I’ve been playing Baldur’s Gate 3 lately, a game you’ve probably heard a lot about already. I’ve been surprised at how fiddly and fussy a game it is, and I cannot believe what a massive success it’s been ‘despite’ that. It feels like a thought experiment made real, one positing that players don’t need everything sanded down and rounded off for them. It’s hard to imagine a game like this making it through the American AAA studio system unscathed (and interesting to contrast some of the design decisions here to the Dragon Age games).
But anyway anyway, what I really want to talk about is, what gloves should you wear for lockpicking???
Solving Your Problems with Dice
First off, let me explain what we’re looking at here.
BG3 implements D&D 5th edition rules, which means a lot of dice based resolution. It does a lot of these rolls in the background, especially in combat. But for checks you attempt outside of combat, you get to actually roll a cute little die around on a dedicated resolution screen (which is always a nice touch).
So you do this for things in dialogue like persuading someone (or using animal handling to calm a giant eagle), and actions like picking locks or disarming traps. When you attempt any sort of skill-based action like that, you roll the famous D20, add some bonuses, and compare the result against the ‘Difficulty Class’ (DC) of the task. If your roll is equal to or greater than the DC, you succeed, and otherwise you fail.
So the formula would be something like this:
IF d20 + modifiers >= dc THEN success ELSE fail
You can get modifiers from a bunch of of sources, including a stat-based bonus, proficiency and expertise, and bonuses from spells and so on. But they all just combine together to an additive number.
More interesting is another mechanic called Advantage. When you “Roll with Advantage”, you roll two die and take the highest value. The gamefeel of this is simply great- you actually get to roll two dice! Seeing two dice roll around has much more visual interest than just adding a little number at the end like you do for modifiers.
And you can stack bonus modifiers with Advantage as well. So the pseudocode formula is maybe something like this:
IF MAX(d20,d20) + modifiers >= dc THEN success ELSE fail
Okay, so now you’re up to speed.
Five Finger Discount Codes
It’s an RPG, and so of course that means stealing everything that isn’t nailed down. But even if we weren’t always pickpocketing people, you come across a lot of locked chests and traps too. All of those use the same stat (Dexterity) and skill (Sleight of Hand) for checks.
So, as we’re outfitting the party’s designated appropriations agent, here’s the question: Which is better, a +1 modifier, or rolling with Advantage? The question is not academic, here’s two gloves you can come across early on:
(And as I was doing some research to write up this post, I saw I’m not the only one to wonder about this particular question)
Intuitively, my guess was Advantage would come out ahead (it’s two dice! Two is more than one!), so I went with the Gloves of Thievery, although I did wonder what the proper probability breakdown was. But then I came across these puppies:
Your ability score (which goes from goes from 1 to 20, or higher) confers a bonus modifier equal to FLOOR[(Ability - 10) / 2]. So those gloves setting your DEX to 18 mean getting a bonus of +4, which is a big bonus over my main character’s natural dexterity (13, so a bonus of +1). So now the question is- What’s better, Advantage, or a +4 modifier? (Or technically, I guess, Advantage and a +1 modifier, or a +4 modifier). Now I’m not so sure that Advantage has the advantage.
As with most problems in life, this means it’s time to open up Excel!
Brute Forcing It (Athletics check, DC 10)
The first thing I did was just brute force it, which is what I usually do. No need to remember probability math when you can just look at every possible outcome!
Make a column goes from 1 to 20, and another of just 1 repeated. Then after the first column, a formula of =A2, and for the second, =B2+1. Then drag those down and repeat till you have 400 rows, and bam!, you’ve got every outcome. This is a simple iterator to generate every permutation, I use this sort of thing a lot when I’m being lazy. From there, we just need a column that takes the MAX of each row. Now we have a table of every single possible outcome of rolling two D20s and taking the highest value.
From there, we can use COUNTIFs on column C to see how many times a number comes up that’s higher than 20, 19, 18, and so on.
