Are Champion fighters better now that we have Crusher/Piercer/Slasher?

[Hael]

But it doesn't actually work. It takes a relatively large number of rounds to have say a 95% chance of using all your dice on precision attack using anything smarter than a use it anytime you miss kind of heuristic.

I’m struggling to see this. The player isn’t coming from this with zero knowledge. Instead he has likely gotten a feel for the optimal heuristic. He has access to the number of rounds the DM tends to play combat, potential meta game knowledge of AC, avg number of fights per short rest etc. In practice a good player will tend to be close to ideal damage maximization if that’s what leads to the optimal outcome (in practise it’s probably CC that matters more but that’s a different issue). Hence he’s going to be close to the best case scenario where if he has x die, then x misses are turned into hits. Alternatively, if there aren’t that many rnds, then a few of those will be converted into straight ripostes or something like that, just to get the dmg out.

[LudicSavant]

But it doesn't actually work.

It works insofar as it allows you to make fewer and narrower assumptions than the alternative, even if the number of assumptions is still not zero.

[Frogreaver]

But it doesn't actually work. It takes a relatively large number of rounds to have say a 95% chance of using all your dice on precision attack using anything smarter than a use it anytime you miss kind of heuristic.

Or to say it another way, you can't actually consider precision attacks to be front loaded because you can't actually front load it. It takes a relatively large number of rounds to ensure you are able to use all 6 and an even larger number if you want to only use precision attack when you have a high probability of converting to a hit.

Thus, number of rounds between short rests cannot be removed from precision attack analysis.



Precision attack would also have a rate of use and if there aren't adequate rounds in the day to use it all the times you want using your preferred heuristic then you can't say the champion needs to 'catch up' to the damage provided by all the uses of it. Which takes us directly back to the question of how many rounds are in the short rest.

To put this in perspective
=Binom.Dist(5,'4*8',0.2,'Cumulative') = 36.0191%

That is give the following assumptions:
1. The 0.2 comes from our Precision attack heuristic is to use precision attack when we miss by 1-4.
2. We make 4 attacks per round
3. We go through 8 rounds of combat
4. We want to see the probability that we use 5 or fewer dice on precision attack.

The result here is that we have a 36.02% chance to use fewer than our 6 superiority dice. Solving for the weighted average number of superiority dice used yields 5.3 superiority dice. That is the actual battlemaster damage from precision attack in this scenario is 11.67% lower than you listed.

It works insofar as it allows you to make fewer and narrower assumptions than the alternative, even if the number of assumptions is still not zero.

They aren't fewer or narrower. They are flat out incorrect.

I’m struggling to see this. The player isn’t coming from this with zero knowledge.

Of course not. I would never assume that. This critique holds even when the player is using a fairly intelligent heuristic.

Instead he has likely gotten a feel for the optimal heuristic. He has access to the number of rounds the DM tends to play combat, potential meta game knowledge of AC, avg number of fights per short rest etc. In practice a good player will tend to be close to ideal damage maximization if that’s what leads to the optimal outcome (in practise it’s probably CC that matters more but that’s a different issue).

I agree with all of this. What are you reading me as saying that makes you think I disagree?

Hence he’s going to be close to the best case scenario where if he has x die, then x misses are turned into hits.

Depends on what you mean by best case scenario. If you mean a scenario where he has uses 6 dice on high probability precision attacks because that's the highest DPR manuever he has then I'd say there's a significant chance of that not happening unless the time between short rests is significantly longer than we would normally consider.

If you mean he's not going to naively waste abilities then I agree. He will be close to the maximum he can output, it's just the maximum he can output isn't typically going to be 6 superiority dice going to high probability precision attacks.

Alternatively, if there aren’t that many rnds, then a few of those will be converted into straight ripostes or something like that, just to get the dmg out.

Assuming he can predict the number of rounds with that degree of certainty and do fairly complex math in his head on the fly then yes. For example, if you have 1 round of combat left is it better to go ahead and use it on a +damage dice maneuver or wait incase you get to use precision attack. At what point does one 'cut their loses' and use the guaranteed damage manuever?

I would say you actually significantly overrate a players ability to predict the number of rounds left in the short rest, you significantly overrate his ability to make complex calculations on the fly to determine the 'best' time to use superiority dice for maximum DPR output. Even though I agree that the player will have a decent idea of these things - it will be far from perfect.

[LudicSavant]

They aren't fewer or narrower. They are flat out incorrect.

What assumption have I made that is incorrect?

