What's the optimal way to reroll dice with the Piercer feat and extra attack?

[Merudo]

Once per turn, the Pierce feat lets you re-roll one piercing damage dice (you must use the new roll).

If you have only one attack, the optimal decision is the following:


Compute the expected damage increase of rerolling each dice, and reroll the die with the greater expected improvement (if all improvement are negative, don't reroll any).

For example if a level 1 rogue attacks with a Light Crossbow (d8) with Sneak Attack (1d6), and rolls a 3 on the d8 and a 1 on the d6, the expected improvement of each dice is

d8: 4.5 - 3 = 1.5
d6: 3.5 - 1 = 2.5

and therefore the d6 should be rerolled at the expected damage increase of doing so is 2.5, which is greater than 1.5.

However, with extra attack, it's not clear how the Piercer feat should be used. If the first attack lands and the damage rolls are low-ish, should a damage dice be rerolled, or should the ability be conserved in case the second attack lands and one of the damage dice is worse?

[Man_Over_Game]

Once per turn, the Pierce feat lets you re-roll one piercing damage dice (you must use the new roll).

If you have only one attack, the optimal decision is the following:



For example if a level 1 rogue attacks with a Light Crossbow (d8) with Sneak Attack (1d6), and rolls a 3 on the d8 and a 1 on the d6, the expected improvement of each dice is

d8: 4.5 - 3 = 1.5
d6: 3.5 - 1 = 2.5

and therefore the d6 should be rerolled at the expected damage increase of doing so is 2.5, which is greater than 1.5.

However, with extra attack, it's not clear how the Piercer feat should be used. If the first attack lands and the damage rolls are low-ish, should a damage dice be rerolled, or should the ability be conserved in case the second attack lands and one of the damage dice is worse?

Nope, just use it as soon as it may be relevant. You have a 50/50 chance of rolling higher than average, so use it as soon as you know you'll earn value out of it. A guaranteed +1 is a lot better than a +2 with a 50/50 chance, because you know how to plan around a consistent +1.

Now, that's an example that represents a doubled amount of value. If are able to get an average of +2 damage from using the Piercer feat from an earlier attack roll, you'd need a +4 difference from a future roll to be worth holding onto (as you have a 50% chance of having the roll not be worth a reroll using Piercer at all). The chances of getting +4 damage from Piercer is really, really small, so just take the safe bet. Here, a bird in the hand is literally worth two in the bush.

Put another way, RNG is never in the player's favor (unless the game is tilted to do so, like how the Wild Magic table is made of mostly-positive effects).

[Hael]

The short answer is that this gets complicated, b/c the answer depends on quite a few external variables. Do you have advantage or have it used up, what is your to hit and what is the AC you are going up against and is this known to you (or is there like an ensemble average AC for your lvl), Elven Accuracy, GWF style?, how many attacks do you have and what die are you looking at.

I don't think there is a closed form solution to this, so the real answer is to simulate it and use that to inform your reroll choice.

[J-H]

I think it'd waste less table time to houserule Piercer to just "You do +2 damage with piercing weapons."

[Dark.Revenant]

It depends on the die size and the chance to hit, but generally:

If this is your last attack for the turn: Just reroll the lowest die that's below the average, if any.
If you have one or two attacks left: Reroll the first time you get a 1 or a 2 on a damage die.
If you have three or more attacks left: Reroll only if you get a 1 on a damage die.

[Man_Over_Game]

I think it'd waste less table time to houserule Piercer to just "You do +2 damage with piercing weapons."

Piercer's a half-feat. You'd want to either change it to +2 once a turn, or +1 per attack. Otherwise, I agree.

[Chronos]

It's not how many attacks you have left; it's how many dice. A high-level rogue, for instance, is rolling so many dice on a single attack that it's very likely that at least one will be a 1. A jouster, by contrast, might be rolling only a single die per attack, but it'll be a d12, so very unlikely to be a 1 (but very beneficial to use the feat, if it is).

Also keep in mind that if you save it for another hit, it's possible that you won't even get that other hit at all. Your chance of hitting is one of the many variables that goes into the calculation.

[Naanomi]

Also depends on what you are fighting. If you suspect 2+ability mod may kill it, you probably don't reroll that 2

[OldTrees1]

Background:
If your damage dice are above average, then it is not worth using Piercer.
You will reroll whichever die is the furthest below its average. (The opening post called this expected improvement)
If you have only 1 attack, then it is worth assigning Piercer to that attack.
If you have multiple attacks but only hit with your last attack, then you only have 1 attack.
If you have 2 attacks and hit with the first attack, then the math begins.

The Math:
1) You have 2 attacks and hit with the first attack. You might or might not hit with the second attack. Let's assume a probability of P that you do hit.
2) What is the biggest difference in expected improvement you might see on the second attack if it hits? Let's call that D for now. If PD < your current expected improvement, then use Piercer on the first attack.

For example if you are a 20th level Rogue Sneak Attacking with a Shortsword and your lowest roll was a 2, then your current expected improvement is 1.5 and your theoretical maximum improvement is 2.5. Is 1.5 < 2.5 P? Aka do you have greater than (not equal to) a 60% chance of hitting?

3) What is the expected chance that you will roll something with a greater expected improvement on the next roll? This is a bit easy to calculate by assuming the inverse. Say we calculating the chance a 20th level Rogue with a Rapier rolls a 1 on at least one of their 10d6 or a 1-2 on the 1d8. We start by calculating the chance they don't by multiplying the chance they don't for each die. (5/6)^10 * (6/8)^1 ~12%. The chance they do something is 100% minus the chance they don't. So their chance is ~88.% Let's call that X=0.88

Now let's check if PDX < your current expected improvement. In this case we were checking for D=2.5, X=0.88, and current expected improvement of 1.5. Is 1.5 < P(2.5)(0.88)? It is only when your chance to hit is greater than 68%.

Of course we can make this even more fine tuned by separating out the probability of rolling a 1 vs rolling a 2 on that next attack.
X2.5=1 - ((5/6)^10 * (7/8)^1) ~= 0.86
X3.5=1 - ((7/8)^1) ~= 0.13
Is 1.5 < P(2.5)(X2.5) + P(3.5)(X3.5)?
Is 1.5 < P(2.5)(0.86) + P(3.5)(0.13)?
Is 1.5 < 2.605P?
Is 58% < Your chance to hit with the next attack?

That was the math for a Rogue with 2 attacks, 10d6 Sneak Attack, and a 1d8 weapon. So an extreme example.