Purpose And Scope
The 2021 edition of "Warhammer 40,000: Kill Team" (KT21 for short) is a skirmish-sized wargame played with miniatures and dice. In KT21, it's a bit complicated to calculate average/expected damage for a shooting attack. Also, it's a bit surprising that improving the accuracy of a weapon by some proportional amount increases the expected damage by an even higher proportion. This post explores the quirks and nonlinearities of expected damage from shooting attacks.
Perhaps the most important thing to realize: number of attack dice, ballistic skill, and save characteristic all nonlinearly impact average damage (even if we ignore crit stuff). Attacking stuff super-linearly affects damage (increasing returns). Save characteristic sub-linearly affects damage (decreasing returns). Damage is hard to reason about.
For instance, doubling the number of attack dice will more than double the post-save damage. How much the post-save damage increases depends on everything else (BS, SV, special rules, auto-successes from cover, etc).
Update: I made my own KT21 Calculator.
Background, How Shooting Works
For calculating damage from a shooting attack, here's an incomplete list of steps that ignores crits and special rules...
- Attacker rolls a number of attack dice equal to their attack ("A") characteristic.
- Each attack die result greater than or equal to the attacker's ballistic skill ("BS") characteristic is a successful hit.
- Defender rolls a number of defense dice equal to their defense ("DF") characteristic.
- Each defense die result greater than or equal to the defender's save ("SV") characteristic is a save and can cancel out a hit.
- If there are any hits remaining after cancellation, those hits do damage.
Example And Empirical Results
If you go to my KT21Calculator (or this Kill Team Simulator), you can simulate shooting attacks with various parameters. It was with a simulator that I first noticed the nonlinear affect of shooting accuracy on damage.
The mechanic of critical hits (and critical saves) slightly complicates evaluation of damage from changes in BS. If BS goes from 5+ to 4+, that doubles the chances of normal hits, but crit hit chance remains the same. An easier thing to evaluate is adding ceaseless to a weapon, which should multiply normal and crit hit chances by 7/6, which is roughly 1.17. You'll find that adding ceaseless multiplies post-save damage by more than 7/6.
So let's pretend a we are using weapon profile {A=4, BS=6+, D=3/4} upon a target with a 4+ save. The pre-save damage is 2.67 and post-save damage is 1.30. If we add ceaseless to this weapon, the pre-save damage is 3.11 and post-save damage is 1.57. Adding ceaseless multiplied the post-save damage by 1.21 rather than 1.17.
Adding ceaseless proportionally increases damage by even more if the BS is improved. Imagine the same weapon profile except with far better BS: {A=4, BS=2+, D=3/4}. Against same target with save=4+, the post-save damages are 6.08 without ceaseless and 7.72 with ceaseless, resulting in a multiplier of 1.27.
Similarly, increasing the number of attack dice gives a larger increase in post-save damage. Going from {A=4, BS=2+, D=3/4} to {A=8, BS=2+, D=3/4} doubles the attack dice, but the post-save damage goes from 6.08 to 16.48, a multiplication of 2.71.
I find these nonlinearities to be unfortunate, because it makes it harder to compare weapon profiles, and even more importantly: it shows that post-save damage is hard to calculate and reason about.
Why Nonlinear?
Why is post-save damage nonlinear and complicated? The short answer: because damage takes the form of subtractive "hits - saves" rather than WH40K's multiplicative "HitRatio * WoundRatio * FailedSaveRatio". In WH40K, in each step of a shooting attack resolution, you roll as many dice as were successes in the previous step; that method leads to where everything is cleanly multiplicative, where doubling attack dice will double final damage. In KT21, the number of defence dice doesn't depend on the number of hits, so you can't just multiply things.
Here's a thought experiment that crisply shows the nonlinearity of "hits - saves": imagine another game similar to KT21, but there are no crits and targets always have exactly 1 successful save. If a weapon profile has A=0 or A=1, the post-save damage will always be 0. But with A=2 or better, you start getting hits, a proportionally "infinite" improvement over 0 damage. Also, going from A=2 to A=3 is a 50% increase in attack dice but gives a 100% increase in damage.