[Frogreaver]

What assumption have I made that is incorrect?

The assumption that precision attacks damage is independent of the number of rounds. In the post you quoted I just showed that assumption was incorrect by way of counterexample (and not a ridiculous out in left field one either, but a fairly typical case study)

[LudicSavant]

The assumption that precision attacks damage is independent of the number of rounds.

Ah, so an assumption I didn't actually make.

[Greywander]

I'm curious how it affects the math if we're a half-orc, seeing as this gives us an extra damage die on crits (which will disproportionally affect Champions). My understanding is that with a greatsword or maul this only adds 1d6 damage, not 2d6, so we'd want to use something like a greataxe to get an extra 1d12 instead. This means using Slasher instead of Crusher, which offers its own utility in combat but will hurt our DPR if we don't have another source of advantage. Alternatively, since there don't seem to be any d12 bludgeoning weapons, we could use a warhammer instead for 1d10 if we really wanted to use Crusher, though a maul might have higher DPR even with the smaller extra crit die.

It just seems like critfisher builds generally want to either be an elf for Elven Accuracy or be a half-orc for Savage Attacks, and seeing as how we're talking about GMW it makes sense to run the tests as a half-orc. I don't expect this to make the Champion jump ahead of the Battle Master, but it might close the gap somewhat.

DPR isn't everything, though. Lategame that regen can be pretty nice, and Tasha's added some juicy fighting styles, making the extra one Champions get a lot more appealing. Battle Masters have their own tricks, as well, so a direct comparison is difficult.

[Frogreaver]

Ah, so an assumption I didn't actually make.

Everyone can read the context of the conversation up to this point. It's apparent to everyone what we were talking about based on our back and forth exchange on the topic.

[Zhorn]

Everyone can read the context of the conversation up to this point. It's apparent to everyone what we were talking about based on our back and forth exchange on the topic.

Clearly I'm not included in that 'everyone'. This current string of posts has me more confused than ever over what you're on about.

[LudicSavant]

I'm curious how it affects the math if we're a half-orc, seeing as this gives us an extra damage die on crits (which will disproportionally affect Champions). My understanding is that with a greatsword or maul this only adds 1d6 damage, not 2d6, so we'd want to use something like a greataxe to get an extra 1d12 instead. This means using Slasher instead of Crusher, which offers its own utility in combat but will hurt our DPR if we don't have another source of advantage. Alternatively, since there don't seem to be any d12 bludgeoning weapons, we could use a warhammer instead for 1d10 if we really wanted to use Crusher, though a maul might have higher DPR even with the smaller extra crit die.

It just seems like critfisher builds generally want to either be an elf for Elven Accuracy or be a half-orc for Savage Attacks, and seeing as how we're talking about GMW it makes sense to run the tests as a half-orc. I don't expect this to make the Champion jump ahead of the Battle Master, but it might close the gap somewhat.

DPR isn't everything, though. Lategame that regen can be pretty nice, and Tasha's added some juicy fighting styles, making the extra one Champions get a lot more appealing. Battle Masters have their own tricks, as well, so a direct comparison is difficult.

Adding in Half-Orc is pretty easy to do in either my DPR Calculator or AnyDice.

The complicated part includes
- the element of player choice and gambling against unknown future events in the Battle Master's decision tree. For example, you hypothetically could decide to only use a smite-like maneuver on a crit... but doing so runs the risk of not actually getting to use all your maneuvers before a short rest arrives.
- the element of team benefits (for example, granting Advantage to teammates with Crusher).
- the impact of things that don't do damage directly, like status effects.

[Frogreaver]

Clearly I'm not included in that 'everyone'. This current string of posts has me more confused than ever over what you're on about.

You presented a method. Ludic praised it for eliminating assumptions. I claimed your method wasn't actually correct because it assumed you could use 6 superiority dice on precision attack despite the number of rounds that were in the short rest. Ludic defended it saying "It works insofar as it allows you to make fewer and narrower assumptions than the alternative, even if the number of assumptions is still not zero." I again claimed it didn't work because the assumptions in it were incorrect. At that point Ludic claims she never made that assumption.

Hopefully that helps.

- the element of team benefits (for example, granting Advantage to teammates with Crusher).

Oh wow. I didn't realize that aspect of Crusher till you just mentioned it. I'm even more impressed with the feat. *When I read it my mind inherently added a "your attacks" to it instead of leaving it as "attacks".

