# For a level 10 Swords bard with Crossbow Expert, Sharpshooter, when is it better for DPR to use Greater Invisibility vs. Swift Quiver?

The scene: A 10th-level College of Swords bard (includes Extra Attack). Crossbow Expert. Sharpshooter. 20 Dexterity. Hand crossbow +1.

Intelligence is his dump stat so he needs to ask a friend what to pick for his Magical Secret.

One is definitely Find Greater Steed, because he is going to ride on a griffon while raining down death with his hand crossbow. The other may or may not be Swift Quiver.

• Greater Invisibility will give me 2 attacks with my action, 1 attack as a bonus action, +10 to hit with advantage.
• Swift Quiver will give me 2 attacks with my action, 2 attacks with my bonus action, +10 to hit.

Ignoring opportunity costs, spell slots costs, defence and anything else other than crossbow damage per round, at what point is it better to use Swift Quiver vs. Greater Invisibility?

• – SeriousBri Jun 27 '20 at 22:07
• The "if you have X% chance to hit use swift quiver, otherwise use greater invisibility" analysis if possible – SeriousBri Jun 27 '20 at 22:17

# Greater Invisibility + Sharpshooter is usually best

I'm sure someone else can come with an analytical solution to this, personally I'm fond of the universiality of monte carlo methods. The general principle, create a simulation of what you want to compare, run the simulation many, many times and average the results.

In this case, all we need is a function which for a set of paramaters (number of attacks, advantage, AC, etc.) simulates that round (rolls to hit and for damage). Run that simulation 100000 times for each set of parameters. So, for each of the four cases (GI or SQ and Sharpshooter or not) we can run that for a set of ACs. My Python code here. Doing that we get damage per round as a function of AC for each case (GI is Greater Invisibility, SQ is Swiftquiver, and a suffix S means with Sharpshooter):

For AC 13 to 22 using greater invisibility with Sharpshooter gives the highest DPR. So unless you're fighting exceedingly low AC enemies or exceedingly high AC that combo is best. If the AC goes 23 or up just use greater invisibility until you hit AC 29 at which point you should start using Sharpshooter again (you're only hitting on crits, so you just want hits to hit as hard as possible). Using swiftquiver and Sharpshooter is only better for ACs 12 or lower, at which point you're basically hitting every non-nat1, so advantage isn't worth the one less attack.

• On the other hand for simple DPR computation (as this) you only need to deduce the analytical expression once. I did it 2 years ago and my code that computes DPR has been used since then haha. But I agree that, due to laziness or whatever else, just simulating it works well. +1 for actually giving the code as well haha – HellSaint Jun 29 '20 at 0:51

# Greater Invisibility is better for targets of AC 13 or greater.

## But swift quiver is significantly better for targets with very low AC.

This problem can be tackled analytically. An advantage of an analytical solution over a Monte Carlo simulation is that anyone can easily tweak the variables and get a new answer for a slightly different situation without having to set up their own simulation. (An advantage of Monte Carlo is that it works equally well for problems of all levels of complexity, but this problem is simple enough to be tackled analytically.)

### Definitions

There are a few variables of interest.

• $$\H\$$, your to-hit modifier.

In your case, with proficiency +4, DEX +5, and a +1 weapon, $$\H=10\$$, unless you use Sharpshooter, in which case $$\H=5\$$.

• $$\AC\$$, the target's armour class.

• $$\D\$$, your expected/mean/average damage on a normal hit.

In your case, you do 1d6+5+1 damage with your +1 hand crossbow. The average value of a six-sided die is $$\(1+2+3+4+5+6)/6=3.5\$$, so your expected damage is $$\D=9.5\$$, or $$\D=19.5\$$ with Sharpshooter.

• $$\C\$$, your expected damage on a critical hit.

Critical hits double the number of dice you roll for damage, so we add another 3.5 damage. $$\C=13\$$, or $$\C=23\$$ with Sharpshooter.

