# 5EMonoclass DPR Comparison: Eldritch Knight Archer vs Melee Arcane Trickster

#### Esker

##### Hero
I was curious how often you would expect to waste superiority dice using your method, so I did a simulation, where I assumed exactly 20 attack rolls between short rests (itself an optimistic assumption, since if rests happen unevenly in the 20 round day, you'll either waste more dice or be forced to use them more aggressively than when the success rate is 80%). Taking a four number range (it doesn't really matter which four, since the probabilities are uniform, and so this generalizes to any enemy AC as long as you're not too close to the floor or ceiling), and counting how many of the 20 rolls fall in that range, obviously the average result is 1/5, or 4, which is ideal for this strategy. But 41% of the time it's less than 4, in which case you wind up wasting one of your dice. And 20% of the time you waste 2. 6% of the time you waste 3.

So, in the spirit of wildly excessive precision and intellectual masturbation, let's actually calculate the expected number of misses you turn into hits per 20 round short rest using this strategy. Taking the "use a die when the result is within 4" strategy, you can expect each rolled die to have an 81.25% chance of turning a miss into a hit. 59% of the time you use all four and wind up with 3.25 misses converted. 21% of the time you only use 3 and wind up with 2.4375 conversions. 14% of the time, you use two, and net 1.625 conversions. And 5% of the time you use only one, getting 0.8125 conversions.

On average, then, you get about 2.7 extra hits per 20 attacks assuming evenly spaced short rests. If the rests are unevenly spaced, the gain is capped at 3.25 during the long stretches, but will go down during the short stretches. So it may be a little lower than that overall. You may be able to find a less stringent threshold for "near miss" that gives you a better result by shifting the average number of near misses above 4 so that the chances of wasting dice goes down*, but that was my point about rolls not being uniformly distributed. But as I speculated, it doesn't make a huge difference (maybe 1.5 DPR or something; pretty close to the approximation error from assuming the rogue always has advantage).

EDIT: I checked; it turns out you do minisculely better by using your dice when you're within 5 rather than within 4, by like 2 total damage throughout the day. And using then when you're within 6 is only minisculely worse than within 4 (by about the same margin).

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#### FrogReaver

##### As long as i get to be the frog
I was curious how often you would expect to waste superiority dice using your method, so I did a simulation, where I assumed exactly 20 attack rolls between short rests (itself an optimistic assumption, since if rests happen unevenly in the 20 round day, you'll either waste more dice or be forced to use them more aggressively than when the success rate is 80%).

Sure that's exactly 6 rounds of combat between rests for our fighter. A little low for an average estimate but close enough.

Taking a four number range (it doesn't really matter which four, since the probabilities are uniform, and so this generalizes to any enemy AC as long as you're not too close to the floor or ceiling), and counting how many of the 20 rolls fall in that range, obviously the average result is 1/5, or 4, which is ideal for this strategy. But 41% of the time it's less than 4, in which case you wind up wasting one of your dice. And 20% of the time you waste 2. 6% of the time you waste 3.

It's a lot more complicated if you really want to divide the day up this way. For example. There exist ways of not missing by 4 where all your attacks hit. In the case where all my attacks hit I'm already vastly out damaging your single attack rogue even though precision itself is adding nothing here. In other words, you want only those cases where the number of hits is about average or below and where you don't miss by 4 or more. Many other cases and the -5/+10 is sufficiently outdamaging the rogue build without any need for the superiority dice in the first place.

#### Esker

##### Hero
Sure that's exactly 6 rounds of combat between rests for our fighter. A little low for an average estimate but close enough.

If it is low, it means a smaller share of attack rolls have the opportunity to benefit from precision attack than in your calculations, so the average damage will be reduced.

There exist ways of not missing by 4 where all your attacks hit.

Yes, this is a good point: when there are fewer than average rolls in the precision attack range, those have to go somewhere, either to automatic hits or to misses where we are not using our dice. So we should consider whether the cases where we have fewer-than-expected triggers for precision attack look different than the cases where we have the expected number or more.

In the ideal case, in a batch of 20 rolls, we have 20%, or 4, in the precision attack range, each of which has an 81.25% to become a hit, and 16 outside of that range. In the case of AC 16 (where the chance to hit if you can't use precision dice is 40%), the 16 that fall outside the precision attack range have a 50% chance of being hits. So, our overall chance to hit is 4/20 * 0.8125 + 16/20 * 0.50 = 0.5625. So, precision attack is giving us an extra 16.25% to our to-hit chance, or an extra 3.25 hits out of 20, as we saw before.