[LudicSavant]

You presented a method. Ludic praised it for eliminating assumptions. I claimed your method wasn't actually correct because it assumed you could use 6 superiority dice on precision attack despite the number of rounds that were in the short rest. Ludic defended it saying "It works insofar as it allows you to make fewer and narrower assumptions than the alternative, even if the number of assumptions is still not zero." I again claimed it didn't work because the assumptions in it were incorrect. At that point Ludic claims she never made that assumption.

Hopefully that helps.

It does not help, because the assumption you falsely claim I made

The assumption that precision attacks damage is independent of the number of rounds.

is not only one that features nowhere in any of my statements, it is also an assumption that is unnecessary in order to measure a number of attacks needed to close a gap.

[Zhorn]

No Frogreaver, I mean I don't follow your reasoning at all. LudicSavant I understand.

Order, front loading or whatever else doesn't matter.
lets make a simple representation
A=5, B=10, C=6
For every attack that B is used, there will need to be four attack where A is used for C to keep pace
AAAAB = 5 + 5 + 5 + 5 + 10 = 30
AAABA = 5 + 5 + 5 + 10 + 5 = 30
AABAA = 5 + 5 + 10 + 5 + 5 = 30
ABAAA = 5 + 10 + 5 + 5 + 5 = 30
BAAAA = 10 + 5 + 5 + 5 + 5 = 30
CCCCC = 6 + 6 + 6 + 6 + 6 = 30

if there were a string of A, then C would initially pull ahead, but a few B's would close that gap. We wouldn't say 'catch up' but it's still the same method.
Every use of B alongside C creates a gap of B-C that will take a number of occurrences of of C alongside A for the differences of C-A to close that gap, whether those B's occur at the start or end or anywhere in-between is irrelevant. The length of a round is irrelevant. The number of rounds is irrelevant. The number of short rests or initiative rolls is irrelevant.

We're working with averages and simplified expected outcomes. There's no need to get overly convoluted with the math.

[Frogreaver]

is not only one that features nowhere in any of my statements, it is also an assumption that is unnecessary in order to measure a number of attacks needed to close a gap.

I would say the 'gap' caused by precision attack can only be computed without an assumed number of rounds if one assumes that the number of rounds doesn't affect the damage precision attack causes. Since you have said you aren't making that assumption - how are you computing the damage 'gap' caused by precision attack?

[LudicSavant]

The number of rounds does have an effect, insofar as you might have better opportunities to use your maneuvers given more die rolls (since you can decide to use a maneuver after seeing the roll). For example, a player might decide that they will only use Menacing Attack if they get a crit (thus adding superiority die *2 damage instead of superiority die *1 with each use), or only use Precision Attack if they miss by 1 (thus making each superiority die convert a miss into a hit with 100% success rate), but if there aren't enough attacks per short rest, they (probably) won't get those opportunities often enough to use up all their superiority dice.

But the method I had in mind would simply choose a given strategy and then evaluate find where its breakpoints are (e.g. "if # of attacks is >X, then strategy A is better" etc). We don't need to actually determine how many attacks there are in a day, only how many attacks there would need to be in order to "catch up."

The real problem comes in when we ask how we can quantify things like Crusher's benefit to allies (giving THEM advantage). Since we'd have to know who those characters are, and how many attacks you are likely to be able to make against the guy you already critted before they die anyways, etc.

[Frogreaver]

The number of rounds does have an effect, insofar as you might have better opportunities to use your maneuvers given more die rolls (since you can decide to use a maneuver after seeing the roll). For example, a player might decide that they will only use Menacing Attack if they get a crit (thus adding superiority die *2 damage instead of superiority die *1 with each use), or only use Precision Attack if they miss by 1 (thus making each superiority die convert a miss into a hit with 100% success rate), but if there aren't enough attacks per short rest, they (probably) won't get those opportunities often enough to use up all their superiority dice.

Agreed!

But the method I had in mind would simply choose a given strategy and then evaluate find where its breakpoints are (e.g. "if # of attacks is >X, then strategy A is better" etc). We don't need to actually determine how many attacks there are in a day, only how many attacks there would need to be in order to "catch up."

As a counterexample consider these 2 strategies.
Strategy 'A' = Use all your dice as fast as possible
Strategy 'B' = Use your dice only when you miss by 1 or crit

Now let's have two cases:
Case '1' = There is a single round in the short rest.
Case '2' = There are 10,000,000 rounds in the short rest.

In Case '1' Strategy 'A' should be clearly better.
In Case '2' Strategy 'B' should be clearly better.

