Let’s talk Dungeons & Dragons!
I’ve been spending a bit of time with Baldur’s Gate 3, rolling dice, passing and failing checks.
Usually, to do an **ability check** you roll a 20-sided die (`d20`

), compare what you rolled with a *difficulty class* (DC) of the check, and if it’s greater than or equal to the DC — you succeed the check, otherwise you fail.

But what if you roll the die **two** times and then pick either the higher or the lower number?
This is called respectively an **advantage** and **disadvantage** in D&D.
It feels that the effects of advantage and disadvantage on the chance of success should be similar.
Yet, it couldn’t be further from the truth!

## Probabilities

Let’s abstract from a physical die and imagine that a one-shot chance of succeeding a check is \(p \in [0,1]\). In other words, \(\mathbb{P}(\mathrm{success})_\mathrm{default}=p\) which gives us a nice straight-line chart.

Now, let’s consider what happens when we add the **advantage**. You fail a check *with advantage* when you fail your one-shot checks both times. Success is a complementary event, which means that

\[ \mathbb{P}(\mathrm{success})_\mathrm{adv} = 1 - (1 - p)^2 = 2p - p^2 \]

Since \(p \geq p^2\) when \(p \in [0,1]\), you get a nice boost to your chance of success compared to the one-shot probability \(p\). This can be illustrated with the following chart:

Finally, when it comes to **disadvantage**, you succeed only when you succeed both your one-shot checks which gives us:

\[ \mathbb{P}(\mathrm{success})_\mathrm{dis} = p^2 \]

The charts for **advantage** and **disadvantage** look *somewhat* symmetrical.
What the fuss is about?

## Relative effect

Things start to get interesting when we look at the *relative* effects of advantage and disadvantage^{1}.

First, let’s look at the change in the chance of success relative to the *one-shot* chance of success:

\[ \mathbb{P}(\mathrm{success})_\mathrm{adv} / p = 2 - p \]

The improvement that comes from the *advantage* is clamped between \(1\) and \(2\), which is a nice bump but nothing extraodinary.
Exactly, as you would expect from a mature game system.

Now, let’s look at the effect of the disadvantage, in particular at *how much worse things get with disadvantage compared to the one-shot probability*.

\[ p / \mathbb{P}(\mathrm{success})_\mathrm{dis} = p / p^2 = 1/p \]

Just by looking at this formula, it’s clear that things are not looking good. And even more so the smaller \(p\) is!

Plotting these charts side-by-side, there’s much less symmetry than originally anticipated:

## Conclusion

In the best/worst-case scenario, when a one-shot chance of success is \(1\) out of \(20\), having **advantage** would *almost* double your chances, but with **disadvantage** you’d be \(20\) times less likely to succeed!
Overall, **disadvantage** has a disproportionally large impact on the odds of success compared to the effect of **advantage**.
This is true for the whole range of one-shot probabilities \(p \in (0, 1)\), with the gap becoming visible at \(p\) around \(\frac{3}{4}\) and apparent at \(p \leq \frac{1}{2}\).

This is an interesting asymmetry in what — at first glance — supposed to be a symmetric game mechanics.
I’m not sure if it’s particularly useful outside of the world of Dungeons & Dragons, but the next time I play Baldur’s Gate 3, I’ll be more serious about picking *disadvantage*-inducing spells to debuff enemies rather than just throwing fireballs at them.^{2}