What is the expected DPR of an animate object spell cast on daggers?

Question

A 9th level bard (CHA 16) casts "Animate Object" on a set of daggers, animating as much as he can.

Assuming 11+ on the d20 is required to hit, what is the expected DPR of the spell?

How does it fare against other damaging spells of the same level?

Answer 1

33.75 damage.

Daggers are Tiny objects. So this question can reduce to "What is the expected DPR of an animate objects spell cast on Tiny objects".

Animate Objects animates 10 of them.

Tiny - HP: 20, AC: 18, Attack: +8 to hit, 1d4 + 4 damage, Str: 4, Dex: 18

Their expected damage when they all hit is 10 * Expected_damage(1d4+4) = 10 * 6.5 = 65.

Their expected damage if they all crit is 10 * Expected_damage(2d4+4) = 90

Hitting on an 11+, they will hit 9 in 20 times and crit 1 in 20 times. So the expected damage is:

65 * 0.45 + 90 * 0.05 = 33.75

Spell comparison

How does it fare against other damaging spells of the same level?

I'm going to exclude non-concentration spells like Cone of Cold from this - in general Concentration spells will produce more powerful effects at the cost of only being able to use one of them at a time, so comparing Concentration and non-Concentration spells is basically apples and oranges.

It's not possible to do a clear mathematical analysis without knowing the saves of the creature(s) that would be faced. But I will provide a more qualitative answer instead.

Area spells

like Insect Plague or Cloudkill, both 5th level Concentration spells that last for 10 minutes (not much more useful than 1 minute, but it could mean their damage is a bit lower than it should be for their level) damage creatures for 5d8 (22.5) or 4d10 (22) in a 20-foot sphere (save for half).

Single-target spells

like Immolation do 8d6 on first casting, and 4d6 on the enemy's turn afterwards (both saves for half). Saves are very relevant here as the damage stops when the creature saves. Immolation is known to be a very low-power spell though, and there aren't very many examples of single-target concentration damage spells.

Bigby's Hand is another comparable spell, dealing 4d8 (18) damage to one target on a successful attack.

The expected damage for that would be:

1. If we stuck with 11+ on a roll hits: 18 * 0.45 + 36 * 0.05 = 9.9 expected damage per round
2. If we assume against the same creature, where a +8 hits on 11+, the +7 Bigby's hand will use (level 9 proficiency = +4, 16 CHA = +3) will hit on 12+, which gives: 18 * 0.4 + 36 * 0.05 = 9 expected damage per round

Conclusion

Based on these spells, I would say that:

• Animate objects is an extremely powerful damage spell against a single opponent, or multiple opponents spread out beyond the range of a area spell, vastly better than comparable others of its level such as Bigby's hand.
• Against multiple opponents close together, area damage spells of the same level can produce a higher DPR, but they are less flexible (tending to not move or move in a way you cannot control).

Answer 2

If 10 animated daggers attack, the DPR is 33.75

Animate Objects

Each dagger is a tiny object, so 10 can be animated at once.

Tiny - HP: 20, AC: 18, Attack: +8 to hit, 1d4 + 4 damage, Str: 4, Dex: 18

Against targets with 19 AC, each dagger hits 10/20 times and crits 1/20 times so the average damage per attack is (1d4+4)*.5 + (1d4)*.05 = 3.375. Assuming all 10 daggers attack, the DPR is 33.75.

Bigby's Hand

The most comparable 5th level spell is, in my opinion, Bigby's Hand, because it functions in a similar way. If you use its clenched fist option against the same target, Bigby's Hand hits 9/20 times and crits 1/20 times so the average damage per attack is (4d8)*.45 + (4d8)*.05 = 9, which is is also the DPR.

Comparison

Animate Object does significantly more damage, but the damage is non-magical, which is often resisted or even ignored.
The animated objects don't have much utility, but they lock down the enemies with lots of potential opportunity attacks.
The animated objects only move 30 feet, and a single AoE may wipe out most of the objects before they even get a turn to act.

Bigby's Hand does significantly less damage, but it's the least resisted type, force.
It can't do opportunity attacks, but has more utility.
It can fly 60 feet and it can't be destroyed easily.
It uses your spell attack, so it's accuracy will increase as you level up.

