How to create dice rolls for multi-attacks in a custom tabletop RPG system

Question

I'm working on a simple tabletop RPG based in a sci-fi universe. The rule set is very basic, since I'll be playing with players with little experience and general tabletop know-how.

Most of it appears to be working fine, based on a test session. The only problem I'm having are attacks that hit multiple times.

Unlike most fantasy RPGs, you can wield weapons that fire e.g. 10 bullets in a single attack. For example, with an assault rifle, you can do a single shot, a short burst or hold down the trigger for fully automatic fire. Single shot is of course the most accurate, burst is less accurate and fully automatic is only accurate for the first few bullets.

I want to know how many bullets hit before calculating the damage. A normal attack roll is a normal 1d20 roll with modifiers against a DC (default 20), e.g. 1d20 + x has to be higher than 20. Afterwards, a damage dice is added for every bullet that hits, e.g. 1d6 for a single bullet, 2d6 for 2 hits, and so on. Characters can wear bullet proof vests that reduce the damage of every bullet, not for the full attack.

On one hand, I could add a "burst damage" and a "full auto damage" stat to the weapons, e.g. a weapon does 1d6 damage single shot, 2d6 burst or 5d6 full auto, but that makes full auto always the superior option, outside of the ammunition/reload mechanics.

I could roll every bullet separately, but that would result in a lot of rolls and/or calculations if I add modifiers to every roll. I thought about reversing it and making the DC DC - modifiers, and just rolling a d20 for every bullet, then counting all that are above the modified DC, but that also favors attacks with lots of bullets, unless I increase the DC for attacks with more bullets by some amount, though what should be the modifiers to achieve the expected result?

My current solution is to have players make a normal attack roll with the weapon specific maximum amount of bullets shot per attack type as negative modifier, and for every point above the DC, one bullet hits, up to the weapon specific number of bullets per attack type. For example, with a normal attack modifier of 10, a roll of 15 and a maximum of 3 bullets to hit with, the player could roll 2d6 for damage (15 (1d20 roll) + 10 (modifier) - 3 (max burst bullets) - 20 (DC) = 2 hits). This works, but it's fairly clumsy and still favors full auto. I thought about increasing the DC of burst to 21 and full auto to 22, to reduce the effectiveness, but it still feels clumsy.

My intention is for single shot to be a safe and reliable choice, burst to be common and full auto to be high risk, high reward. How can I transform that into dice roll rules with a small amount of individual rolls?

Ideally:

• With a high modifier (>15), full auto should have the highest expected damage per attack.
• With a medium modifier (10-15), burst should have the highest expected damage per attack.
• With a low modifier (<10), single shot should have the highest expected damage per attack.

Answers that use math to model the expected damage would be preferred. You can use 1d6 as damage per bullet, 3 bullets per burst and 6 bullets per full auto attack.

Answer 1

There may of course be many ways of achieving an effect like you want, but here is my suggestion:

One bullet hits for every point above the DC.

If every point above the DC is one hit, capped by the number of bullets fired, we need only vary the modifier for the various fire modes to achieve a system in which every mode is sometimes useful.

To disambiguate language, I'll use modifier for the to-hit bonus of the modes of fire, and alter the DC for any other possible modifiers. Note that other positive modifiers are equivalent to lowering the DC.

We can then talk about To-Hit(TH) and DC, and say that the amount of bullets that hit is equal to the following: \$\textit{min}({MAX}, \textit{TH}-\textit{DC}+1)\$ where MAX is 1 for single fire, 3 for burst and 6 for auto. The +1 is there to ensure TH=DC is still a hit.

Now, let's say that we give a +15 modifier to single fire, +10 to burst and +5 to full-auto. You will of course need to fiddle with these numbers depending on what typical DCs are and what you want to be the exact balance between modes of fire.

Example with DC=20

If we call the expected amount of hits X, we can calculate it as follows for the three modes of fire:

1. Single fire. \$X = P(1d20+15 \geq 20) = 0.8\$

2. Burst fire. \$X = P(1d20+10 = 20) + 2 \times P(1d20+10 = 21) + 3\times P(1d20+10 \geq 22) = 1.5\$

3. Full-auto. \$X = \Sigma_{i=0}^4 [(i+1) \times P(1d20+5 = 20+i)] + 6 \times P(1d20+5 \geq 25) = 1.05\$

So, if 20 is a typical DC, this would often make burst the optimal mode of fire, with auto being high-risk and single fire sacrificing damage for more hit chance.

As you wanted, lower DCs make auto better, and higher DCs make single fire better, as we'll see now.

Results with different DCs

This table gives the expected number of hits for each fire mode with varied DC. To get the expected damage just multiply by 1d6 or 3.5.

\begin{array}{r|lll} \text{Fire mode} & \text{DC 12} & \text{DC 15} & \text{DC 20} & \text{DC 25}& \text{DC 28} \\ \hline Single & 1 & 1 & 0.8 & 0.55 & \underline{0.4} \\ Burst & 2.7 & 2.25 & \underline{1.5} & \underline{0.75} & 0.3 \\ Auto & \underline{3.45} & \underline{2.55} & 1.05 & 0.05 & 0 \\ \end{array}

Underlined is the fire mode with the highest expected damage for each DC.

You will still need to tweak the numbers to achieve the balance you want, but this system is capable of creating roughly the desired result while only requiring a single d20 to roll to-hit.