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This answer to a previous question about crossbow builds without Crossbow Expert is relevant to me; my Rogue is about to reach 4th level.

If you only hit on a 20, the Crossbow Expert feat (PHB, p. 165) nearly doubles your chance to hit.
If you hit on a 2, Crossbow Expert is simply +4.5 damage (2d6+6 vs. 1d8+4) over a light crossbow.

Is there a place between these extremes, where using an ASI to increase Dex by 2 is better for DPR than taking the Crossbow Expert feat? One hand crossbow or one light crossbow.

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There is no point at which +2 Dex increases your DPR more than gaining an extra attack as a bonus action via Crossbow Expert, if you are always spending your bonus action on more damage, and if you would just be attacking once with a light crossbow in the case where you don't have Crossbow Expert.

For the following, I'm assuming that pre-level 4 you are already at 16 Dex from the numbers in the question. I'm also assuming you'll be able to use your sneak attack on any target you hit.

As you note, the effects of +2 to Dex are +1 to hit and +1 to damage, while Crossbow Expert grants a second chance to hit for your usual attack damage. So, to compare, we'll be using the following stats:

  • Level 4 with Crossbow Expert:

    • +5 to hit (+3 from 16 Dex, +2 from proficiency)
    • Two attacks at 1d6+3 each
  • Level 4 with +2 Dex & light crossbow:

    • +6 to hit (+4 from 18 Dex, +2 from proficency)
    • One attack at 1d8+4
  • Sneak Attack for both is 2d6 if any hits occur on the turn, limited to once per turn.

Let's start by looking at the best-case scenario for +1 to hit- when you otherwise would require a 20 to hit, and the +1 to hit effectively doubles your hit chance by letting you also hit on a 19. So, for this example, let's say the target has AC 25.

Vs 25 AC

Crossbow Expert

You would have 2 5% chances to critically hit for 2d6+3 damage each, and a 9.75% overall chance of Sneak Attack occurring for 4d6 (since you have a 9.75% chance of at least one attack hitting, which is when Sneak Attack would occur, and its damage would double from the usual 2d6 due to the hit being critical).

With 3.5 being the average result of a d6, this works out to an average expected DPR against a 25 AC target of:

((2d6+3) * 0.05 * 2) + (4d6 * 0.0975)

(((2 * 3.5) + 3) * 0.05 * 2) + (4 * 3.5 * 0.0975) = 2.365 expected DPR

+2 Dex

You would have 1 10% chance to hit for 1d8+4 damage, plus your Sneak Attack for 2d6 more. Essentially, 5% of your rolls will be for 'normal' damage of d8+4+2d6, and 5% of your rolls will be for critical damage of 2d8+4+4d6, with the rest being misses.

With 4.5 being the average result of a d8, this works out to an average expected DPR against a 25 AC target of:

((d8+4+2d6) * 0.05) + ((2d8+4+4d6) * 0.05)

((4.5 + 4 + (2 * 3.5)) * 0.05) + (((2 * 4.5) + 4 + (4 * 3.5)) * 0.05) = 2.125 expected DPR

As you can see, the Crossbow Expert feat provides a higher DPR in this case- 11.3% higher, or so. This gap widens as you get closer to the middling ranges of hit chances (~50%), as that favors two chances to hit more than a flat +1 to hit once- mainly because of how big the impact of two chances to land your one sneak attack per turn is. Let's do the math again with the target having 23 AC instead, just to demonstrate that the gap is widening as AC begins going in the direction of more reasonable ranges:

Vs 23 AC

Crossbow Expert

You would have 2 15% chances to hit the target for 1d6+3, and an overall 27.75% chance for at least landing one attack so that Sneak Attack can occur. Each attack that lands has a 1/3 chance of being a critical hit. Assuming you use Sneak Attack on the first attack that hits, that means that 1/3 of the times Sneak Attack goes off will be on a critical hit.