Thus proving that the additional damage caused by using a specific strategy is dependent upon the the number of rounds in the short rest. This implies that any computed 'gap' you need to 'catch up' by is going to change depending on the number of rounds per short rest

[LudicSavant]

As a counterexample consider these 2 strategies.
Strategy 'A' = Use all your dice as fast as possible
Strategy 'B' = Use your dice only when you miss by 1 or crit

Now let's have two cases:
Case '1' = There is a single round in the short rest.
Case '2' = There are 10,000,000 rounds in the short rest.

In Case '1' Strategy 'A' should be clearly better.
In Case '2' Strategy 'B' should be clearly better.

Thus proving that the additional damage caused by using a specific strategy is dependent upon the the number of rounds in the short rest

This isn't a "counterexample" because it already agrees with the concept I just explained.

This implies that any computed 'gap' you need to 'catch up' by is going to change depending on the number of rounds per short rest

And this part appears to be wrong. You do not need to know that there are X rounds in the short rest in order to determine whether or not you would catch up in Y rounds.

For example, let's say Strategy C is better in the long term, and Strategy D better in the short term -- their performances vary based on how many rounds there are. You could then determine exactly how many rounds it would take for strategy C to exceed Strategy D. You do not need to actually know how many rounds you will get in order to calculate how many rounds it would take.

[Frogreaver]

No Frogreaver, I mean I don't follow your reasoning at all. LudicSavant I understand.

Order, front loading or whatever else doesn't matter.
lets make a simple representation
A=5, B=10, C=6
For every attack that B is used, there will need to be four attack where A is used for C to keep pace
AAAAB = 5 + 5 + 5 + 5 + 10 = 30
AAABA = 5 + 5 + 5 + 10 + 5 = 30
AABAA = 5 + 5 + 10 + 5 + 5 = 30
ABAAA = 5 + 10 + 5 + 5 + 5 = 30
BAAAA = 10 + 5 + 5 + 5 + 5 = 30
CCCCC = 6 + 6 + 6 + 6 + 6 = 30

In that sense sure but that's not what I'm talking about. As one of my posts above showed, in 8 rounds of combat with 4 attacks per round and using precision attack on a miss of 1-4 you are only averaging 5.3 uses of precision attack. Your calculation would assume 6 uses of precision attack. Thus the gap you are setting is higher than the actual gap. Meaning you are over estimating the number of rounds the champion actually needs to catch up. But more importantly the average # of uses of precision attack will change as your total number of rounds changes. For example, in 4 rounds using the same strategy you will only use 3.16 uses of precision attack. You would have assumed 6 uses here as well. It's that assumption of 6 uses even when you've not made enough attacks to be at 6 uses that is the front loading i'm talking about.

We're working with averages and simplified expected outcomes. There's no need to get overly convoluted with the math.

Trying to simplify like you are is leading you to not calculating the expected outcomes correctly. One doesn't need to be overly convoluted. One needs to be accurate though even if it's not quite as simple as one would like.

This isn't a "counterexample" because it already agrees with the concept I just explained.

I could have swore we were saying different things there. And your comment below leads me to believe that as well.

And this part appears to be wrong. You do not need to know that there are X rounds in the short rest in order to determine whether or not you would catch up in Y rounds.

You do when Y depends on X.

[LudicSavant]

You do when Y depends on X.

It is possible to determine that a task would require an hour to complete without knowing whether or not an hour will be available.

[Frogreaver]

It is possible to determine that a task would require an hour to complete without knowing whether or not an hour will be available.

I didn't say every Y depended on X so I'm not sure what that has to do with a claim that this Y depends on this X.

[LudicSavant]

I didn't say every Y depended on X so I'm not sure what that has to do with a claim that this Y depends on this X.

Yet again:

This implies that any computed 'gap' you need to 'catch up' by is going to change depending on the number of rounds per short rest

Per your own suggestion, there is some number of rounds where Strategy B exceeds strategy A. We can determine how many rounds that is, without knowing whether or not we will actually play that many rounds.

Your own "counterexample" supports the very concept that people have been trying to tell you.

[Frogreaver]

Yet again:

Per your own suggestion, there is some number of rounds where Strategy B exceeds strategy A. We can determine how many rounds that is, without knowing whether or not we will actually play that many rounds.

We cannot.

Normally when we compare we get something like:

f(x) = p(x) + q(x)
g(x) = r(x)

• s.t. x is an integer representing the number of rounds
• s.t. p(x) = ax where a is a real number
• s.t. q(x) = c where c is a real number
• s.t. r(x) = bx where b is a real number and b>a

Then we can say for what x is g(x) > f(x) and this is relatively simple to solve.

