This post might be a little different from what I usually write, but I nerd-sniped myself into calculating this and I want to share it with people. In the game of Dungeons and Dragons, you frequently have to roll a 20-sided die to see if something works. As with most things, higher is better. There are a few ways you can boost the result of a roll with, but in this article we will be looking at two: Advantage and Proficiency.

First, proficiency is a straight bonus to each roll. This bonus starts at 2, and grows to up to 6 as your character grows. For advantage you roll two dice and take the higher result. The question I’m looking to answer: given the choice between having either advantage or proficiency, which one is better?

The reason that this question came up, is that there are two feats that improve your chance of maintaining concentration when you get hit by an attack: Warcaster and Resilient (Constitution). Due to licensing rules, I cannot include them here, but for the purposes of this post, the former grants you advantage on saving throws to maintain concentration, and the latter grants you proficiency in (among others) those very same throws. For anyone at Wizards of the Coast reading this: this article falls under fair use, and thank you for not suing me.

Assuming your character doesn’t have proficiency in Constitution saving throws already, which one should you pick?

## Expected outcomes

To determine which one is better, you can compute the expected value for both options, and determine which one is higher. The expected value is the sum of all outcomes multiplied by their respective probabilities, or in other words, $\sum_{x \in X} x P(x)$ where $X$ is the set of all possible outcomes and $P(x)$ the probability of $x$ occurring.

For proficiency, the outcomes are all of equal probability and are equal to the result of our 20-sided die plus our proficiency bonus. Assuming a fair die, this is equal to $bonus + \sum_{i = 1}^{20}\frac{i}{20} = 10.5 + bonus$.

We can compute the same thing for having advantage, but here our distribution is slightly more complicated. There are 400 possible outcomes for rolling two 20-sided dice. The probability to roll a particular number $n$, we can count the number of combinations to do so, and divide that by 400 to reach the probability of getting $n$ as the final result.

Since advantage gives us the maximum of two rolls, there are three cases that can give us $n$. The first case is where both dice roll an $n$; this occurs once. The second case is where the first dice rolls $n$ and the other is lower, and the third case is the opposite. Since there are $n - 1$ numbers lower than $n$ on the die, this happens $2(n - 1)$ times, for a total of $2(n - 1) + 1 = 2n - 1$ options. I’ve plotted these probabilities below.

If you then plug the numbers, you will get an expected outcome of $13.825$. So as long as your proficiency bonus is less than $3.325$, advantage should be better, right?

## Binary thresholds

In general, higher may be better, but in practice, you are rolling to see if you succeed at some particular task, which you either do or don’t. In this case, we are not as interested in the actual result of the dice roll, merely whether it is higher or not than some thresholds. So now the question: how does that compare? Well, it compares like this:

Here we can see that having advantage is slightly better around the middle of the spectrum and slightly worse at the extremes. This slight edge becomes less and less as the proficiency bonus increases, and a proficiency of at least 5 is objectively better. This level of proficiency is reserved for late-game level characters only, so you won’t get to enjoy it as often.

Now is the part where we remember why we computed all of this: saving throws to maintain concentration. The threshold for this saving throw is $\max(10, \frac{\text{damage taken}}{2})$. Hits of over 20 damage are rare, especially for lower level characters, so the threshold will generally be right around where the optimum for advantage is. This gets shifted slightly when you consider that most characters will have a small bonus to constitution, but the effect remains.

## An example

Graphs and math aren’t the clearest to everyone, so let’s just look at a concrete example. We will be looking at two brave adventuring Wizards, Albus and Bigby, who both sport a wonderful +2 to Constitution. Albus took up Resilient (Constitution) while Bigby took Warcaster. They are both fairly average, so they don’t roll for their hit points and instead just have $8 + (4 + 2) * (\text{level} - 1)$ of them.

When we first encounter our adventurers they’re at level 5 fighting a goblin army with some more martial oriented friends. They’re both doing their best slowing the armada but unfortunately two of them got through and stabbed them for a perfectly average $5$ damage. Now they need to make a DC 10 saving throw to maintain concentration. Albus gets to use his +3 proficiency and his +2 constitution, so he needs to roll a $5$, which gives him a $\frac{20 - 4}{20} = 80\%$ chance of making it. Bigby just adds his constitution, so he needs to roll an 8, but he gets to do so with advantage. His odds are $\frac{15 + 17 + \dots + 37 + 39}{400} = 87.75\%$. These are good odds for the both of them, but Bigby has the edge.

The second time we see our friends they are at level 9 and engaged in magical fisticuffs. They are both once again concentrating on their own spells but they are hit by 4th level fireball (presumably since they never learned not to stand next to eachother) and take an average 32 fire damage, requiring another saving throw, now of DC 16. Albus’s proficiency improved to 4 (note: higher than 3.825) so he needs to roll a 10, giving him $\frac{20 - 9}{20} = 55\%$ chance of success. Bigby needs to roll a 14 now after applying his constitution bonus. This gives him a chance of $\frac{27 + 29 + \dots + 37 + 39}{400} = 57.75\%$ to succeed. Despite Bigby’s expected advantage now being lower than Albus’s proficiency, he still has a slight edge over Albus in maintaining concentration when taking a hit that took over half of their measly 56 average hit points.

Finally, the wizarding duo is squaring off against the BBEG of their adventure, still at level 9. Having learned from their previous mistakes, they no longer stand next to each other, but now are in a perfect line to be in the path of a 6th level lightning bolt, taking an average 40 damage, requiring a DC20 save. Bigby tries his luck again with his advantage, but the odds for him rolling a 18 are merely $\frac{35 + 37 + 39}{400} = 27.75\%$. Since Albus only has to roll a 14, his odds are a much more solid $\frac{20 - 13}{20} = 35\%$. Surprisingly, with the same proficiency and the same modifier, Albus can both be at an advantage and at a disadvantage compared to Bigby.

## In conclusion

If you just look at making your saving throw to maintain concentration, Advantage will win over Proficiency in the lower levels, and will slowly be outclassed in the later levels. This lets you choose your own winner, depending on the type of game you’re playing.

If you look at the other aspects of the two Feats, Resilient is just more bland, and doesn’t add much in the way of character while Warcaster does. This may be true and may also just be my opinion. Then again, why would anyone play role-playing games for the role-playing?

This article was edited to include a section with examples based on feedback on Reddit. If you have any other comments, feel free to leave them there.