D&D 5E Quick Great Weapon Style math question

[Zelc]

As printed, GWF adds +1.33 damage on 2d6, +0.83 damage on 1d12, and +0.8 damage on 1d10.

With your change, GWF adds +1 damage on 2d6, +0.75 damage on 1d12, and +0.7 damage on 1d10.

If you change it to "turn any below-average roll into the average roll", it adds +1 damage on 2d6, +1.25 damage on 1d12, and +1 damage on 1d10. This is my recommendation.

If you change it to "you can reroll any damage die", it adds +1.5 damage on 2d6, +1.5 damage on 1d12, and +1.25 damage on 1d10.

If you change it to "roll 1 extra weapon die, drop lowest", it adds +1.46 damage on 2d6, +1.99 damage on 1d12 (its expected value is 0.03 better than 2d6), and +1.65 damage on 1d10. (Rolling 2 extra d6's adds +2.34 damage on 2d6)

 

[Coredump]

Given that a number of people feel the "reroll if you roll a 1 or 2 on a damage die" bit can slow things down,

Really....??? It takes what, an additional 4 maybe 5 seconds? Okay, lets be crazy and say its 10 seconds..... is that really a concern?


As for your question... it will work, just decide ahead of time if you are rounding up or down.

 

[Quickleaf]

As printed, GWF adds +1.33 damage on 2d6, +0.83 damage on 1d12, and +0.8 damage on 1d10.

If this is accurate, and I'd need to see maths to be sure, then [MENTION=1288]Mouseferatu[/MENTION] the obvious change is to just make Great Weapon Fighting grant +1 damage and be done with it. No re-rolling and makes it more fair for wielders of different heavy weapons.

 

[Mouseferatu]

If this is accurate, and I'd need to see maths to be sure, then [MENTION=1288]Mouseferatu[/MENTION] the obvious change is to just make Great Weapon Fighting grant +1 damage and be done with it. No re-rolling and makes it more fair for wielders of different heavy weapons.

Oh, that's definitely the simplest way, and possibly the most mechanically sound. But I have two objections (which, I freely admit, are nitpicky as all hell ).

1) It increases the maximum, not just the average. I like the idea of an ability that lets you average higher but not hit harder.

2) Just adding +1 is... Well, it's kinda boring. Saying "You get average damage if you roll low" may amount to the same thing, but for some reason--and no, I can't justify this with any sort of logic --it just feels more interesting to me.

Maybe it's just because there are so many simple numerical bonuses already? I dunno.

 

[Coredump]

Oh, that's definitely the simplest way, and possibly the most mechanically sound. But I have two objections (which, I freely admit, are nitpicky as all hell ).

1) It increases the maximum, not just the average. I like the idea of an ability that lets you average higher but not hit harder.

2) Just adding +1 is... Well, it's kinda boring. Saying "You get average damage if you roll low" may amount to the same thing, but for some reason--and no, I can't justify this with any sort of logic --it just feels more interesting to me.

Maybe it's just because there are so many simple numerical bonuses already? I dunno.

Aside from people on the internet making claims.... have you experienced GWF actually slowing down game play? I just can't see it adding more than a few seconds to the round. *Maybe* a full minute or two to the entire session if multiple people have it.

And rerolling your 1-2 is part of the *fun* of the ability, I don't see a point in saving time by cutting out the fun.

 

[Cernor]

I think you're on sound math doing this. I mean if you were running the probabilities, you would assume a statistical average for the re-rolled damage dice anyhow, wouldn't you? I think the maths are more complex than I'm making them, but I’ll do my best. Note: I am not a professional statistician.

The only question is if this gives an unfair advantage to certain weapons. I'm thinking greatsword/greataxe (2d6 damage, avg 7, so rolling a '1' isn't possible) vs. a reach/versatile weapon (1d10 damage, avg 5.5).

According to my dirty maths:

Great Weapon Expert with greatsword/greataxe gives you a net +0.2777 damage bonus per round.

[SBLOCK=Greatsword/greataxe Maths]
Let’s say your average PC taking the Great Weapon Expertise feat is likely to use a greatsword or greataxe which deals 2d6 damage.

The odds of rolling a 2 (1, of course, is impossible) on 2d6 is 2.77% each time you roll.

Usually we do DPR (damage-per-round) calculations for monsters over 3 rounds, so let’s do the same here. Also, let’s assume that the character with this feat is making two attacks per round, so either they are 5th level with Extra Attack, make an opportunity attack, or have haste cast on them; I think that’s a pretty reasonable assumption.

So two attacks per round, over the course of 3 rounds - what are the odds of rolling a 2?

2.77% * 6 = 16.62%

This means that in about 1 out of every 6 combats (which we’re assuming last 3 rounds each) you’re going to see a 2 rolled on 2d6 (statistically speaking).

And we’re going to assume that when you re-roll your 2d6 you get a statistically average result of 7.

Which works out to getting a +5 bonus to damage (from a 2 to a 7) once every 6 combats, or once every 18 rounds.

And that works out to getting a +0.27777 bonus to damage each round.[/SBLOCK]

Great Weapon Expert with a reach/versatile weapon gives you a net +0.26666 damage bonus each round.

[SBLOCK=Reach/versatile weapon Maths]
Now the same situation, but they’re wielding a reach weapon or a versatile weapon in two hands which deals 1d10 damage.

The odds of rolling a 1 or 2 on 1d10 is 20%.

So two attacks per round, over the course of 3 rounds - what are the odds of rolling a 1 or 2?

