### comparisons of combat rules in tabletop games

i'd like to look at statistically equivalent methods of handling combat in tabletop games, and draw out their differences with respect to how they affect the play experience.

## roll to hit and for damage

this is the method used in the platonic *dungeons & dragons *game. let there be some target that has 1-8 hit points (average: 4.5), and an attacker that can cause the target to lose 1-6 hit points (average: 3.5). furthermore, let there be a 50% chance of the attacker hitting the target. taking into account that half of all hits are misses, the average amount of damage dealt by the attacker is 1.75 hit points. it will statistically take **2.57 rounds** for the attacker to totally defeat the target. furthermore, there is an initial **21.88%** chance that the attacker will defeat the target in one round.

if the target instead had 1-6 hit points, as in the original 1974 *dungeons & dragons*, the chance becomes **29.17%** and it will take **2 rounds** on average to defeat the target. it becomes apparent here that since on average d8 ≅ d6 + 1, and since there is a 50% chance to-hit, the target's "lifespan" is increased by 1 / 2 rounds when switching from d6 to d8 for hit points.

this way is basically the 'default' method, so i'm going to hold off from making comments until i have compared it with other methods.

## roll for damage only

this method was popularized (if not originated) by *into the odd*. let us assume as before that there is a target with 1-8 hit points, and that the attacker can cause the target to lose 1-6 hit points. as it were, this method has a 100% "to-hit" chance. we can expect the attacker to defeat the target in **1.29 rounds**, and there is an initial **43.75%** chance that the attacker will defeat the target in one round.

keeping the same values as before, combat seems much deadlier. if you wanted to retain the same countdown towards defeat while removing to-hit rolls, you necessarily have to increase the target's hit points. let the target now have 2-16 hit points, so that now it will be defeated by an attacker of 1-6 in **2.57 rounds**. in this respect, we have made this method and the one with non-zero miss chance statistically equivalent. however, at an average of 9 hit points, there is a **0% **chance that the attacker will defeat the target at a damage rate of 1-6.

there is a blog post that i can't find for the life of me, that the to-hit roll is simply abstracted hit points since it prolongs the lifespan of the target. however, even when the countdown to defeat is statistically equivalent between these two methods, they necessarily result in different experiences of play. it seems to me that the purpose of the to-hit roll is not just to prolong the lifespan of the target but, literally and directly, to introduce the likelihood that a hit fails altogether. that sounds obvious, but it's a distinction worth drawing out with different implications for play than simply increasing hit points.

## roll to hit only

this method is found in the board game *dungeon!*, likely to avoid extra bookkeeping for a light take on the dungeon crawl experience. i've also seen it used in games which, besides reducing bookkeeping, also want to have a more brutal (and perhaps realistic) feel to combat: either you hit and defeat the target, or you don't. if we want it to take **2.57 rounds** to defeat some target, we need a to-hit chance of 38.9%. this chance of defeating the target is always the same round to round.

you might notice, then, that the purpose of having hit points is none other than to act as memory: the more you hit without defeating the target, the higher the chance you will defeat the target each subsequent round. having no hit points means that the chance of defeat is constant per round.

when you increase the number of total hits that can be sustained from 1 to any other number, you run into the same effect as increasing hit points: there becomes a zero chance of the target being defeated in one round if that target is not being hit multiple times.

## the end

we have seen that the functions of losing hit points and rolling to-hit are different, even if in their own way they prolong the lifespan of a target. we can better differentiate these methods according to their literal effects:

- method 1: the more you successfully hit a target, the higher the chance you will defeat them
- method 2: the more you hit a target (100%), the closer you are to defeating them
- method 3: there is a % chance of hitting a target and defeating it
- method 3b: a target has a number of successful hits it can sustain before defeat

yes, thank you! ava actually messaged me soon after i posted this to say it was her post haha, i was embarrassed

ReplyDeleteI'm curious if you could maybe elaborate a bit on what you mean by saying that variable/constant damage/hp doesn't matter? Unless this is speaking specifically to low-hp scenarios?

ReplyDeleteI'm mostly asking as I've found these various methods (regardless of statistics) to drastically effect the emotional pacing of combat and thus worthy of further investigation.

Generally I've found that more Variable damage systems lend to more emergent play experiences (where players let loose the beyblade of their character and get to watch and narrate the results). Whereas more Fixed systems (especially with tighter damage curves) offer players more direct agency and are better suited to "Combat as a tactical challenge" wherein designers can build challenging choices where players are able to make informed combat decisions whilst the clock is always ticking towards one side or the other losing a war of attrition.

(That said I'm also realizing when I talk about this I'm talking about very low variance in the random damage- As I can see the argument that high variance damage serves a near identical function. Though I'd still attest it helps keep combat moving forward to a finite conclusion when time at the table is often and all-too-precious resource.)

But maybe the point of this article totally went over my head, so let me know what your thinking was!

hi there, thank you for your question! it was definitely something i said without thinking too much about it, because you're right that whether damage is constant or variable really affects the outcome of combat (as randomness would do to anything else!). i think what i meant was that with respect to the schema i outlined, i.e. methods 1-3, the math does not change much if you make hit points or damage constant or variable :) they have different effects on the game, but they can be understood as having no effect specific to any of the methods.

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