What is the Expected Damage Gain from this Part of my Homebrewed Gunslinger Feat?

Question

Preamble:

One of the players in a campaign I will be DM’ing for will be playing a fighter archetype called the Gunslinger (made by Matthew Mercer for the RPG show Critical Role). Since the player loves the risk-reward of the Sharpshooter feat, I’ve decided to try my hand at homebrewing something similar - but with more risk and more reward.

In order to satisfy this I have tried to increase the damage gain (with respect to Sharpshooter), but also add some drawbacks and such. I have omitted all of the drawbacks in this feat that are not relevant to this question. This feat is not balanced with the information below, this question is only to ask what the expected damage increase is. If you're not familiar with the Misfire mechanic I've placed the definition of it at the bottom of this question.

The Feat:

Before you make an attack with a Firearm that you are proficient with, you can choose to make a Buster Shot:

• Increase misfire number by 5 for this shot
• add an additional number of damage dice equal to double the original damage dice (the new total is triple the original)
• If the attack is a hit, the target takes the new total damage
• [ some drawbacks that don’t affect the damage gains ]

Buster Shot Example (Pistol 1d10 with Misfire 1, Dex mod +0):

• Misfire number becomes 6 [1+5]
• New damage becomes 3d10 [ 1d10 + 2*(1d10) ]
• If the shot hits and does not misfire the target takes 3d10 damage
• [drawbacks that don’t affect the damage gain]
• If it was a critical hit, it deals 2*(3d10) = 6d10 damage

The Question:

How much would using the “Buster Shot” as described above increase my Gunslinger’s expected damage to the target, assuming that the fight lasts for 5 rounds (or another set number of rounds, if you prefer to use that in your calculations)? The reason it needs to be more than one round is to account for the attacks the Gunslinger would miss out on from (perhaps) fixing their weapon, as pointed out by @Nickmagus in their answer below.

The same type of analysis was done on the Sharpshooter/GWF Feat in the answers to this question ( Are class features, Abilities, and feats that allow −5 to attack to get +10 to damage mathematically sound? ).

It would be preferred if answers could include more than one base damage test case, but if the math proves to be too complex let’s say the Gunslinger’s base damage is 10 points.

Clarifications & definitions:

Misfire: Whenever you make an attack roll with a firearm, and the dice roll is equal to or lower than the weapon's Misfire score, the weapon misfires. The attack misses, and the weapon cannot be used again until you spend an action to try and repair it. To repair your firearm, you must make a successful Tinker's Tools check (DC equal to 8 + misfire score). If your check fails, the weapon is broken and must be repaired out of combat at half the cost of the firearm.

Also, a free online version of the Gunslinger by Matt Mercer can be found here: https://www.dropbox.com/s/82o72v47ddc8lzz/Gunslinger%205E.docx?dl=0

Answer 1

We can fairly safely assume that the misfire chance increase doesn't change the expected damage since you won't be hitting much on a roll of 1-6 anyway. That being said the change in damage is easy to calculate. For any dN (d6,d10, etc.) the expected value of 1dN is (N+1)/2 (1d6=3.5, 1d10=5.5, etc) and the expected value of MdN is M×(ev of 1dN). Therefore the expected value of 1d10 (a regular hit) is 5.5 while 3d10 (buster) is 16.5, 2d10 (a regular crit) is 11 while 6d10 (a buster crit) is 33.

For the actual misfire rules (misfire causes loss of next round and chance of loss of rest of combat). Let me first state an equation in plain english for the damage each round (where n for the first round is 0, n for the second is 1, etc.)

$$ (\text{"chance you hit"}\times;\text{"dmg ev"}+\text{"chance you crit"}\times\text{"dmg ev"})\times(1-\text{"chance you misfired last round"})\times(1-\text{"chance you misfire"}\times\text{"chance you fail to fix"})^n $$

For N rounds where Y=ev of damage dice, and X=number needed to hit, M=misfire number of weapon, and T is the tinker skill (assuming M+5 is greater than X, and T is greater than 8+M)

• with feat: $$ 3 \times \left(Y\times \frac{20-X+1}{20}+\frac{Y}{20}\right)\times \left(1+\sum^{N-1}_{n=1}\left[\left(1-\frac{M+5}{20}\right)\times\left(1-\frac{M+5}{20}\times\frac{8+M-T-1}{20}\right)^n\right]\right) $$

• without feat: $$ \left(Y\times\frac{20-X+1}{20}+\frac{Y}{20}\right)\times\left(1+\sum^{N-1}_{n=1}\left[ \left(1-\frac{M}{20}\right) \times \left(1-\frac{M}{20}\times\frac{8+M-T-1}{20}\right)^n \right]\right) $$

for 5 rounds with M=1, X=11, Y=5.5 and T=5 we have expected 31.75 damage with the feat and expected 14.3 damage without it.

note $$\sum_{n=1}^{N-1}\left[x^n\right] = x+x^2+x^3+x^4+\ldots+x^{N-1}$$