However, in the case of precision attack the function would instead be

f(x) = p(x) + q(x)
g(x) = r(x)

• s.t. x is an integer representing the number of rounds
• s.t. p(x) = ax where a is a real number
• s.t. 0.2 is the probability the attack roll lands on a value where you will use precision attack (note this represents the chosen 'use when miss by 1-4' heuristic and can be changed but will yield different results)
• q(x) = h*d * innerproduct(y , z ) where y is the vector <1,2,3,4,5,6,6,6,....> and z is the vector <binomal.dist(1,nx,0.2,0), binomal.dist(2,nx,0.2,0), ....>
• s.t. h is the chance precision attack turns a miss to a hit
• s.t. d is the damage done when you hit
• s.t. n is the attacks per round
• s.t. r(x) = bx where b is a real number and b>a

I am not aware of any method that can be used to solve r(x) > p(x) + q(x) given the above equations.

[bid]

To put this in perspective
=Binom.Dist(5,'4*8',0.2,'Cumulative') = 36.0191%

That is give the following assumptions:
1. The 0.2 comes from our Precision attack heuristic is to use precision attack when we miss by 1-4.
2. We make 4 attacks per round
3. We go through 8 rounds of combat
4. We want to see the probability that we use 5 or fewer dice on precision attack.

The result here is that we have a 36.02% chance to use fewer than our 6 superiority dice. Solving for the weighted average number of superiority dice used yields 5.3 superiority dice. That is the actual battlemaster damage from precision attack in this scenario is 11.67% lower than you listed.

Now that is useful. Thank you for showing me how to use this tool.

Seems about right for champion when I try a sanity check:
=Binom.Dist(0,'4*8',0.05,'Cumulative') = 20% of never rolling 19,
=Binom.Dist(1,'4*8',0.05,'Cumulative') = 52% of rolling 19 no more than once.
=Binom.Dist(5,'4*8',0.05,'Cumulative') = 99.5% of rolling 19 no more than 5 times.

I tried and it seem 0.2 is not 1-4 but 1-5 {.208 = (12+11+10+9+8)/12/20}
=(5×12−5×(5-1)÷2)÷12÷20 is the general excel equation for your precision attack heuristic.

Lets see how it looks around that point:
=Binom.Dist(5,'4*8',0.208,'Cumulative') = 32% of not spending all 6 with off-by-5
=Binom.Dist(4,'4*8',0.208,'Cumulative') = 17% of not spending 5 with off-by-5
=Binom.Dist(5,'4*8',0.238,'Cumulative') = 19% of not spending all 6 with off-by-6
=Binom.Dist(4,'4*8',0.238,'Cumulative') = 9% of not spending 5 with off-by-6
=Binom.Dist(5,'4*8',0.262,'Cumulative') = 12% of not spending all 6 with off-by-7
=Binom.Dist(4,'4*8',0.262,'Cumulative') = 5% of not spending 5 with off-by-7
You really have to take bad options to be likely to spend most SD that fast. It might be better to waste some as +1d12 damage if you want to maximize your damage output and know the day will be short.


OTOH, you don't have to spend all your SD to do better...
Trying an extremely low spending limit:
=Binom.Dist(2,'4*8',0.05,'Cumulative') = 77% of rolling 19 no more than twice.
=Binom.Dist(2,'4*8',0.208,'Cumulative') = 2% of not spending 3 with off-by-5
You have 77% of rolling 19 less than 3 times and be limited to (2d6)*2 or 14 damage.
You have 2% of doing less than 3 SD and be limited to (2d6+3)*2 or 20 extra damage.

You really need enough combat round to spend all your SD or champion never has a chance to catch up.
EDIT: silly me used level 3 improved critical instead of the level 15 one. not fixing it further than this caveat.

[SLOTHRPG95]

@Frogreaver I think you and @Ludic are talking past each other further and further with your abstracted statements. If I understand you correctly, your original objection to @Zhorn's methods was that he assumed (implicitly) that all 6 superiority dice would be used for precision, hence maximizing the initial gap between Champion's damage output and the BM's. You pointed out that the number of attacks/rounds before all 6 can be used for precision is not fixed, and hence the damage gap is not fixed for relatively small numbers of attacks.

