Effective HP versus: d20

The most famous RPG is Dungeons & Dragons, and its core mechanic is rolling 1d20, adding a modifier, and comparing it to a target number such as Armor Class (AC).

It’s often said that rolls in 5e D&D feel “swingy”, and often in a way that implies that this is bad. If this level of “swinginess” is indeed bad, there’s a simple way that a d20 system could be made less so: if we multiplied all the modifiers by 10, the d20 would mean almost nothing, and the modifiers almost everything. Even a more reasonable multiplier of 1.5 or 2 would make the game feel less “swingy”. Likewise, we could make the system less swingy by replacing the d20 with a smaller die, such as d10 (and subtracting 5 from ACs and DCs).

The designers of 5e explicitly chose to do exactly the opposite of this under the doctrine of “bounded accuracy”, which restricted the scale of modifiers, especially AC, compared to previous editions. I don’t think “swinginess” was a mistake, but I also don’t think it a design goal of 5e — it was a side-effect of this design decision.

Effective hit points

The bounded accuracy article rightly focuses on AC, but let’s put some more concrete numbers to it. We’ll consider a concept of effective hit points (EHP). While this is a common phrase, its meaning depends on context. In this context, we’re going to define it as the mean amount of damage needed to destroy a target, including damage lost to misses.

If the chance to hit the target is p, it will take a mean of 1 / p attacks to hit the target once, as determined by the geometric distribution. Therefore, the target’s HP is effectively multiplied by 1 / p. We’re going to call 1 / p the EHP multiplier. In turn, the chance to hit is equal to the tail distribution of the dice and modifiers we are rolling, which can be seen using the “At Least” button on AnyDice. In summary:

  • Take the tail distribution (“At Least”) in AnyDice.
  • Compute the reciprocal to get the EHP multiplier.
  • Plot it.

This will help us visualize the value of each point of modifier.

Note that EHP multiplier doesn’t behave exactly like getting more raw HP. A very large EHP multiplier corresponds to a very small chance to hit. This tends to make combat less pleasant to play:

  • If it takes many hits to destroy a target, a very small chance to hit means that a very large number of attacks will be needed to destroy it, dragging combats out.
  • If it takes only a few hits to destroy a target, combat becomes determined by random lucky hits.

Therefore, we will tend to want to keep the EHP multiplier from getting too large.

So why bounded accuracy? If we graph EHP versus the roll needed to hit on a d20, we get this:

As the roll required to hit approaches 20, the EHP suddenly starts shooting to the moon. If it weren’t for the rule that natural 20s always hit, a 21 needed to hit would result in being completely unhittable, with infinite EHP.

There’s also a severe discontinuity in the value of each point of AC. If we are only hit on a 19+, our effective hit points are already 500% what they were at a 10+, and just one more point of AC will double this to 1000%. The next point after that either has infinite effect (if natural 20s don’t always hit) or zero effect (if they do).

Finally, such low chances to hit are generally not fun to play.

How can a RPG avoid these problems? One way is to restrict the modifiers so that we generally stay out of the right tail where these problems occur, hence bounded accuracy. For a d20 system, I think this is the right choice.

This series

But are there other solutions? If we could find some distributions that didn’t have this extreme behavior at the right end of the graph, that could allow us to more “safely” scale up our modifiers and hopefully make the system feel less swingy (if we want to).

In this series, I plan to cover the following:

  • What are some other possible probability distributions?
  • What do their corresponding EHP curves look like, especially towards the right side (increasing AC)?
  • How can we implement those probability distributions using physical dice without making things too complicated?

For this series, I’ll stick to roll + modifier versus target number (aka “roll over”) systems with binary hit/miss outcomes. This is by no means the only type of system, but it’s simple to use and the simplest to analyze, which is why I chose it for this first series.

We’ll continue next time with sums of standard dice, or XdY. If you’re in a hurry, you can skip to the fourth part of this series for the main payoff.

Series index

  1. Effective HP versus: d20 <- You are here
  2. Effective HP versus: XdY
  3. Effective HP versus: exploding dice
  4. Exploding dice: adding standard dice
  5. Exploding dice: opposed rolls

Articles about probability in RPG design.

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