### Normal attacks

The chance to hit at all on a normal roll is

$$\frac{21-AC+H}{20}$$

for $$\20 \ge AC-H \ge 2\$$. If $$\AC-H\$$ is greater than or equal to 20, you can only hit on a natural 20, which is always a critical hit. If $$\AC-H\$$ is less than or equal to 2, then you can only miss on a natural 1, which is always a miss. We can represent this by forcing the quantity $$\AC-H\$$ to always be between 2 and 20 inclusive, by replacing all instances of $$\ AC-H\$$ with min(max(AC-H,2),20). I'll leave this substitution as implied in my derivations here and only apply it when plotting.

We want to separate out the normal hits from the critical hits. The chance for a critical hit is $$\1/20\$$. The chance for a normal hit is the chance to hit minus the chance for a critical hit,

$$\frac{21-AC+H}{20} - \frac{1}{20} = 1 - \frac{(AC-H)}{20}.$$

The expectation value (or average/mean value) of an event is the sum of the values of each outcome (damage) times the probability of each outcome. The expected damage for a single normal attack is

$$D \left(1 - \frac{(AC-H)}{20}\right) + C \frac{1}{20}.$$

The probability to hit with advantage (where we roll twice and take the highest result) can be calculated by considering the chance to miss twice. Hitting with advantage is the same as not missing twice in a row with normal rolls.

(An alternative formulation is to consider the chance to hit on the first die and miss on the second, plus the chance to hit on the second die and miss on the first, plus the chance to hit on both dice. However, this is more complicated for no benefit.)

The chance to miss on a single d20 roll is 1 minus the chance to hit,

$$\frac{AC-H-1}{20}.$$

Because die rolls are independent events, we can simply multiply probabilities together, so the chance of missing twice is that value squared. The chance of hitting with advantage is

$$1 - \left(\frac{AC-H-1}{20}\right)^2.$$

As before, we want to separate out the chance for a normal hit and a critical hit. The chance to not get a critical hit is $$\19/20\$$, so the chance to get a critical hit with advantage is

$$1 - \left(\frac{19}{20}\right)^2 = \frac{39}{400}.$$

The chance to get a normal hit with advantage is

$$\left(\frac{19}{20}\right)^2 - \left(\frac{AC-H-1}{20}\right)^2 = \frac{361 - (AC-H-1)^2}{400}.$$

Therefore, the expected damage for a single attack with advantage is

$$D \left(\frac{361 - (AC-H-1)^2}{400}\right) + C \frac{39}{400}.$$

### Damage per round

We have two different scenarios. We can choose to make three attacks with advantage (using greater invisibility), or four attacks with normal rolls (using swift quiver). (Another permutation is whether or not we use Sharpshooter, but that just changes the values for $$\H\$$, $$\D\$$ and $$\C\$$.) We can get our damage per round by multiplying our damage per attack by the number of attacks.

With swift quiver, our expected damage per round is

$$4 \left(D \left(1 - \frac{(AC-H)}{20}\right) + C \frac{1}{20} \right).$$

With greater invisibility, our expected damage per round is

$$3 \left( D \left(\frac{361 - (AC-H-1)^2}{400}\right) + C \frac{39}{400} \right).$$

It now remains to substitute in values to determine which is better in our scenario, and by how much.

We find that when you require a roll of 8 or more to hit, greater invisibility is better than swift quiver. When you require a roll of only 7 or less to hit, you are sufficiently likely to hit that the extra attack is better than increased hit chance.

With greater invisibility, you want to use Sharpshooter when the target's AC is 22 or less. With swift quiver, you want to use Sharpshooter when the target's AC is 21 or less. In both cases, you also want to use Sharpshooter when the target's AC is 29 or more (when you can only hit on a natural 19 or 20 on a normal roll, or a natural 20 on Sharpshooter).