What Type Of Nonlinear?
There are many types of nonlinear functions, so it's very vague to say that BS has a super-linear effect on post-save damage. I did a toy example with two attack dice, two defense dice, and no crit mechanics. For a given save characteristic, average post-save damage is a quadratic function of BS.
Expected number of damaging hits: -2*h^2*s^2 + 2*h^2*s + 2*h*s^2 - 4*h*s + 2h. "h" is hit chance for each attack die and "s" is save chance for each defense die. For an 'h' oriented view: h * (2 - 4*s + 2*s^2) + h^2 * (2*s - 2*s^2 ). And if s=1/2, then we get (h - h^2) / 2...so we know I made an error somewhere, dang.
I also did a toy example with two attack die and one defense die, and it's expected-number-of-damaging-hits formula was h^2*s - 2*h*s + 2h = h*2*(1-s) + h^2*s.
It seems reasonable to conclude that for any number of attack dice, the formula will contain "h" terms with exponents up to that number of attack dice (similar for defense dice). So, for 5 attack dice and 3 defense dice, the formula would contain a h^5*s^3 term and terms with lower exponents.
How Can Better Understanding Be Used?
How do we use this knowledge to our benefit? There are a few ways:
- Fusillade's splitting of attack dice really weakens each attack. You should downgrade your valuation of Fusillade. This reddit post discusses the very limited circumstances where fusillade is useful.
- Special rules, better BS, and more attack dice are better than you may think. Putting Relentless on a boltgun {4,3+,3/4} can almost double the damage against a Custodes Guardian.
- Rely on intuition less and specialized calculators more.
- Remember more specific facts (ugh).
Addendum1: Adding Attack Dice Vs Subtracting Defense Dice
Glass Half Dead (Andrew Lynch) asked about the difference in damage between a weapon with A=5 and a weapon with A=4 and AP1. In more general terms, what are the effects of rolling one more attack die vs rolling one less defense die? He realized that the answer depends on the BS of the attacker and the Sv of the defender. For example, subtracting a Sv=2+ die is intuitively more potent than adding a BS=6+ die. But what about when the BS and Sv characteristics are closer and more subtle effects come to dominate?
Here are some concrete scenarios all with BS=3+, NormDmg=3, and CritDmg=4 (like a bolter). All damage calculations done by ktcalc...
Looking at the BS=Sv=3+ scenario, we see a natural higher potency of a marginal attack die over a marginal defense die. I think a BS=3+ die is more impactful than a Sv=3+ die because defense dice are "wasted" more frequently than attack dice. A die success is "utilized" if that success affected the final damage and is "wasted" if that success doesn't affect the final damage. The average impact of a die depends both on the die's chance of being a success AND
the chance the success actually affects the final outcome.
For instance, if an attacker rolls no successes, then any defender successes are wasted in that they don't reduce the damage. Likewise, if an attacker rolls a normal hit and the defender rolls a crit, that crit did not live up to its full potential to reduce final damage by the critical damage amount. Perhaps most tantalizingly, if an attacker rolls a crit and the defender rolls a normal save, that normal save was wasted.
Attack successes can be wasted too, but it is far less often than for defense successes; the only scenario is where the attack did zero damage.
Therefore, an extra BS=3+ die is more likely to make an impact than a removed Sv=3+ die. I would guess that there are some scenarios where we could get an extra BS=3+ die to be more impactful than a removed Sv=2+ die, and here is one: {A=3, APx=0, AvgDmg=1.06} beats {A=2, APx=1, AvgDmg=0.96}. I just had to start out with fewer attack dice.
Hello! Thanks for the analysis, I just started KT and is was helpful. Just one comment : you're missing a scenario where attack sucesses are wasted. It's when you overkill the target. It's a very common occurence which may lead to trying to spreading out threats but the math behind it are very situational so difficult to model.
ReplyDeleteYou are correct, another category of wasted successes is overkilling the opponent. I think this softens my conclusions and adds some caveats. For instance, against a defender with 1 wound, your kill chances are probably best maximized with one fewer defense die rather than one more attack die. I hope to update this post with your excellent point. Thank you.
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