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 feels “swingy”, and often in a way that implies that this is bad. If it is indeed bad, there’s a simple way that a d20 system could be made less swingy: if you 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, you could make the system less swingy by replacing the d20 with a smaller die, such as d10+5.

The designers of 5e explicitly chose not to do 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.

To-hit numbers and effective hit points

So why bounded accuracy? The linked article rightly focuses on AC, but let’s put some more concrete numbers to it. Consider the effect that the roll needed to hit has on effective hit points (EHP). EHP is inversely proportional to the chance to be hit. In turn, the chance to hit is equal to the tail distribution, which can be seen using the “At Least” button on AnyDice.

If we graph EHP versus the roll needed to hit on a d20, taking a 50% chance to hit as our baseline, we get this:

The uniform distribution has a very non-uniform effect on EHP.

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 at a massive 500%, and just one more point of AC will double this to 1000%. However, with the rule that natural 20s always hit, the next point of AC has absolutely zero effect.

How can a RPG avoid these problems? One way is to restrict the modifiers so that we generally stay in the center range where we don’t have these problems, 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?

We’ll continue next time with sums of standard dice, or XdY.

Articles about probability in RPG design.

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