly better than this algebra shows. 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.

 

[JoeGKushner]

Depends on the character. If it's a paladin, ranger or other fighter type that can't select Weapon Specialization, then it's a wait for the old 9th level feat slot and Improved Crit. Otherwise it's W.S. first then improved crit.

 

[DragonLancer]

If you are playing a fighter then I'd suggest working towards the improved critical over a +2 damage. But thats me. The more crits you can pull off the better IMO.

 

[Driddle]

(Assuming) average damage is X...
you have Y numbers in your threat range ...
and your critical multiplier is Z. ...

Calculate (X-2)Y(Z-1). If the number is more than 40, go with Improved Critical; if less than 40, Weapon Specialization....

I looked over the weapon table to see when you might get (X-2)Y(Z-1)>40. Basically... never.

Well. There ya go.
I believe the increased crit option is an illusion that appeals to people who enjoy the thrill of rolling more dice, under the false belief they're doling out bunches of damage.

 

[Mirage_Patrick]

A lot depends on the game. One with lots of undead and constructs makes crits worthless, hence the + damage is much better

 

[Particle_Man]

In 3.0, Improved Crit would have been worth it if you had a vorpal weapon. Now, WS is almost always better, unless you have a magical weapon or ability that lets you do something simply back-breakingly AWESOME on a crit.

 

[iwatt]

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