The Mathematics of Expertise

What is Expertise?

Warning! This article contains mathematics!

Of all the various mechanics of 5e, ability checks are certainly my favorite one. They are a simple tool that can solve many conflicts. Player vs creature, player vs environment, or sometimes, player vs player. To check what a character knows of the Killer Rabbit, the DM asks them to roll History. They roll a d20, add the relevant ability score (here Intelligence), and then, if they are proficient, their proficiency. If they roll high, they learn that the Killer Rabbit is a mighty enemy, even stronger than an adult dragon or a vampire. Indeed, only a group of courageous and determined adventurers could take it on, and hope to survive to tell the tale.

Expertise make it that instead of adding your proficiency, you add it twice. This nifty feature is only available to Rogues and Bards. Hence, Rogues and Bards are not only the best at skills in general — Rogues get four, Bards get any three and Jack of All Trades — but they also get Expertise in two skills, up to four at later levels. The introduction paragraphs of the Bard and the Rogue justify it as well:

The greatest strength of bards is their sheer versatility. […] They have a wide-ranging knowledge of many subjects and a natural aptitude that lets them do almost anything well. Bards become masters of the talents they set their minds to perfecting, from musical performance to esoteric knowledge.

Player’s Handbook, p51

They have a knack for finding the solution to just about any problem, demonstrating a resourcefulness and versatility that is the cornerstone of any successful adventuring party. […] Rogues devote as much effort to mastering the use of a variety of skills as they do to perfecting their combat abilities, giving them a broad expertise that few other characters can match.

Player’s Handbook, p94

The mechanics fit perfectly the flavor. Taking it further, the Bard is a genius, and the Rogue has great wits, both above everyone else. Meanwhile, a Wizard cannot ever be better at Arcana than a Bard with Expertise. Same issue with Cleric and Religion. Or Fighter with Athletics. Especially in fact grappling checks: the best grappler is a Rogue! This is not simply a matter of mechanics, an unfortunate side effect so to speak, but intended.

Of course, those are mostly tropes; a Wizard doesn’t have to be good at Arcana to be a great Wizard. But the selection of skills available to each class reflect those tropes. For instance, a Fighter get access to Acrobatics, Animal Handling, Athletics, History, Insight, Intimidation, Perception and Survival. Meanwhile, a Rogue get access to Acrobatics, Athletics, Deception, Insight, Intimidation, Investigation, Perception, Performance, Persuasion, Sleight of Hand and Stealth. The Rogue’s text explicitly points to this limited set of skill:

Many rogues focus on stealth and deception, while others refine the skills that help them in a dungeon environment, such as climbing, finding and disarming traps, and opening locks.

Player’s Handbook, p94

Yet, because of how the skill system works, a Bard or a Rogue can pick expertise in any skill they are proficient in. This includes the ones they get from their background or race. Furthermore, Rogue and Bard can easily cover the most useful adventuring skills (Athletics, Acrobatics, Stealth, Perception, Investigation). Of course, the other skills have their use and can sometimes provide an amazing result — see my example of the Killer Rabbit. But those skills rarely decide life and death for the party, compared to combat and dungeoneering.

Expertise’s dominance somewhat mitigated at the first level by the ability dependence of each class. A Rogue could have expertise in athletics, but while they might have a +1 or +2 in Strength, the Fighter could have +3 at first. In general, the system works.

And you get one expertise, and you too, and you too…

Wizards proposed a solution with one of their Unearthed Arcana, where some feats grant expertise. However, WotC did not include it in Xanathar’s Guide to Everything.  Given WoTC policy about UA, this probably means there was no popular support. Most likely, players felt Expertise was too unique of a class feature to be given like that.

Yet, as I am running a game with little combat, the power of skill proficiency and expertise increase comparatively. From level 9 to 12, Expertise means a bonus of +12 with a +4 in an ability mod. At first, I thought this was an issue only with my type of game. But I found out that others shared that sentiment, even those running a fair amount of combat.

When I discussed the topic with some people, I noted that most agreed Expertise was a fine, simple and elegant mechanic. But at the same time, a few noted that it made ability checks trivial, and involved big modifiers. And they disliked that, because they were fierce supporters of the Bounded Accuracy principle. However Bounded Accuracy applies to Combat. The numbers are balanced so that a party of level X can take on an encounter of CR X, no matter their build, by the simple power of their normal attacks and cantrips. Even if Expertise explodes all boundaries, the spirit of Bounded Accuracy doesn’t apply to it. After all, the mechanics of the game focus on combat.