Conclusion

Overall Animate Objects vastly outstrips other spell of its level for damaging a single target. However, it's not an appropriate spell to fight against the many monsters that resist or are immune to non-magical damage. It's also a bad choice against enemies that have AoE: I've personally seen 10 rapiers be animated and then killed by a breath weapon before any of the animated objects had a turn.

Answer 3

Let's use Anydice

I've turned to my favourite tool, anydice, to help do a statistical analysis on the spell.

I'm using the following anydice script (which is itself copied and modified from this helpful blog post):

\**** Damage calculator ****\

function: attackroll ROLL:n vs AC:n {
if ROLL >= 20 { result: CRITDAMAGE }
if ROLL = 1 { result: 0 }
if ROLL + ATTACKMOD >= AC { result: DAMAGE }
result: 0
}

ATTACKMOD: 8
ATTACKTIMES: 10
DAMAGE: 1d4 + 4
CRITDAMAGE: 2d4 + 4

output ATTACKTIMESd[attackroll d20 vs 19] named "10 tiny animated objects vs. AC 19"
output ATTACKTIMESd[attackroll [highest 1 of 2d20] vs 19] named "10 tiny animated objects vs. AC 19 with advantage"

The "table" output for this script is a bit difficult to parse, so I recommend looking at the "graph" view instead. Hopefully it is obvious how one might alter the script to compare against different target ACs or with differently sized objects.

I have also included an output which assumes the objects attack with advantage. If your DM is using flanking rules, it's probably safe to assume the dagger swarm will be able to reliably flank with itself if it surrounds the enemy target. It also applies if you're getting the objects to attack a prone or restrained target.

So let's examine the numbers...

Expected Damage

Against a target with an AC of 19 (such that our daggers are hitting exactly half the time) we see the following results (with the expected damage/standard deviation in parentheses):

10 tiny animated objects vs. AC 19 (33.75 / 11.12)

10 tiny animated objects vs. AC 19 with advantage (51.19 / 10.16)

We note that if the daggers can attack with advantage, that improves your expected damage by just over 50%. If you play with the target AC, the effect of advantage diminishes as the target becomes easier to hit. In most cases, though, having advantage will make the spell significantly more powerful.

Comparison to other spells

Let's compare this to a spell like Cloudkill, which does 5d8 damage or save for half. In a given round the expected damage to a creature which fails its save is 22.5; 11.5 for a passed save. Assuming a hypothetical creature which is saving against the spell exactly half the time, the average damage is 17. That's barely half the expected damage output of Animated Objects against a single target in similar circumstances (hitting half the time). A similar spell at 5th level, Insect Plague, is slightly worse - dealing 4d10 damage with a save for half, for an expected DPR of 16.5. A one-shot (rather than ongoing) damaging 5th-level spell, Flame Strike, does 8d6 total damage with a save for half, so averages to 21 damage - still outclassed by a single round of attacks from the dagger swarm.

However, these other spells are all AoE effects, and if affecting just two enemies in a turn, their DPR already draws level with that done by the daggers. (The daggers might also get to make opportunity attacks on moving enemies - but the possible increased damage that might do is equalled if the AoE spell affects a couple more enemies.) Though animate objects is clearly superior against a single target, against a group of any size, these same-level AoE spells become more effective - assuming the rough equivalence of saving throws and attack rolls.

Conclusion

A swarm of tiny animated objects is a comparatively effective single target damage spell, but compared to AoE spells of the same level like Cloudkill or Insect Plague, it doesn't do obscene damage. It's definitely a good spell to bust out when you want to put maximum hurt on a single target - perhaps especially effective against enemy spellcasters, who are apt to have lower AC and would potentially be forced to make multiple concentration checks per round if maintaining their own spells. However, it's less effective than similarly-levelled AoE spells for dealing with groups of enemies, and has counters that those spells do not (the daggers themselves are quite vulnerable to AoE spells, though they're actually pretty resilient - but obviously not invulnerable - in physical combat).

In situations where you can get the daggers to attack with advantage - especially if you're using flanking rules and as such can probably get advantage reliably - the power of the spell is a lot greater, and the increased hit and crit chance pumps up damage considerably. In those cases the dagger swarm is a clearly superior option for raw single target damage to other spells of similar level - who wouldn't want to get to do ~50 damage with a bonus action for 10 rounds?