(d6+3 * 0.15 * 2) + (d6 * 0.05 * 2) + (2d6 * 0.2775) + (2d6 * 0.2775 * (1/3))

Breaking this down a bit, just for clarity's sake:

  • (d6+3 * 0.15 * 2): You have 2 chances of hitting for d6+3 with 15% accuracy.
  • (d6 * 0.05 * 2): On each of your two attack attempts, you have a 5% chance of dealing d6 extra damage due to a crit.
  • (2d6 * 0.2775): You have a 27.75% chance of landing any attacks at all, allowing you to Sneak Attack once for 2d6 extra damage.
  • (2d6 * 0.2775 * (1/3)): Since your landed hits have a 1/3 chance of being critical hits, 1/3 of your Sneak Attacks will deal an extra 2d6 damage.

((3.5 + 3) * 0.15 * 2) + (3.5 * 0.05 * 2) + ((2 * 3.5) * 0.2775) + ((2 * 3.5) * 0.2775 * (1/3)) = 4.89 expected DPR

+2 Dex

You would have 1 20% chance to hit the target for 1d8+4, plus 2d6 for your Sneak Attack. Each attack that lands has a 1/4 chance of being a critical hit, so 1/4 of your Sneak Attacks will be on critical hits for double damage.

((d8+4 + 2d6) * 0.2) + ((d8 + 2d6) * 0.05)

Breaking this down a bit, just for clarity's sake:

  • ((d8+4 + 2d6) * 0.2): 20% chance to hit for d8+4 damage, plus a Sneak Attack for 2d6.
  • ((d8 + 2d6) * 0.05): 5% of the time, you also deal an extra d8 + 2d6 damage due to a critical strike.

((4.5 + 4 + (3.5 * 2)) * 0.2) + ((4.5 + (3.5 * 2)) * 0.05) = 3.675 expected DPR

Yep, just like we thought, the DPR gap has widened even more in Crossbow Expert's favor, with it now being ~33% higher than +2 Dex.

For thoroughness' sake, let's check the other end of things, where a 3+ on the D20 is a hit and the +2 Dex would bring that to a 2+ (target AC of 8):

Vs 8 AC

Crossbow Expert

You would have 2 90% chances to hit the target for 1d6+3, and an overall 99% chance for at least landing one attack so that Sneak Attack can occur. Each attack that lands has a 1/18 chance of being a critical hit. Assuming you use Sneak Attack on the first attack that hits, that means that 1/18 of the times Sneak Attack goes off will be on a critical hit.

(d6+3 * 0.9 * 2) + (d6 * 0.05 * 2) + (2d6 * 0.99) + (2d6 * 0.99 * (1/18))

((3.5 + 3) * 0.9 * 2) + (3.5 * 0.05 * 2) + (2 * 3.5 * 0.99) + (2 * 3.5 * 0.99 * (1/18)) = 19.365 expected DPR

+2 Dex

You would have 1 95% chance to hit the target for 1d8+4, plus Sneak Attack for 2d6 more. Each attack that lands has a 1/19 chance of being a critical hit, so 1/19 of your Sneak Attacks will be on critical hits for double damage.

((d8+4 + 2d6) * 0.95) + ((d8 + 2d6) * 0.05)

((4.5 + 4 + (3.5 * 2)) * 0.95) + ((4.5 + (3.5 * 2)) * 0.05) = 15.3 expected DPR

So even at the opposite end of things, this gap still exists, favoring Crossbow Expert at 27% higher than +2 Dex. You can see that the gap is smaller than it was at 23 AC, as even with only one attack you're still likely to get Sneak Attack off.

Now, let's do one last example right in the middle ranges of AC when the benefits of 2 attacks over one are near their peak:

Vs 16 AC

Crossbow Expert

You would have 2 50% chances to hit the target for 1d6+3, and an overall 75% chance for at least landing one attack so that Sneak Attack can occur. Each attack that lands has a 1/10 chance of being a critical hit. Assuming you use Sneak Attack on the first attack that hits, that means that 1/10 of the times Sneak Attack goes off will be on a critical hit.