I am still very sensible to the argument that Expertise trivializes Ability checks. Let us see if that is true.

What is relative increase and why it matters

Here is another important fact I cannot understate enough. You need to look at both the absolute increase and the relative increase to gauge the power of a bonus. For instance, increasing your Dexterity from +2 to +3 increases your odds by 5% on any roll where you add your Dexterity modifier. What about relative increase? This table shows its importance.

A 5th level with a Dex+3 has a 75% chance to hit AC12, while someone with a Dex +4 has 80% chance. The absolute increase is 5% over the board. Yet the relative increase is quite different, at AC 10, it is of 6%, but 11% at AC18, a significant boost. Hitting AC between 1 and 12 doesn’t really matter. It’s easy, and that’s it. What matter are the most difficult cases. Hence, in a system with Bounded Accuracy as one of its design principles, ASI is a powerful feature.

Now, let’s take a look at Expertise’s mathematical impact. Unlike AC, the DC of an Ability Check can climb up fairly easily in certain situations, even at low levels. It’s also often a one-time thing. When you fail an ability check — one that is not used in combat like grappling — you are unlikely to repeat it immediately.

The Mathematics of Expertise

Because we are dealing with uniform probabilities with some slight modifications, a lot of the results yield the same conclusion. Expertise is strong. But getting to see the overall picture is still useful. And with some other results that don’t follow a uniform law, we can refine our findings.

You can find the Google Spreadsheets recapitulating all results here.

In the rest of this article, we use the following short-hands:

  • Tom is the skilled character, Kira is the expert character.
  • A primary skill is a skill with a high ability modifier (+3 for level 1-4, +4 for 5-8, +5 for 9-20).
  • A tertiary skill is a skill with a low ability modifier (+1 for level 1-4, +2 for 5-20).

Ability checks DC

The Player’s Handbook suggests the following difficulty scale for Ability Check DCs:

Difficulty Very Easy Easy Normal Hard Very Hard Nearly Impossible
DC 5 10 15 20 25 30

But thanks to the power of spreadsheets, we look at the full 10-30 DC range.

Weighed Distribution

Not all DCs are equal. Some occur more often than others. For this reason, we weigh DCs according to the following rules:

  • The highest weight, 5, matches the 50% DC for the tertiary skill of Tom.
    For instance, at level 1, Tom might be proficient in Stealth with a Dexterity modifier of +1. He would have 50% chance to hit 15, hence we put the more weight on 15. Coincidentally, this is also what the Player’s Handbook call normal.
  • The nearby weights are set to 3, 2 and 1.iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
  • Then the tails are more arbitrary according to the level range, ranging from 1 to 0.10 in weights.

Here is an example of how the distribution of weights looks, for a proficiency equal to 5:

Our choice is arbitrary, but this distribution favors Tom and not Kira. At the end tail of the distribution, Tom has very little chance of success, while Kira still has decent chances.

Expertise across the levels

First, let’s take a look at how Tom and Kira’s skills progress as their level do.

Quick glance from the Mega-Sheet

No surprise, we can see how Expertise shines more and more while proficiency increases. At proficiency 4, Kira has a 50% chance at hitting a DC23, at proficiency 6, a DC 27. Kira reliably hits average DCs, if not always at higher levels, and has a great shot of hitting 25+.

First Decile, Median, Last Decile

We look at the first decile, the last decile and the median of DCs.

For a tertiary skill

How to read that table:

For instance, Tom has a 90% chance of hitting a 5 with a proficiency of +2 and a mod of +1. Or Kira has a 50% chance of hitting a 24 with a proficiency of +2 and a mod of +2.

These graphs show the evolution of the first decile, median and last decile in function of proficiency. Kira is red, Tom is blue.

The point where the slope changes is the result of an increase of the Ability modifier by 1.

For a primary skill

Just a simple commentary: thanks to expertise, Kira with a proficiency of +4 hits 15 with 90% chance, and 23 with 50% chance!

These graphs show the evolution of the first decile, median and last decile in function of proficiency. Kira is red, Tom is blue.