When using Sharpshooter optimally, greater invisibility is superior to swift quiver when the target's AC is at least 13. At your level, this should be most enemies which pose any danger.

### Better by how much?

Of course, there are factors other than damage per round which you will want to consider. In order to be able to weigh DPR against other factors, you need to know how much better one option is against another.

This can be achieved by subtracting the swift quiver case from the greater invisibility case. We will assume that we are using Sharpshooter optimally (by taking the maximum of the Sharpshooter and non-Sharpshooter cases).

Greater invisibility has the greatest difference at AC 19, with 6.8075 extra damage compared to swift quiver. The difference is at least 6 damage for ACs from 17 to 21 inclusive. Above this Sharpshooter stops being useful, so damage differences become less.

Conveniently, greater invisibility is most effective (compared to swift quiver) in the mid-range of armour classes, and remains effective for targets of very high AC.

Swift quiver provides substantially more DPR (up to 15.4225 more damage for AC 7 or less) for targets of low AC. For AC 10, the difference is 5.91625. For AC 8 (e.g. a zombie), the difference is 11.9612. If your campaign includes many low AC creatures, then swift quiver would be better against them. But if your typical targets are even lightly armoured you will want greater invisibility.

• At college (many years ago) I studied both mathematics and advanced mathematics (and physics which was basically more maths). This answer shows why I dropped out. – SeriousBri Jun 28 '20 at 9:25
• The much better for targets with low AC heading should probably be qualified to say you're talking about ACs well below 10, which is pretty rare to find. When most people read "low AC", they're probably thinking like 11 or 12, at which point they're close and it would be misleading to say "much". TL:DR: suggest changing it to "significantly better for targets with AC far below 12". The largest relative difference is like 30% at AC7 which does justify a "much", but "significant" downplays the importance of targets with such low AC for cases where DPR is a good measure. – Peter Cordes Jun 28 '20 at 22:35
• Also, such low AC targets often don't have a lot of HP either, like your Zombie example, so an extra attack lets you take out 4 instead of 3 if you're lucky and one-shot their 22 HP each time. (Or if an AoE knocked some health off first). Elder Black Pudding has AC7 with 130HP, but most enemies have at least AC 10 or 11. – Peter Cordes Jun 28 '20 at 22:40
• Another use-case for Swift Quiver would be if you already have advantage, e.g. from Hold Monster or Faerie Fire. Then Invis gives you nothing. – Peter Cordes Jun 28 '20 at 22:46
• @PeterCordes Noted, and subheading changed accordingly. AC 10 is around when swift quiver's benefit compares to greater invisibility's peak benefit. But yes, anything below 10 is typically unusual. As for other sources of advantage, swift quiver, hold monster, faerie fire and greater invisibility all require concentration, so unless you can rely on allies (which is beyond this question's scope) you can only use one. – BBeast Jun 29 '20 at 0:27

## Greater invisibility is usually better.

Here are the DPRs at ACs 12-25 for the two different spells:

For higher armor classes, greater invisibility is decisively better, and for lower armor classes, it is nearly equal to negligibly worse.

Now, a somewhat rigorous proof that greater invisibility is superior to swift quiver for damage output against targets having AC between 13 and 25, inclusive. The math works out a bit differently when the target AC is so high only criticals can land. This case is ignored.

### First, we define our (derived) constants:

$$\HIT = 5\$$. This is the bonus to hit on each attack.

$$\DAM = 19.5\$$. This is the average damage per hit, the average of 1d6+16.

$$\CRIT = \frac{3.5}{20}=.175\$$. This is the additional damage per attack we can expect from critical hits without advantage.

$$\CRITADV=3.5\cdot\frac{39}{400}=.34125\$$. This is the additional damage per attack we can expect from critical hits with advantage.