20% * 6 = 120%

This means that in 1 out of every 5 combats (which we’re assuming last 3 rounds each) you’re going to see a 1 or 2 rolled on 1d10 (statistically speaking).

And we’re going to assume that when you re-roll your 1d10 you get a statistically average result of 5.5.

Which works out to getting a +4 bonus to damage (from a 1.5 to a 5.5) once every 5 combats, or once every 15 rounds.

And that works out to getting a +0.2666 bonus to damage each round.[/SBLOCK]

I like it... Other than the probabilities, which are multiplicative rather than additive.

With 6 attacks over 3 rounds (using the 1d10 weapon for simplicity), you have a 36% chance of having rolled a 1 or 2 on round 1, a 60% chance of having rolled a 1 or 2 after round 2, and the odds after round 3 are about 75%.

So assuming the reroll occurs once in 3 combats out of 4, and gives a +4 (average) damage boost when it applies, we get a +12 damage added over the course of 12 rounds which works out to a 1 DPR boost.

That seems like a lot... Especially if you add in the probabilities of it happening multiple times.

 

[Zelc]

If this is accurate, and I'd need to see maths to be sure

You can calculate the probability of rolling any result on an individual die as follows:

In order to get a result of 1 or 2, you have to roll a 1 or 2 first, then reroll a 1 or 2. Thus, pdf(x) for x = 1 or x = 2 is just 2/s*1/s where s = the number of sides on the die.

In order to get a result higher than 2, you can either roll it straight up, or roll a 1 or 2 first then reroll that result. Thus, pdf(x) or x > 2 is 1/s+2/s*1/s.

For the expected value, sum up the x*pdf(x). For 2d6, multiply it by 2.

The base average for 2d6 is 7, 1d12 is 6.5, and 1d10 is 5.5. Subtract that from the expected value after the change for the increase.

 

[Dausuul]

One method would be to have 1s be average round down, and 2s round up - so a 1 on a greataxe would be a 6, and a 2 a 5.

That's a good idea; keeps the average exactly the same. If I were trying to reduce the number of rolls involved, I'd probably go with this. Another way to think of it:

On a d4, if you roll 1 or 2 on a die, you get +1 to that roll.
On a d6, if you roll 1 or 2 on a die, you get +2 to that roll.
On a d8, if you roll 1 or 2 on a die, you get +3 to that roll.
On a d10, if you roll 1 or 2 on a die, you get +4 to that roll.
On a d12, if you roll 1 or 2 on a die, you get +5 to that roll.

 

[Mouseferatu]

And rerolling your 1-2 is part of the *fun* of the ability, I don't see a point in saving time by cutting out the fun.

For some people, absolutely. For others, not so much.

IME, the ability doesn't slow things down at all early in the evening. But toward the end of the game, when everyone's getting tired and the adrenaline's running down and the caffeine's wearing off, even simple math and rerolls start to take longer for some players. Once we get there, just saying "A 1 or 2 on one of your d6s for your greatsword becomes a 4" (or whatever system one ends up going with) is actually quicker.

Won't apply to everyone. Won't apply to every game. And I wouldn't make it a house rule in every campaign. I'm just trying to figure out what the balanced options are.

 

[Bacon Bits]

Given that a number of people feel the "reroll if you roll a 1 or 2 on a damage die" bit can slow things down, is there any solid mathematical reason not to change that to "treat any rolls of 1 or 2 on your damage die as if you'd rolled the average for that die"?

I know it's not going to average exactly the same, but am I correct in thinking it ought to be near enough for most practical purposes? Do you see any hidden downsides? (I'm foolishly posting this after 4 AM, so if I'm overlooking anything blatantly obvious, I blame sleep deprivation.)

Theoretically, it's just fine. Indeed, this is the easiest way to calculate the average of a GWF damage die:

A 1d6 has an average of 3.5. ( 1 + 2 + 3 + 4 + 5 + 6 ) / 6 = 3.5.

If we change it so that the values of 1 and 2 are instead the average for a 1d6, we get: (3.5 + 3.5 + 3 + 4 + 5 + 6) / 6 = 4.167 (4 and 1/6).

Thus 2d6 with rerolling 1s and 2s once is 8.33 (8 and 1/3). This is great for theoretical and expected damage calculations.

For practical, in game use, it's not that useful. Partially because you're not accurately modeling the damage (3 damage average is what you get with 1d5, not 1d6) but mostly because rolling dice is fun. I could see if if you're using a VTT, though.

If you really want to eliminate dice re-rolling, you could use a deck of cards. You'd need 4 of each card 3-6 and 1 of each card 1-2 for 26 cards total to represent 1d6 with one reroll of 1s and 2s. Of course, you'd have to riffle shuffle 7 times, draw a card, put the card back, riffle shuffle 7 times, and then draw a card, but you'd never need to re-roll!

If you wanted to model the greatsword with GWF with one deck and one draw, you'd need 324 cards. (Card value × number of cards): 2 × 1, 3 × 2, 4 × 9, 5 × 16, 6 × 32, 7 × 48, 8 × 56, 9 × 64, 10 × 48, 11 × 32, 12 × 16. Having played a Battle of Wits deck before, however, I think most people would consider this unacceptably difficult to shuffle.

If you also have GWF and Savage Attacker, you'll need a staggering 104,976 cards to accurately model the outcomes with a single draw: 2 × 1, 3 × 8, 4 × 135, 5 × 640, 6 × 2,816, 7 × 8,064, 8 × 15,232, 9 × 25,088, 10 × 24,192, 11 × 18,688, 12 × 10,112. Average damage 9.46.

[It's a very, very, very slow day at work.]