D&D 4E Charop DPR and nova: accuracy vs. damage

[crunchy4e]

I am interested in an optimizer's view of accuracy vs. damage in 4e. I ask because the classic rogue guides rate Backstabber as a gold feat in heroic tier, but some of the sample builds they give don't even include Nimble Blade anywhere in the 30-level progression. The average damage increase for Backstabber is +2/3/5 by tier, while Nimble Blade is a flat +1, so certainly by Epic or even Paragon I can see why Backstabber takes precedent, but in Heroic, is there a reasonable argument to be made for Nimble Blade over Backstabber?

A nearly-identical question is the relative merit of Sly Flourish vs. Piercing Strike. If we restrict to melee hits and ignore style feats, Sly Flourish easily does +3-5 damage over Piercing Strike, but if targeting reflex gives a 2-3 point accuracy buff, is that worth it?

Obviously there is a limit here -- 100% accuracy for tiny damage (Magic Missile) isn't as good as 60-70% accuracy for much larger, better-scaling damage. There is also a gameplay aspect, since a 5% chance of doing 100 damage and a 100% chance of doing 5 damage might have the same expected value, but the variance brings excitement (and potentially frustration). Mathematically, I'm more interested in both pure expected DPR and nova-round value for a single-attacking class, since I assume Avengers and Rangers could take a hit on accuracy before a Rogue would want to.

Furthermore, for example, if a player hits only on a roll of 16 or higher, then a +1 accuracy bonus increases hit-rate by 20%, while if a player hits on a roll of, say, 4 or higher, a +1 accuracy bonus increases hit-rate by only 6%. This might mean that accuracy matters marginally more for, say, Barbarians than it does for a traditionally more accurate class like Avengers, Rangers, or Rogues -- independently of a barbarian's often-massive damage bursts.

In short, the question could be boiled down to the sliding and conditional relative worth of 1 point of accuracy and 1 point of damage.

Secondarily I am interested in how 1 point of defense would be included in the calculus. E.g., Draji Palatial Practice (DSCS) gives the target an full round of automatic -2 penalty to attack rolls. How does an automatic -2 accuracy debuff, regardless of hit or miss, compare to simply targeting Reflex (via Piercing Strike) in the first place at roughly +2 accuracy over AC?

 

[crunchy4e]

Here are some thoughts, but I'm interested if anyone disagrees or can find an issue with the line of thinking:

As a preliminary note, an n-sided die with an even number of sides averages (n+1)/2. So d6 averages 3.5, d12 averages 6.5, and so on.

Step 1: ignoring crits

Expected attack damage is E(damage | hit)P(hit). E.g., a 1d4 + 4 attack averages 6.5 on a hit, so the expected DPR is 6.5*(hit chance). Abbreviate this as dh -- d for expected damage, h for hit probability (0 to 1). Ignore crits for now. (I'll include them later.)

Increasing accuracy by 1 increases hit probability by 1/20, or 0.05. This changes dh into d(h+0.05) = dh + 0.05d, increasing DPR by d/20. Increasing expected damage by 1 (e.g. increasing a die size) changes dh into (d+1)h = dh + h, increasing DPR by h.

Therefore, if h > d/20, then given the choice, it would be better to increase damage by 1. If d/20 > h, it would be better to increase accuracy.

Example: a level 1 rogue with 18 dexterity does 1d4 + 4 on several at-will powers. With advantage, a base rogue does an additional 2d6 damage (or 7 on average). Thus, the expected at-will damage with CA is 1d4 + 11, or 13.5. Backstabber increases this expected damage to 15.5, while Nimble Blade increases the hit rate by 5%.

Thus: (d+2)h = dh + 2h, and d(h+0.05) = dh + d/20 = dh + 13.5/20 = dh + 0.675.

So as a possible answer to my own question, if crits are ignored, Backstabber beats Nimble blade as long as 0.675 > 2h. I certainly expect most rogues (or even any non-optimal PC) to have a hit rate better than 0.675/2 = 33.75%.

Step 2: including crits

A standard critical strike at level 1, taking maximal damage, would be 1d4 + dex mod + 2d6 = 4 + 4 + 12 = 20. With Backstabber, that increases to 24.

A standard swing crits 5% of the time, and the rest of the hits do standard damage. I.e. damage is no longer dh, but 0.05*(crit damage) + (h-0.05)d.

With Nimble Blade and no Backstabber, this becomes 0.05*20 + ((h + 0.05) - 0.05)*13.5 = 1 + 13.5*h. With Backstabber and no Nimble Blade, this becomes 0.05*24 + (h - 0.05)*15.5 = 1.2 + 15.5h - 0.775 = 0.45 + 15.5h.

