Utility Closet: Balance Calculations for Damage, Armour and Penetration

Balancing damage and armour in GURPS is complicated. GURPS Pyramid 3/77 gave us the Combat Effectiveness Rating, or CER, to help mitigate this with a system that mimics the way CR works in Dungeons & Dragons. It's imperfect, obviously, since a single number can't accurately encapsulate how much of a threat a monster or bad guy is, but it's a welcome addition for rule-of-thumb balance in a system where one of the common worries for new GMs was combat balance. While reading through GURPS Ultra-Tech over the few years I've been playing the system, I've put together a few easy formulae to help understand the interactions between damage, armour divisors, penetrating damage and injury that can help a gamemaster understand what kind of threats they're placing in their players' path, so I'm sharing them with you down below.

Warning! This blog is finally living up to its namesake. Under the break you'll find mathematics as complicated as a square root! This is because we need to calculate a standard deviation, although I've done all the hard work and these calculations require absolutely no understanding of statistics beyond knowing what a mean or average is. All of these can be done with the basic calculator on your phone or computer. Bear in mind also that I have dyscalculia – if I can do it, so can you!

For these equations, N refers to the number of damage dice, M refers to the multiplier and A refers to the adds. For example, 6d+2 has N = 6, M = 1 and A = 2. As another example, 6d×3 has N = 6, M = 3 and A = 0. Remember that M will never be zero – if it's not stated, M = 1.

Stats 101
Calculating the mean damage of a weapon is simple, since the average result* of a six-sided dice is 3.5. Use this formula:

Remember operator precedence – do all the multiplication first, then add A. With one extra step you can work out the mean penetration, or the DR required to stop half of all attacks:

Likewise, working the same equations backwards allows you to work out the mean dice of damage needed to be able to penetrate a piece of armour half of the time:

In the last equation, the armour divisor is the divisor of the attack you're figuring out. You can omit it if you're not sure what kind of attacks the armour will come up against, and divide the mean dice to penetrate by it later.

Getting Tricky
This isn't the whole story, since dice rolls are variable. The easiest measure of the variability of a dice roll is the standard deviation. As a rule of thumb, 68% of all rolls will be within one standard deviation of the mean and 95% of all rolls will be within two standard deviations of the mean. If you have trouble grasping what this means, read the Examples section below. The fact that GURPS uses six-sided dice exclusively makes calculating the standard deviation very easy:

For ease of comparison, you can work out the "typical damage range" of a weapon. The minimum of the typical damage range is the mean damage minus the standard deviation and the maximum is the mean damage plus the standard deviation. 68% of all rolls will fall within the typical damage range, so this is a good ballpark for it's most common range of damage.

You also can work out the typical penetration range. This is the same as the typical damage range but each value is multiplied by the weapon's armour divisor. If you're looking for armour to protect against a specific weapon, this is an important value! If you have a DR equal to the maximum of the typical penetration range, only 16% of shots can ever penetrate your DR, and these will only do minimal damage.

Examples
A laser pistol (GURPS Ultra-Tech p. 115) deals 3d(2) burn damage. It's mean damage is 3 × 3.5 = 10.5, which is enough to take an unarmoured person from full health to unconsciousness in one shot! The mean DR it can penetrate is 10.5 × 2 = DR 21. The standard deviation is √(3 × 35 ÷ 12) = 2.95. Rounding this up to 3 points, the typical damage range is 7.5–13.5 points. Multiplying these by 2, we get the typical penetration range as 15–27 points of DR. If a target has DR 27, only 16% of shots from a laser pistol will penetrate and at maximum damage, these shots will only ever deal 4.5 points of damage!

The Commando Battlesuit (GURPS Ultra-Tech p. 183) has DR 105. The mean dice of damage required to penetrate is 105 ÷ 3.5 = 30d. This is quite a lot! But at TL10, armour divisors are common. Lasers have a (2) armour divisor so a laser weapon needs 30 ÷ 2 = 15d to penetrate on an average roll. Gauss weapons perform even better, requiring only 30 ÷ 3 = 10d to penetrate. At this point, the Gauss Minigun, dealing 10d(3) pi-. can penetrate half the time, while the Gauss HMG and Portable Railgun, dealing 16d(3)pi- and 5d×3(3) pi- respectively, can both reliably penetrate the battlesuit.

* – Technically, I'm referring to the expectation value, although in everyday speech this is often called the average.