What's better? Attacking with rapier or two dagger attacks?

Question

I'm looking for an answer supported by math and logic, not role-playing.

I have a level 1 rogue (level 2 at the end of the last session) that tries to use Sneak Attack whenever possible, and he has a rapier and two daggers. (He doesn't use the higher damage short swords because the rapier and daggers were the standard starting equipment and there's been no place to get short swords as of yet.) I began to wonder if one is technically a better option than the other since, as far as I can tell from my math, the damage range is exactly the same. I am keeping in mind the rules about Two-Weapon Fighting (from the 5e SRD):

When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand. You don’t add your ability modifier to the damage of the bonus attack, unless that modifier is negative. If either weapon has the thrown property, you can throw the weapon, instead of making a melee attack with it.

In my case, the rogue has a +3 dexterity mod, so on a hit both have a range of 4-11 (or 5-17 with Sneak Attack). The question is, then, is it better to attack once knowing that if you hit, you stand a chance of getting full damage...or better to attack twice, knowing that if you miss the first time you at least get a second shot? Does using Sneak Attack change any of the decision-making? (In my mind, it doesn't.) Or is there really no difference at all?

Answer 1

Dual wielding daggers does scale way better with your sneak attack than using only one rapier.

At first if all attacks hit the damage calculation would be:

$$ \text{Two Daggers}: 2d4+\text{DEX}+\text{SNEAKATK} \\ \text{Rapier}: 1d8+\text{DEX}+\text{SNEAKATK} $$

With Dex = 3 and Sneakatk = 1d6 that would be:

$$ \text{Two Daggers}: 6-17 \text{(Avg. 11.5)} \\ \text{Rapier}: 5-17 \text{(Avg. 11)} $$

Variable Hit Chances:

The problem is, that not all attacks hit.

We know that daggers and rapiers use the same attack bonus, thus having exactly the same chance to hit.

$$ \text{Two Daggers}= \text{Hit Chance}\times(1d4+3)+\text{Hit Chance}\times(1d4)+\text{SNEAKATK}\\ \text{Rapier}= \text{Hit Chance}\times(1d8+3)+\text{SNEAKATK} $$

Calculation for Sneak Attack:

If we assume, that every hit is eligible for sneak attack, we can add the damage to the daggers if any of the two attacks hit. If my math does not fail me that calculates as follows: (For two Daggers)

$$ \begin{align} \text{One Attack Misses} &= (1-\text{Hit Chance}) \\ \text{Both Attack Miss} &= (\text{One Attack Misses})^2 \\ \text{Sneak Attack Chance (NOT Both Attack Miss)} &= 1-(\text{Both Attack Miss}) \\ &= 1-(1-\text{Hit Chance})^2 \end{align} $$

Also let us talk in average damage for ease:

$$ \text{Two Daggers}: \text{Hit Chance}\times(8)+(1-(1-\text{Hit Chance})^2)\times(3.5) \\ \text{Rapier}: \text{Hit Chance}\times(7.5) + \text{Hit Chance}\times(3.5) $$

Base Damage:

Regardless of HitChance the daggers deal more damage in average. As the HitChance can never be less than 0.05:

$$ \text{Hit Chance}\times8 > \text{Hit Chance}\times7.5 $$

Sneak Attack damage:

The damage would be equal in both cases, so the relevant part is the trigger chance. Assuming the attacks are eligible for a Sneak Attack, the relevant part would be whether you hit or not.

When attacking twice, your chance to hit is higher than when only attacking once.

$$ (1-(1-\text{Hit Chance})^2) \geq \text{Hit Chance} $$

Assuming 0 < HitChance ≤ 1

Proof (Thanks to @Glen_b):

$$ 1-(1-H)^2 = 1-(1-2H+H^2) = 2H-H^2 \\ = H+H(1-H) \geq H \forall 0\leq H \leq 1 $$ with equality only possible at the endpoints and a maximum difference at $$H=0.5$$

Here is a graph to visualize the chances that you can apply your sneak attack damage:

Y = Sneak Attack Chance; X = Hit Chance per Attack; Red = Two Daggers; Blue = Rapier

Thus, two daggers are better with and without sneak attack.

Here is another graph visualizing the average damage for both weapons (Assuming +3 Dexterity modifier, 1d6 sneak attack, no magical bonuses, all attacks eligible for sneak attack.)