If 0.45 + 15.5h > 1 + 13.5h, then Backstabber beats Nimble Blade on average, including crits. This is equivalent to saying 2h > 0.55. I certainly hope a rogue with advantage, or any PC in 4e, hits more often than 0.55/2 = 27.5% of the time.

The reason the cutoff is lower than before is that accuracy does nothing to improve critical strike damage, but increasing damage dice does.

Conclusion

This should answer some basic questions and set a template for Sly Flourish vs. Piercing Strike type questions, although it's hard to include debuffing in the mix. Draji Palatial gives an automatic -2 attack debuff regardless of whether it hits, but it's hard to compute the relative merit of the party having +2 defenses against single opponent vs. the expected damage tradeoff of, say, a more accurate Piercing Strike.

Thoughts? Corrections? Disagreements?

 

[Joshua Randall]

Paging @MwaO

 

[Neonchameleon]

One of the big issues is that the Rogue in 4e is already incredibly accurate especially if you take into account they should always have Combat Advantage. And if you hit you hit and that's the end of things while most targets survive the rogue's attack.

To take an example for an average level 2 monster in 4e vs a 4e rogue the rogue should be hitting on 3s.

• AC = 14 +level = 16 for an average monster (14 for artillery and brutes, 18 for soldiers)
• A level 2 rogue with a dagger and inherent bonuses should be at +13 to hit.
  • Dex 18 (+4) (more will be dex 20 than dex 16), dagger +3, Rogue Weapon talent +1, Level 2 +1, Light Blade Expertise +1, +1 dagger/Inherent bonus +1, Combat Advantage +2

When you're hitting on 3s then you've an 18/20 chance of hitting. A +1 to hit over that increases your damage per round by 1/18. Which means that if a rogue is doing 18 points of damage per attack with sneak attack that +1 to hit is worth an average of one point of damage.

But even though you're more likely to be hitting on 5s than 3s in reality as most monsters you'll fight will be a couple of levels above you (so +1 to hit is only worth about 1/16 of your damage rather than 1/18) rogues have "pocket accuracy". Because so much of their damage is made up of their sneak attack rogues tend to save things like their action surge to give them a second chance of landing that sneak attack if they miss in an early round of combat so, if the rogues only use it in the first three rounds of combat and can only use it one combat in two we're almost treating it like an advantage situation here. (Minor action attacks like Low Slash also give a second chance of landing that sneak attack).

As for sly flourish vs piercing strike, this is where monster roles and other powers come in. Piercing Strike isn't just worth an average of +2 to hit (not that that would be bad). Piercing strike is an extra tool in your toolbox, and one that is most important against the enemy type you have the hardest time hitting. Against soldiers piercing strike is worth an average of + 4 to hit because soldiers don't have their reflex defence boosted the way their AC is (and frequently have a low one because the heavy armour weighs them down). Not every foe is a soldier and it's frequently worth nothing against artillery (which you hit on 2s anyway) or skirmishers (many of whom are agile but going to hate your other at will giving you mobility).

If I could only have one at will and had to always just spam that I'd probably rate a melee only sly flourish as being at least competitive with piercing strike - but sly flourish doesn't do something a standard [2W] encounter power doesn't. But as a tool in your toolkit sly flourish has the same problem as e.g. deft strike or 95% of your encounter powers against high AC foes. Meanwhile you can look at your foes and guess times when Piercing Strike will be an effective +4 to hit - and at that point it will do far more damage than your encounter powers.

 

[MwaO]

In general, the more straightforward way of looking at it is:
How much damage do you do on average with a punch? Pretend for a second that it is 20 damage.
Say you hit on a 5. So 15 regular punches = 300 damage.
Add +1 = hit on a 4. So 16 regular punches = 320 damage.
Can you do more than 20 damage with those 15 hits via some other way? Say you have a choice between 2d8 damage or 2d6 damage. That would be +2 damage on average with those 15 hits for +30 damage. So most of the time, Backstabber is going to be better than Nimble Blade in Heroic.
Now, let's make 40 damage. 600 damage vs 640 damage. But Backstabber is just 3d8 vs 3d6 or +45 damage. And now that we're attacking multiple times on our turn, that's a much more narrow difference a significant amount of the time.

Of course, you probably can have both on a pure Rogue by that time, but…I am not typically a fan of Backstabber as a feat choice in any case, because it is in essence, a +2/3/5 damage extra usable once per turn feat.