I’m going to skip over the rest of this part. But if I was just answering this question quickly, this is really all I’d do (or even less, in reality I didn’t even label the columns). The results of this looked interesting enough that I wanted to generalize it and do out the actual math (tables are quick to make, but get clunky fast). So that’s what we’ll look at in a second. But I wanted to make sure to mention doing this step! Simply generating and counting is always quick and easy, and you can use it check your more complicated work later on too.
A Better Than Average Advantage
So if not brute forcing it, how can we tell which is better? Some of the wiki pages and posts about this suggest comparing the expected value of the two scenarios, which would be the ‘average outcome’ of the two kinds of dice rolls. There’s a really great breakdown on this wiki here, which goes into a great deal of detail (maybe even too much detail):
But the premise here is simple enough. What’s the ‘average’ roll on a normal D20? Well we just add up all the numbers and divide by twenty. And hey, they’re sequential, so it turns out we can actually just add the first and last and then divide by two (for why that works… the screenshot above points towards why, but it’s not important, just one of those pleasant elegancies).
Average for D20 = (20+1)/2 = 10.5
Deriving the calculation for the highest of two dice is a bit trickier (see the wiki above for the calculation steps, or search around for ‘expected value of rolling two dice and taking the highest value’), but whether we calculate it or just cheat and average the brute force table we made before, we’ll find that:
Average for Highest of two D20s = 13.825
And so there we have it. The difference between these two is 3.325, so… rolling with Advantage is like getting a modifier of +3.325, right? It’s better than +1,+2 and +3, but worse than +4? That wiki page states it confidently:
A ha, but is it though?
Probability Distribution in Detail
Saying it averages to +3.325 isn’t wrong, but it turns out that “is equivalent to” is obscuring quite a lot of interesting nuance. I’ve seen this repeated elsewhere in answers to this question, using average outcomes (the expected value) to show which is better. But I think it’s always dangerous to work with averages if you haven’t looked at the underlying distribution.
(I’m going to try and go a bit quickly on the math here, but you can check out my spreadsheet to see the calculation, and I’ll link out to some resources.)
My first thing I did was make this as a table, but let’s look at the resulting chart for now. First, what’s the probability of an un-modified D20 beating a DC check of 1, 2, … 20 etc? We’re really asking, what’s the probability of rolling at least a 1, at least a 2 and so on. Well, the chance of any given number is 1/20 = 5%, so it’d be a cumulative sum. It’s a 5% chance to roll a 20, 10% to roll a 19 or 20, 15% or roll an 18, 19 or 20, and so on. (I use a more general solution in my spreadsheet, you can check out this lovely academic paper here for details on the actual formula).
Anyway, if we do that out, it’d look like this:
So this tells us the probability of succeeding a DC 1 check is 100%, and for DC 20 it’s 5%, and it’s just a linear interpolation between those two points. It’s a very simple linear relationship, each level of DC we remove just adds a straight 5% bump.
Let’s graph Advantage on there too so we can compare. We can be clever about the formula here. “What is the probability of rolling at least a 5 on two D20s?” seems a bit complex because you’re dealing with an OR. We could roll higher than 5 on one die, or the other, or both, which seems tricky to account for. But we can actually flip it around and calculate what it isn’t. Rolling ‘at least a 5 on two dice’ means NOT rolling lower than a 5 on BOTH dice, which is actually much simpler to calculate (see this great answer on stackexchange for details). If we didn’t care about the formula and just wanted the percentages, we could also just head over to Jasper Flick’s AnyDice and run “output [highest of d20 and d20]” as well.
Anyway, once we have that, let’s add it to our chart:
Whoa, it’s nonlinear! I wasn’t expecting that. My rule of thumb is nonlinear is usually more interesting than linear, and we’ll see that is the case here.