Y = Damage; X = Hit Chance per Attack; Red = Two Daggers; Blue = Rapier

Opportunity Costs

Through OPs comment on another answer, I know his rogue is level 2. This means he could use his bonus action to Dash, Disengage or Hide.

Later on, depending on your subclass and table rules, you might gain Feats or other class features like Fast Hands which require your bonus action. So, why would I take the Two Daggers instead of the Rapier? This would trade in positioning possibilities or other maybe useful opportunities for a measly 0.5 damage.

While the damage difference might appear to be 0.5 in average, the real advantage of dual wielding comes with the sneak attack chance.

Look at it this way:

If you have only 50% hit chance the difference would be

$$ (0.5\times8+0.75\times3.5) - (0.5\times(7.5+3.5)) = 1.125 $$

The daggers now deal 1.125 damage more in average.

Same scenario, but without sneak attack:

$$ (0.5\times8)-(0.5\times7.5) = 0.25 $$

Only a 0.25 damage difference. This gap increases drastically as sneak attack damage increases.

So your goal is to get your sneak attack damage through. If your first attack hits, your rapier deals in average 2 damage more than the dagger, but if the first attack misses you deal no damage with the rapier, while dual wielding gives you another attack to possibly deal your sneak attack damage.

Why is this so important?

When looking at your weapon attack damage, how does it scale?

• Your Dexterity modifier might go up (but would benefit both weapons equally)
• Your sneak attack damage increases with your level (better for daggers, because of higher hit chance if you attack twice)
• You may get a magic weapon

The 2 Avg. damage difference from the damage dice, if the first attack hits looses its relevance while the increased chance of sneak attack gains in relevance.

Let's look at a level 5 rogue who used is ASI on Dex (+4) and has now 3d6 Sneak attack damage. With a 50% hit chance: [the numbers in square brackets are without sneak attack]

$$ \begin{align} \text{One Dagger: } 0.5\times(2.5+4)+0.5\times(10.5) = 8.5 \text{ [3.25]}\\ \text{Two Daggers: } 0.5\times(2.5+4)+0.5\times(2.5)+0.75\times(10.5) = 12.375 \text{ [4.5]} \\ \text{Rapier: } 0.5\times(4.5+4)+0.5\times(10.5) = 9.5 \text{ [4.25]} \\ \text{Rapier+1: } 0.55\times(4.5+4+1)+0.55\times(10.5) = 11 \text{ [5.225]} \end{align} $$

Y = Damage; X = Hit Chance per Attack; Red = Two Daggers; Blue = Rapier

Even a Rapier+1 does not reach the Avg. damage of two daggers.

Here an level 20 (+5 Dex, 10d6 Sneak attack damage) example:

Y = Damage; X = Hit Chance per Attack; Red = Two Daggers; Blue = Rapier

The damage gap increases, but to get the most out of your turn I would recommend the following:

If your first attack with dual daggers hit, accept that you might have dealt more damage with a rapier and use your bonus action to get away (improve your survivability). If this is no option you can attack another time to maybe deal 1-4 extra damage.

If the first attack misses, attack another time, this is where dual wielding shines.

Answer 2

(Quick math note: The average value of a dice roll for any single die is ((1 + max dice value) / 2). d4 is 2.5, d8 is 4.5, etc.)

For raw damage, the two daggers are better. In a turn of combat, two dagger swings hit for:

(d4+3) + (d4) + (1d6 from Sneak Attack)

Which averages out to:

(2.5 + 3) + (2.5) + (3.5) = 11.5

Meanwhile, a rapier hits for:

(d8+3) + (1d6 from Sneak Attack)

Which averages out to:

(7.5) + (3.5) = 11

So the daggers have a minor edge of an extra .5 average damage over the rapier. The daggers have an additional huge benefit for damage as well, in that since you only get 1 sneak attack/turn, having two chances to land it massively increases the odds you do so in contrast to the rapier's single attack.

Let's say you have a 60% hit chance (require a 9+ to hit against a particular target).