But what about the times when we only get 3 rolls in the mid-range? The missing one has exactly the same distribution of possible values as the 16 we assumed didn't trigger precision attack before, so we get a to-hit chance of 3/20 * 0.8125 + 17/20 * 0.50 = 54.6875%. With 2 in the mid-range, it's 53.125%, and with 1 it's 51.5625. Of course, if there are more rolls in the "within 4" range than we have precision dice, those are automatic misses. Overall, the chance to hit over N rolls between short rests is given by the formula

[sum for n = 0 to n = N] P(n rolls within 4) * [min(n, 4) * 0.8125 + (N - n) * 0.50 + max(n-4, 0) * 0.00]

where n is the number of rolls that meet our criterion for triggering precision attack.

P(n rolls within 4) is given by the binomial distribution over N rolls with a "success" probability of 0.20.

With some help from R, this is easy to compute:

AC <- 16 # enemy AC
atk<- 3 # attack bonus after -5 penalty
m <- AC - atk # smallest natural roll to hit w/o precision
N <- 20 # Number of attack rolls per short rest
t <- 4 # Maximum distance when we will use a die
n <- 0:N # Number of attack rolls within t of a hit

# Chance of n rolls of N within t
p_n <- dbinom(n, size = N, prob = t / 20)
# Chance to hit when using precision attack
p_prec <- sum(1/t * (1/8) * (8-t+1):8)
# Chance to hit when not triggering precision attack
p_other <- (20-m+1) / (20 - t)

to_hit <- sum(p_n * (pmin(n,4)/N * p_prec + (N-n)/N * p_other))
to_hit
[1] 0.5341341

Actually, this didn't take action surge into account: we're assuming 23 attack rolls, not 20. So, changing N to 23, we get

to_hit

[1] 0.524508

Slightly lower to-hit, but of course more attacks. With the 16.5 damage per attack, that's

dph <- 16.5
daily_attacks <- 3 * N
total_damage <- daily_attacks * to_hit * dph
total_damage
[1] 597.1524
DPR <- total_damage / 20
[1] 29.85762

So, like I said, about 1.5 below what we'd get if we had exactly the "right" number of uses of precision attack every rest. For other ACs, here's what the table looks like with 7 rounds per short rest.

total_per_sr DPR
11 305.4 43.6
12 285.6 40.8
13 265.8 38.0
14 246.0 35.1
15 226.2 32.3
16 206.4 29.5
17 186.6 26.7
18 166.8 23.8
19 147.0 21.0
20 127.2 18.2

In the case where all my attacks hit I'm already vastly out damaging your single attack rogue even though precision itself is adding nothing here. In other words, you want only those cases where the number of hits is about average or below and where you don't miss by 4 or more. Many other cases and the -5/+10 is sufficiently outdamaging the rogue build without any need for the superiority dice in the first place.

I'm not sure what you're saying here... Precision attack is responsible for more than half of the fighter's edge over the rogue, which we now know how to quantify (thanks, by the way, for spurring me to think about this!). Without superiority dice, the fighter's DPR (vs AC 16) over 20 rounds, assuming three action surges, is just

23 * 3 * 0.40 * 16.5 / 20 = 22.8

which is only about 27% better than the rogue if we assume the rogue gets advantage 75% of the time. (The fighter will get advantage from party members some of the time too, so it's a bigger edge than that, but y'know)

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#### FrogReaver

##### As long as i get to be the frog
If it is low, it means a smaller share of attack rolls have the opportunity to benefit from precision attack than in your calculations, so the average damage will be reduced.

Yes, this is a good point: when there are fewer than average rolls in the precision attack range, those have to go somewhere, either to automatic hits or to misses where we are not using our dice. So we should consider whether the cases where we have fewer-than-expected triggers for precision attack look different than the cases where we have the expected number or more.

In the ideal case, in a batch of 20 rolls, we have 20%, or 4, in the precision attack range, each of which has an 81.25% to become a hit, and 16 outside of that range. In the case of AC 16 (where the chance to hit if you can't use precision dice is 40%), the 16 that fall outside the precision attack range have a 50% chance of being hits. So, our overall chance to hit is 4/20 * 0.8125 + 16/20 * 0.50 = 0.5625. So, precision attack is giving us an extra 16.25% to our to-hit chance, or an extra 3.25 hits out of 20, as we saw before.

But what about the times when we only get 3 rolls in the mid-range? The missing one has exactly the same distribution of possible values as the 16 we assumed didn't trigger precision attack before, so we get a to-hit chance of 3/20 * 0.8125 + 17/20 * 0.50 = 54.6875%. With 2 in the mid-range, it's 53.125%, and with 1 it's 51.5625. Of course, if there are more rolls in the "within 4" range than we have precision dice, those are automatic misses. Overall, the chance to hit over N rolls between short rests is given by the formula

[sum for n = 0 to n = N] P(n rolls within 4) * [min(n, 4) * 0.8125 + (N - n) * 0.50 + max(n-4, 0) * 0.00]

where n is the number of rolls that meet our criterion for triggering precision attack.

P(n rolls within 4) is given by the binomial distribution over N rolls with a "success" probability of 0.20.