(d6+3 * 0.5 * 2) + (d6 * 0.05 * 2) + (2d6 * 0.75) + (2d6 * 0.75 * (1/10))

((3.5 + 3) * 0.5 * 2) + (3.5 * 0.05 * 2) + (2 * 3.5 * 0.75) + (2 * 3.5 * 0.75 * (1/10)) = 12.625 expected DPR

+2 Dex

You would have 1 55% chance to hit the target for 1d8+4, plus your Sneak Attack for 2d6 more. Each attack that lands has a 1/11 chance of being a critical hit, so 1/11 of your Sneak Attacks will be on critical hits for double damage.

((d8+4 + 2d6) * 0.55) + ((d8 + 2d6) * 0.05)

((4.5 + 4 + (3.5 * 2)) * 0.55) + ((4.5 + (3.5 * 2)) * 0.05) = 9.1 expected DPR

In the middle ranges of hit chances, you can see Crossbow Expert has a ~39% DPR lead over +2 Dex.


Below is a chart using the same stats as above to calculate the expected DPR against a target of varying armor classes. The orange line is the Crossbow Expert rogue, and the purple line is the rogue who opted for +2 Dex and is using a light crossbow instead.

Chart of expected rogue DPRs for the two different builds


Of course, this is all purely in regards to comparing one light crossbow shot vs 2 hand crossbow ones- the gap is smaller if we compared, say, going Crossbow Expert vs. using two shortswords and Two-Weapon Fighting with +2 Dex instead. Plus, the route of +2 Dex has non-DPR benefits of initiative, ability checks, and AC that should be taken into account, as well as the fact that rogues have a wide variety of other uses for their bonus actions. But the scope of this question is about a Crossbow Expert hand crossbow user vs a light crossbow user with +2 Dex instead, so I'll leave it at that.

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    \$\begingroup\$ I'd just point out that you only take the second shot when the first one misses, thus maximizing your sneak attack hits while preserving as many bonus actions as possible, it only reduces the expected DPR by about 2, so you'd still be outpacing the Light Crossbow guy, just by a little less, and still get most of the benefits. \$\endgroup\$ – Darth Pseudonym Oct 21 '18 at 18:42
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    \$\begingroup\$ Oh, also worth mentioning: Unless you're depending on always having a friend in melee with your target, you'll need to be using Hide to get those sneak attacks in, in which case the single shot benefits slightly more because you'll lose advantage if the first attack misses (unless you have the Skulker feat instead of Crossbow Expert, which is a whole other discussion), and you'll need Cunning Action to re-hide each turn (but that depends on having reliable cover, which is only situationally available). \$\endgroup\$ – Darth Pseudonym Oct 21 '18 at 18:44
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    \$\begingroup\$ @DarthPseudonym I would not assume that you only use the bonus attack when the first misses at all, especially in a DPR sim. At range, you have less need of the Cunning Actions than a rogue up close in melee who's trying to weave in and out of their front line. And, since your bonus attack damage still gets +Dex, it's far more impactful than the bonus attack of two-weapon fighting. Also, having an enemy with at least one of my allies in melee is, in my experience, vastly more common (nigh-100%) than having a source of cover my DM allows me to repeatedly use to hide in the same place with. \$\endgroup\$ – CTWind Oct 21 '18 at 21:27
  • \$\begingroup\$ (Besides that, attacking twice is part of the comparison that the question asked for- "(2d6+6 vs. 1d8+4)") \$\endgroup\$ – CTWind Oct 21 '18 at 21:29
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    \$\begingroup\$ @DarthPseudonym, when you are far enough from the enemies, the best thing you can do with a bonus is mostly take another shot, even if DPR is not your highest concern. \$\endgroup\$ – András Nov 15 '18 at 10:15

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