With the daggers, you have two 60% chances to land your one sneak attack for the turn. That means you have to get both 40% chances of not landing attacks in order to lose out on sneak attack's damage, which means an 84% chance of landing sneak attack with two dagger swings in contrast to one rapier attack's 60%:

(1 - (.4 * .4)) = 0.84

So, continuing with the 60% hit chance, the two dagger's average damage per turn is:

((d4 + 3) * .6) + ((d4) * .6) + ((1d6) * .84)

Which averages out to:

(3.3) + (1.5) + (2.94) = 7.74

So 7.74 damage per turn.

For the rapier, since there's only one attack per turn we can simply directly multiply the above 11 by the 60% hit chance and get an average damage per turn of 6.6. So the two dagger swings gets you an average of ~1 more damage per turn, and that gap grows as your sneak attack dice pool increases with level- about an extra 1 damage per sneak attack die:

3.5 (average sneak attack die value) * (0.84 (two-weapon fighting chance of sneak attack in a turn) - 0.6 (single weapon fighting chance of sneak attack in a turn)) = 0.84 extra average damage per sneak attack die for two-weapon fighting over single weapon fighting.

That all said, there are other major considerations to take into account before declaring two-weapon fighting to always be the winning strategy. Namely, that rogues have a ton of other uses for their single bonus action due to the Cunning Action class feature.

For example, one strategy that can be used to keep the rogue safe is using your Action to attack, then using your bonus action to disengage and stay ~15ft out of melee range. That way, assuming you have another ally adjacent to the target, the target would have to take an opportunity attack from your ally to come after you. The rogue's one attack in this scenario would be better off using the rapier than the dagger. That said, if you were using light weapons, you would still have the choice of disengaging or going for the offhand attack/sneak attack after a miss with the main Attack action.

Another strategy is having a rogue with the Greenflame Blade or Booming Blade cantrips from the Sword Coast Adventurer's Guide sourcebook. They use their action to cast said cantrips rather than take the Attack action, so they can't two weapon fight on the same turn they cast the cantrip. As a result, there's not much reason to use the lower-damage dagger over the rapier for the attack made as part of the cantrip.

Answer 3

Expected result answer

A rapier does 1d8 damage (4.5 mean), a dagger does 1d4 (2.5 mean) so a hit with a rapier will do 2 more damage than a hit with a dagger (or 4 on a critical). If your chance to hit is \$p\$ you will expect to do \$2p + 0.1\$ more damage with a rapier (allowing for a critical). Your expected damage with a second dagger is \$2.5p + 0.125\$.

The question is, for what values of \$p\$ is \$2.5p + 0.125 > 2p + 0.1\$? A little bit of algebra will tell you for \$ p > -0.05 \$. Now, since a 20 always hits, the smallest value of \$p\$ is 0.05: so, it is always better to use 2 daggers.

Including sneak attack only makes 2 attacks better as it increases the changes of sneak attacking.

Of course, two short swords are better than both.

Full probability distribution answer

If you want to do this thoroughly you need to factor in your chances of hitting, missing and critical hitting with each of your attacks.

This anydice program does this:

AC: 16
DEX: 5
PROF: 2
DAGGER: 1d4
RAPIER: 1d8
SA: 1d6

function: first A:n for AD:d second B:n for BD:d{
  if A = 20 & B = 20 {result: 2dAD + 2dBD + 2dSA + DEX}
  if A = 20 & B + DEX + PROF >= AC & B != 1 {result: 2dAD + 2dSA + 1dBD + DEX}
  if A = 20 {result: 2dAD + 2dSA + DEX}
  if A + DEX + PROF >= AC & A != 1 & B = 20 {result: 1dAD + SA + 2dBD + DEX}
  if A + DEX + PROF >= AC & A != 1 & B + DEX + PROF >= AC & B != 1 {result: 1dAD + SA + 1dBD + DEX}
  if A + DEX + PROF >= AC & A != 1 {result: 1dAD + SA + DEX}
  if B = 20 {result: 2dBD + 2dSA}
  if B + DEX + PROF >= AC & B != 1 {result: 1dBD + SA}
  result: 0
}

output [first d20 for DAGGER second d20 for DAGGER]

output [first d20 for RAPIER second 1 for 0]

Please play around with the parameters, however, for all the ones I tried, the daggers are better.

Answer 4

Why not use a rapier and a dagger.... you choose to only use your dagger if you miss with the rapier, keeping you bonus action. This gives you 2 chances to deliver the SA damage.