With some help from R, this is easy to compute:

AC <- 16 # enemy AC
atk<- 3 # attack bonus after -5 penalty
m <- AC - atk # smallest natural roll to hit w/o precision
N <- 20 # Number of attack rolls per short rest
t <- 4 # Maximum distance when we will use a die
n <- 0:N # Number of attack rolls within t of a hit

# Chance of n rolls of N within t
p_n <- dbinom(n, size = N, prob = t / 20)
# Chance to hit when using precision attack
p_prec <- sum(1/t * (1/8) * (8-t+1):8)
# Chance to hit when not triggering precision attack
p_other <- (20-m+1) / (20 - t)

to_hit <- sum(p_n * (pmin(n,4)/N * p_prec + (N-n)/N * p_other))
to_hit
[1] 0.5341341

Actually, this didn't take action surge into account: we're assuming 23 attack rolls, not 20. So, changing N to 23, we get

to_hit

[1] 0.524508

Slightly lower to-hit, but of course more attacks. With the 16.5 damage per attack, that's

dph <- 16.5
daily_attacks <- 3 * N
total_damage <- daily_attacks * to_hit * dph
total_damage
[1] 597.1524
DPR <- total_damage / 20
[1] 29.85762

So, like I said, about 1.5 below what we'd get if we had exactly the "right" number of uses of precision attack every rest. For other ACs, here's what the table looks like with 7 rounds per short rest.

total_per_sr DPR
11 305.4 43.6
12 285.6 40.8
13 265.8 38.0
14 246.0 35.1
15 226.2 32.3
16 206.4 29.5
17 186.6 26.7
18 166.8 23.8
19 147.0 21.0
20 127.2 18.2

I'm not sure what you're saying here... Precision attack is responsible for more than half of the fighter's edge over the rogue, which we now know how to quantify (thanks, by the way, for spurring me to think about this!). Without superiority dice, the fighter's DPR (vs AC 16) over 20 rounds, assuming three action surges, is just

23 * 3 * 0.40 * 16.5 / 20 = 22.8

which is only about 27% better than the rogue if we assume the rogue gets advantage 75% of the time. (The fighter will get advantage from party members some of the time too, so it's a bigger edge than that, but y'know)

By the way, good work on this. I thought I had taken everything into account for precision attack and apparently didn't.

#### Greg Benage

Forgive the necro, but by RAW, you have to roll separate initiative for the owl. This drops a nuke on the ATs ability to use the familiar to reliably produce advantage. If the owl wins initiative, it can move in and ready a help action -- but it can't use Flyby. If the AT wins, he/she can move in and ready an attack/spell, but then can't use Cunning Action to disengage or use Uncanny Dodge to mitigate damage.

Granted, in AL, I always ask and every single DM I've had says, "Nah, just have the owl go on your turn." My AT is 11th level.

#### Esker

##### Hero
Forgive the necro, but by RAW, you have to roll separate initiative for the owl. This drops a nuke on the ATs ability to use the familiar to reliably produce advantage. If the owl wins initiative, it can move in and ready a help action -- but it can't use Flyby. If the AT wins, he/she can move in and ready an attack/spell, but then can't use Cunning Action to disengage or use Uncanny Dodge to mitigate damage.

You're right that you are supposed to roll separate initiative for the owl, and we actually do at my home table, but it's not as bad as you describe: the owl can take the help action as long as they go before you. And if they go after you they can help for your next attack. So at most you're missing out on one round of advantage, which probably happens a little over half the time (since the rogue will have a higher initiative modifier). Figuring three round combats, that's still significant though: maybe 1/5 of your attacks are in the first round of a combat where your owl hasn't had a turn yet. So maybe 75% is a little too high as an estimate of how often you have advantage.

#### Greg Benage

You're right that you are supposed to roll separate initiative for the owl, and we actually do at my home table, but it's not as bad as you describe: the owl can take the help action as long as they go before you. And if they go after you they can help for your next attack. So at most you're missing out on one round of advantage, which probably happens a little over half the time (since the rogue will have a higher initiative modifier). Figuring three round combats, that's still significant though: maybe 1/5 of your attacks are in the first round of a combat where your owl hasn't had a turn yet. So maybe 75% is a little too high as an estimate of how often you have advantage.

I don’t know: The owl can’t help your rogue, specifically. If it takes the help action on an enemy, the next character to attack that enemy gets advantage. Maybe it’s the rogue, maybe not. Especially challenging in “boss fights” where you most need that DPS, but may often force the party to choose between tactically superior focus-fire and your rogue’s need for advantage.

#### Esker

##### Hero
The PHB text on using the help action in combat says:
Alternatively, you can aid a friendly creature in attacking a creature within 5 feet of you. You feint, distract the target, or in some other way team up to make your ally's attack more effective. If your ally attacks the target before your next turn, the first attack roll is made with advantage.

I read the phrase "a friendly creature" to mean that you decide whose attack you are helping. So it doesn't have to be the next in the initiative order, or even the next to attack that target.