We can see straight away that Advantage is better than rolling a normal D20, obviously. But it isn’t better in the same way all throughout. A +3 modifier would just give us a +15% bump across the whole line. But with Advantage, it’s giving a huge +25% bump in the middle, but a much smaller increase towards the extremes. The chance increase of beating a DC 10 check is massive (80%, versus 55%), but for a DC 20 check it’s much less noticeable (10% vs 5%).
This makes a certain amount of intuitive sense:
On the low-end, the increase is small, because you’re already very likely to succeed, but it can’t push the probability above 100% (unlike a modifier) because there’s always a small chance you roll low on both dice.
On the high-end, it’s hard to roll a 20, and getting two chances to do it does double your odds, but it’s still relatively unlikely.
- Which leaves the middle. The middle is getting such a big increase because we’re getting both die’s rolls, giving us three ways to succeed. We could win on one die and lose on the other, or lose on one die and win on the other, or win on both dice. So we get to sort of ‘double count’ the chances of success, and those combinatorically stack in the middle range.
This seems pretty cool to me, from a design standpoint, because it means rolling with Advantage’s benefits are situational. That means the answer to ‘Which is better?’, is the universal best answer: ‘It depends.’
Let’s add in a comparison to some simple bonus modifiers to our chart to see:
Cool! So you can see here that bonuses just move the line up. That means they can intersect the nonlinear Advantage line. If we compare a +4 bonus to Advantage, we can see that the +4 bonus is definitely better at the high-end (25% chance to beat DC 20, vs. 10% for Advantage). But it’s actually worse for a big swathe of the middle DCs.
So this means rolling with Advantage is not really equivalent to a +3.325 bonus, is it? +3.325 would always be better than +3 and always be worse than +4. But even though that’s what it is ‘on average’, rolling with Advantage is sometimes better and sometimes worse.
Critical Consideration
So far I’ve been ignoring another interesting rule, which is critical hits and misses. Rolling a ‘natural 20’, which means the die value before modifiers, is always a success, and rolling a ‘natural 1’ is always a failure. This means that the success rate is never actually 100%, as I’ve shown it above. It actually tops out at 95% for the modified rolls, because no matter how big the modifier, rolling a 1 is still a failure. But what about for rolling with Advantage? It’s not immune to critical failures, but it makes them very unlikely- to fail a DC 1 check, you would need a roll a 1 on both dice, which is a 0.25% chance (1 in 400). That is a lot less than 5% (1 in 20).
This means that Advantage is actually preferable in situations where the DC might be low but you don’t want to risk a failure, like picking pockets or disarming traps.
Rolling with Disadvantage?
By the way, there’s another mechanic called ‘Rolling with Disadvantage’, where you roll two dice and take the lower of the two. Here’s some questions to think about (and, for fun, try to guess what the answer will be before you look it up or calculate it): What do you think the shape of the Disadvantage line will be on that graph? What do you think is worse, rolling with Disadvantage, or a -4 Modifier?
Optimal Glove Strategy
So now we can finally answer the strategy guide question we were asking before (in case you were here for the pro tips). Purely in terms of lockpicking/pickpocketing (so we can ignore other enchantment benefits), here’s the optimal strat:
- Gloves of Thievery (rolling with Advantage) is always preferable over Gloves of Power (+1 bonus)
- They’re even preferable over Unlucky Thief’s Gloves (+2 bonus), except against a DC check of exactly 19, 20 or 21.
- (Rolling with Advantage is better for DC 18 and lower, and any higher it’s better because of the higher chance of a critical success)
- Gloves of Dexterity (+4) is preferable only if your natural Dex bonus is +0 or less, or your natural Dex bonus is +1 and the DC is between 19 and 23
I don’t really care about the result so much though, it’s more just the interesting implications of this. Having both a linear and nonlinear distribution of outcomes creates situational applications, which seems like a good design lesson.
Now you try!
Here’s a copy of my spreadsheet with the calc for everything discussed so far:
And I tried making an interactive Desmos thing for this (direct link), although it didn’t turn out that great:
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