Specifically, you could model the required number of attacks/rounds with a negative binomial distribution: use of precision is a "failure" and you want to know the number of successes before hitting six failures, with total number of attacks then being successes plus six. Given that the Champion will always do better as number of attacks tends to infinity, the BM needs to pull ahead early, or it'll just always be behind. If the BM uses a superiority die every time an attack is missed by 6 or less, then the distribution for the number of attacks before all superiority dice are used is 6 + NB (6, .3), which has a mean value of 20 attacks.

Although I think your original objection is sound in theory, in practice at 20th level there won't be many inter-SR periods where you don't get off at least 20 attacks. That's just three rounds if you action surge twice. Hence, at least at 20th level, which was under discussion given 6d12 sup. dice and four attacks per round etc., the variable gap size becomes a non-issue. You don't need 10M rounds to optimally use your sup. dice for DPR, you only need (on average) about three. Of course, this will vary at other levels but it's too late at night to do a breakdown for all levels based on size of sup. dice, number of dice, number of attacks per round, etc.

Hope this clarifies some confusion on all sides.

[LudicSavant]

We cannot.

If you could not determine a number of rounds where B exceeds A, then you could not make statements like the following:

In Case '1' Strategy 'A' should be clearly better.
In Case '2' Strategy 'B' should be clearly better.

If you believe that you cannot determine a number of rounds where B would exceed A, how did you determine that B would exceed A in case 2?

[Frogreaver]

If you could not determine a number of rounds where B exceeds A, then you could not make statements like the following:



If you believe that you cannot determine a number of rounds where B would exceed A, how did you determine that B would exceed A in case 2?

Why did you ignore the rest of my explanation?

[LudicSavant]

Why did you ignore the rest of my explanation?

You didn't actually explain why anything I said is wrong. As SLOTH pointed out it looks like you're just talking past me.

[Frogreaver]

@Frogreaver I think you and @Ludic are talking past each other further and further with your abstracted statements. If I understand you correctly, your original objection to @Zhorn's methods was that he assumed (implicitly) that all 6 superiority dice would be used for precision, hence maximizing the initial gap between Champion's damage output and the BM's. You pointed out that the number of attacks/rounds before all 6 can be used for precision is not fixed, and hence the damage gap is not fixed for relatively small numbers of attacks.

Yes, thank you.

Specifically, you could model the required number of attacks/rounds with a negative binomial distribution: use of precision is a "failure" and you want to know the number of successes before hitting six failures, with total number of attacks then being successes plus six. Given that the Champion will always do better as number of attacks tends to infinity, the BM needs to pull ahead early, or it'll just always be behind. If the BM uses a superiority die every time an attack is missed by 6 or less, then the distribution for the number of attacks before all superiority dice are used is 6 + NB (6, .3), which has a mean value of 20 attacks.

I'll have to read up on the negative binomial distribution. I had glanced at it before but didn't think it would help on this problem.

Just to compare your results with my weighted average from the binomial distribution, I only get 5.2 dice used for those same parameters. I tested with using a superiority dice whenever an opportunity arose to use one with those parameters and I got your 6. However, that doesn't align with how superiority dice actually work as no matter how many opportunities you have to use them at most you can only use 6 and so I'm even more convinced the negative binomial distribution won't be useful here.

Although I think your original objection is sound in theory, in practice at 20th level there won't be many inter-SR periods where you don't get off at least 20 attacks.

I think you need many more rounds than that to reach 'effectively 6 dice used'.

That's just three rounds if you action surge twice. Hence, at least at 20th level, which was under discussion given 6d12 sup. dice and four attacks per round etc., the variable gap size becomes a non-issue. You don't need 10M rounds to optimally use your sup. dice for DPR, you only need (on average) about three. Of course, this will vary at other levels but it's too late at night to do a breakdown for all levels based on size of sup. dice, number of dice, number of attacks per round, etc.

Hope this clarifies some confusion on all sides.

I think you have, though I'm curious to hear if you reach the same conclusions I just did about the negative binomial distribution.

You didn't actually explain why anything I said is wrong. As SLOTH pointed out it looks like you're just talking past me.

That's not what Sloth said.

[LudicSavant]

That's not what Sloth said.

Here's what Sloth said.

@Frogreaver I think you and @Ludic are talking past each other further and further with your abstracted statements.

[Frogreaver]

Here's what Sloth said.

"@Frogreaver I think you and @Ludic are talking past each other further and further with your abstracted statements."

Yes, and this statement is not the same as what you claimed he said.

"As SLOTH pointed out it looks like you're just talking past me."