Absolute and Relative Increase

Now, let us measure how much better Kira is than Tom.

Absolute Increase

Expertise proves to be quite reasonable at the lowest levels, as it does not guarantee success for average DCs (see numbers above). But absolute increase shouldn’t be our only metric, as I explained earlier. We need to look at relative increase to see how much better Kira is than Tom.

Relative Increase

We look at the relative improvement for the Primary Skill and the Tertiary skill.

Expertise has a much higher value than an ASI number wise. The advantage increases exponentially with proficiency. But this needs to be considered in the context of both the existence of 18 skills, and the fact that some of those are much more useful in combat than others.

By what margin does expertise outshine proficiency?

In 5e, the designers have given a lot of thoughts to balance so that no class absolutely outshines others. The Rogue is a master of stealth; the Bard can support the whole team without stealing their thunder; the Paladin is a rock, yet they can’t do what Wizards can. Obviously, someone with expertise is going to outshine someone simply proficient, all other things equal. Yet, outshining doesn’t mean a perfect absolute domination. Because of the volatility of the d20, it is possible for the one simply proficient to beat the expert. What are the odds of that?

What are the odds for someone proficient to roll better than the expert?

While we could look at the odds of Tom beating Kira (and you can find them in the spreadsheet), what truly matters are the odds of Tom beating Kira, knowing Tom succeeded. Indeed, what good is it to have beaten the expert, if you fail the task? Let’s look at the probability of Tom beating Kira, conditioned by Tom success.

Tertiary Skill

Primary Skill

Tom can still contribute…at least around the average DC

The numbers were a little surprising to me, I expected that Tom would have had a harder time beating Kira. Don’t be confused by the fact that the higher the DC, the highest the odds of beating the expert. Since we are assuming Tom’s success, Tom has to roll a high value like an 18 to be able to beat high DCs. Kira on the other end is rolling between 1 and 20. What are more interesting are all those 40-60% odds at the lower frontier. When we consider Tom’s odds at beating average DCs and add this information, we can conclude that Tom still has its chance to contribute. He is not useless. Of course, Kira has a high chance of killing it, but that is the intended power of the Expertise feature.

For higher DCs, the high odds are deceptive, since in the first place, Tom is very unlikely to reach those numbers. Meanwhile, Kira has a serious chance of doing so. For instance, Kira, with a proficiency of +4 and a mod of +5 beat 23 with a 50% chance.

Reliable Talent

Rogues are in a unique situation because of Reliable Talent. In normal games, that’s perfectly fine. The rogue gets to feel cool, they are the go-to skill solving machine. And when they are not proficient in a particular skill, another character gets to save the day.

However, in a game where ability checks are pro-eminent, this is the equivalent of having an auto-hit 90% of the time (if not 100% given those high DCs) on certain skills. Meanwhile, at level 11, a Fighter has a 50% chance of hitting an AC of 19. If you run one of the few games that rely a lot on ability checks, then Reliable Talent is a detriment. With it, a single character could take care of almost all the skill challenges that have the most impact on play. I would consider that un-fun. There will be no choice but to adapt. For instance, you could forbid the rogue from rolling when they are not proficient, or deliberately challenge skills at which the rogue is bad at.

 Expertise trivializes ability checks in the late tiers of play

In a game where ability checks occur at a “normal” frequency, or where characters are between level 1 and 4, Expertise is a non-issue. Sure, Kira outshines Tom. But that is the power of the feature, and Tom can still compete with her, from times to times.

If you have a game that relies much more on ability checks than combat relatively to the standard, then Expertise is an extremely strong feature as we saw above. No other boosts to attack rolls or saving throws compare with it.  In my game, the problem is mitigated by the fact that narration and players’ ideas play a large role in how the game advance. Even if Expertise trivializes certain skill checks, those are often minor in the grand scheme of things. What matters more than any roll is role-playing.

Of course, I insist again, most tables won’t ever feel this is a problem. Especially when you consider that, according to WotC statistics, the vast majority of game don’t go beyond level 7.

Alternative to Expertise

Of course, now is the time to explore alternatives to Expertise for the few of us out there that would find a use for it. I already have some ideas about it. But this is an article I shall write later, I hope you look forward to it!

Next week

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