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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?

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    \$\begingroup\$ As a general note on DPR calculations: remember that any "overkill" damage in D&D is lost, so if your target has 1hp left doing 20hp of damage to them is no better than doing 1hp. This effect increases the benefit of reliability in damage over maximum theoretical damage. Thus, in this case, the benefit of two daggers over a rapier is higher than simple calculations suggest. \$\endgroup\$ – Jack Aidley May 15 '17 at 9:29
  • \$\begingroup\$ Yes, there's a reason. I'll update the question. \$\endgroup\$ – David Robinson May 15 '17 at 16:48
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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 Hit Chance with two Attacks vs. one Attack

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 DamageGraph

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 HigherLvlDmg

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 HighestLvlDmg

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.

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  • \$\begingroup\$ FYI: double backslash (line 1 \\ line 2, where line 2 can be on an actual new line) is the newline character inside $$ blocks, \text{Hit Chance} will output plain text, \times outputs the multiplication × symbol. \$\endgroup\$ – doppelgreener May 14 '17 at 21:31
  • \$\begingroup\$ This meta announcing MathJax also has helpful links to on- and off-site MathJax references. (Though they are scattered among the question, answers, and comments....) \$\endgroup\$ – nitsua60 May 15 '17 at 1:28
  • \$\begingroup\$ Where do you want to link in the plot? In pretty sure there's a way to do it, but I'm not sure where you want it. \$\endgroup\$ – SevenSidedDie May 15 '17 at 1:39
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    \$\begingroup\$ I think this good answer would be even better with some discussion (based on your experience) of how significant the expected ~1-pt. bump in damage is, coming at the opportunity cost of the other things a rogue might do with their bonus action. \$\endgroup\$ – nitsua60 May 15 '17 at 12:48
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    \$\begingroup\$ @nitsua60 I think the best option would be to use two daggers and if the first attack hits you can/should use the bonus action for something different. If the first attack misses you still have the chance to deal the stealth attack damage (which is the majority of the damage in higher levels). I will edit and argue when I am back home later. \$\endgroup\$ – Thyzer May 15 '17 at 12:54
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(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.

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    \$\begingroup\$ It looks like OP only has 1d6 for SA: "on a hit both have a range of 4-11 (or 5-17 with Sneak Attack)." It doesn't effect your (excellent) logic any, but it tweaks the numbers that underly the opportunity cost-benefit analysis some. (I.e. increased chance to land the SA is less impactful if SA is less damaging.) \$\endgroup\$ – nitsua60 May 14 '17 at 21:18
  • \$\begingroup\$ @nitsua60 Good spotting, will tweak after my current game session's over. \$\endgroup\$ – CTWind May 14 '17 at 21:20
  • \$\begingroup\$ @ nitsua60 You're correct! My rogue leveled up to 2 at the end of the session. The question is a bit level specific, but I am very interested in how it scales. \$\endgroup\$ – David Robinson May 14 '17 at 22:49
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    \$\begingroup\$ +1 for the other uses of bonus action for rogues. Although the OP heavily implied that DPR was the criterion for "better", they also asked for answers based on logic. Logic dictates that the combatant that survives ultimately does more damage. \$\endgroup\$ – keithcurtis May 15 '17 at 2:15
  • \$\begingroup\$ It should be worth noting that by virtue of multiple dice rolls, using two weapons also has the added benefit of a more centered damage distribution (i.e. smaller chance of dealing very little damage, at the cost of smaller chance of doing a lot of damage), and is thus inherently more predictable (and less risky). \$\endgroup\$ – tomasz May 15 '17 at 12:05
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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.

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  • \$\begingroup\$ Is the idea that it simulates both dagger attacks vs. the rapier attack padded out by a 'null' attack? \$\endgroup\$ – nitsua60 May 15 '17 at 1:30
  • \$\begingroup\$ @nitsua60 exactly \$\endgroup\$ – Dale M May 15 '17 at 1:36
  • \$\begingroup\$ @bastijn these comments don't appear to be suggestions to Dale nor requests for clarification--it seems you've got the makings of your own "use booming blade!" answer in there. As such I'm deleting them from this post. Feel free to post your own answer, and if necessary an elected moderator can reproduce the contents of these comments for your use. \$\endgroup\$ – nitsua60 Nov 11 '17 at 14:55
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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.

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  • \$\begingroup\$ "You choose to only use your dagger if you miss with the rapier" is allowed by what rules exactly? \$\endgroup\$ – kviiri Nov 11 '17 at 10:23
  • \$\begingroup\$ @kviiri this answer might be discussing the trade-off between using a bonus action to make a second attack vs. all the other useful things a rogue might do with it. @ peachkadubrey: this answer's drawing flags for being "very low quality." I think you'll find it gets a better reception if you explain why {rapier+dagger} is a preferable to either rapier or two daggers. \$\endgroup\$ – nitsua60 Nov 11 '17 at 14:51
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    \$\begingroup\$ @nitsua60 I may have parsed the answer incorrectly, but it's wrong nevertheless as two-weapon fighting requires both weapons to be light. \$\endgroup\$ – kviiri Nov 11 '17 at 14:57

protected by doppelgreener Nov 11 '17 at 14:27

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