### And then our (derived) variables:

$$\AC\$$ = Armor Class. We will derive expectation functions of armor class for each scenario, and graph them.

$$\k=21+HIT-AC\$$. Dividing this by $$\20\$$ gives the probability an individual attack hits the target without advantage.

$$\k'=\frac{40k-k^2}{20}\$$ . Dividing this by $$\20\$$ gives the probability that in individual attack lands with advantage.

### Swift Quiver

With a casting of swift quiver, we make $$\N=4\$$ attacks. Our expected damage per round is:

$$\N(DAM)\bigg(\displaystyle{\frac{21+HIT-AC}{20}}\bigg)+N\cdot CRIT=101.925-3.9AC\$$

### Greater Invisibility

This one is a little bit trickier, as the probability calculation for advantage is pretty wild.

Here we are making $$\N=3\$$ attacks. Our expected damage per round is:

$$\N(DAM)\bigg(\displaystyle{\frac{k'}{20}}\bigg)+N\cdot CRITADV=...\text{algebra}...=54.25875+1.755AC-.14625AC^2\$$

### Compare

It remains only to graph these functions against each other:

As you can see, greater invisibility has higher expected damage per round than swift quiver for ACs 13-25.

Apologies to mobile users. The equations may run off the side of the screen. Rotating your device to landscape should display everything correctly.

# Sharpshooter with Greater Invisibility is better

In the majority of use cases, Sharpshooter with Greater Invisibility yields higher damage. Interactive view (compare damage values)

Against enemies with 12 AC or less, using Sharpshooter with Swift Quiver is slightly better. We are talking 1 more damage at 12 AC, 3 more at 11 AC, and 6 more at 10 AC.

Against enemies with 13 AC or more, using Sharpshooter with Greater Invisibility is better. 1 more damage at 13 AC, 2 more at 14, more at 7 damage at 19 or 20 AC. At best that's ~30% better.

At your level and above, it won't be common to see enemies with AC 12 or less, and they are likely to not be much challenge for your party to dispatch. You will likely see ACs around 17-18 at your current level, and they will get to 20 as you level up. You are unlikely to see ACs over 20 in a normal game.

• @Someone_Evil Huh, not sure why google sheets is doing that. I republished it with static data, that's about all I can think of. If it times out / breaks again I'll just make a public facing sheet. – user-024673 Jun 30 '20 at 0:21

Another spell that is in the conversation is Haste. If you have Crossbow Expert already, then Haste provides the same number of attacks as Swift Quiver for a spell two levels lower:

Turn 1, you either use your bonus action to cast Swift Quiver and make two attacks from extra attack, or you use your action to cast Haste, use your hasted action to make one weapon attack, and use your bonus action to make another from Crossbow Expert. Either way, two attacks.

Turns 2+, you either make two attacks with your action and two with your bonus action from Swift Quiver, or three with your action and one from your bonus action from haste. Either way, four attacks.

While Greater Invisibility is still more damage on most ACs, Haste has the benefit of being one spell level lower than it.

• Well, if you are putting the "one level lower" into account, then it's hard to dismiss all other utilities from Greater Invisibility, like the fact you are, well, invisible, making you invulnerable to many spells (that require seeing the target), being attacked with disadvantage (which is consistently better than +2 AC) and overall out-of-combat utility (although then regular invisibility is just as good). Nonetheless, it is a good point on how bad Swift Quiver actually is in this scenario: Haste is better at everything other than the Lethargy effect. – HellSaint Jun 30 '20 at 3:48
• I would also make a point that Haste is a more interesting spell to take as Magical Secret, since it is not a Bard spell. Greater Invisibility can be taken regularly as a normal spell since it's already on the list anyway. – HellSaint Jun 30 '20 at 3:49
• Hmm, Haste is interesting, because doubling the movement speed of the mount also gives the ability to get in and out of range each turn with the flying mount now being a rocket ship. Normally I dislike Haste (because stun) but it is definitely a consideration! – SeriousBri Jun 30 '20 at 12:31