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My DM does not like Advantage/Disadvantage. While I disagree with his reasons on this, he has decided to replace it with a +2/-2 rule and I respect him (mostly) enough to follow his rules.

I recently realized that this probably has significant impact on my champion fighter, who gets critical on 19/20. Since I will still need to roll natural 19/20, and natural 19s and 20s will be less likely without the second die from advantage, I'm interested in knowing mathematically how much this house rule affects me.

Assuming the fighter is using a d8 weapon (war pick), what is his average damage output including a % chance for critical in using both Advantage/Disadvantage as well as the +2/-2 rule?

AnyDice would be appreciated so I can tweak as needed. I can't seem to wrap my head around how to do the math!

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Assumptions

I'm going to be making the following assumptions, based on what you've already provided:

  • 3rd level (since you get a crit on a 19 or 20)
  • 16 Strength (no ASI to bump up to 18)
  • Fighting a CR 3 creature (for base math)
  • Average damage is 7.50 (4.50 from the die +3 Str mod)
  • Average crit damage is 12.00 (4.50 per die +3 Str mod)
  • Attack bonus is +5 (+2 prof, +3 Str mod)
  • The enemy has AC 13 (per the DMG guidelines on page 274)
  • DPR calculations are:
    • Crit damage = Crit % x average crit hit damage
    • Normal damage = Hit % x average hit damage
    • hit % = 100 – [miss %] – [crit %]

Note that any flat modifier to the damage total won't change with a crit, since you only double the dice rolled, not the modifiers added.

Champion Fighter

Per the DMG page 274, a CR3 creature has an average AC 13, meaning you need to roll an 8 or higher.

With Advantage

AnyDice can tell us our miss chance and our crit chance. From there, we know our hit chance.

  • A normal roll of 1d20 will have a 35% miss, 10.00% crit (1.20 DPR), and 55.00% normal hit (4.13 DPR) for a total DPR of 5.33
  • A roll with advantage will have a 12.25% miss, 19.00% crit (2.28 DPR), and 68.75% normal hit (5.16 DPR) for a total DPR of 7.44
  • A roll with your DMs +2 rule will have a 25.00% miss, 10.00% crit (1.20 DPR), and a 65.00% normal hit (4.88 DPR) for a total DPR of 6.08

With Disadvantage

AnyDice can tell us our miss chance and our crit chance. From there, we know our hit chance.

  • A normal roll of 1d20 will have a 35% miss, 10.00% crit (1.20 DPR), and 55.00% normal hit (4.13 DPR) for a total DPR of 5.33 (unchanged)
  • A roll with disadvantage will have a 42.25% miss, 1.00% crit (0.12 DPR), and 56.75% normal hit (4.26 DPR) for a total DPR of 4.38
  • A roll with your DMs -2 rule will have a 45.00% miss, 10.00% crit (1.20 DPR), and a 45.00% normal hit (3.38 DPR) for a total DPR of 4.58

Rogue

How does this change affect other classes, specifically those who rely on bonus damage dice? I'm using a rogue for this example since sneak attack is easy enough to calculate, but a paladin falls under the same heading with their smite spells and the like.

We use the same percentages and base damage (assume Dex and a rapier) for our fighter, but we add sneak attack damage. That's 2d6 at level 3, so with advantage we add +2d6 (7) on a hit and +4d6 (14) on a crit. Attacks without advantage don't get sneak attack damage added in, so the disadvantage numbers from above carry over (I know you can get sneak attack damage without advantage, but we'll ignore that for simplicity).

With Advantage

  • A normal roll of 1d20 will have a 35% miss, 5.00% crit (1.30 DPR), and 60.00% normal hit (8.70 DPR) for a total DPR of 10
  • A roll with advantage will have a 12.25% miss, 9.75% crit (2.54 DPR), and 78.00% normal hit (11.31 DPR) for a total DPR of 13.85
  • A roll with your DMs +2 rule will have a 25.00% miss, 5.00% crit (1.30 DPR), and a 70.00% normal hit (10.15 DPR) for a total DPR of 11.45

With Disadvantage

  • A normal roll of 1d20 will have a 35% miss, 5.00% crit (1.30 DPR), and 60.00% normal hit (8.70 DPR) for a total DPR of 10
  • A roll with disadvantage will have a 42.25% miss, 1.00% crit (0.12 DPR), and 56.75% normal hit (4.26 DPR) for a total DPR of 4.38 (identical to the fighter, as no advantage means no sneak attack)
  • A roll with your DMs -2 rule will have a 45.00% miss, 5.00% crit (1.20 DPR), and a 50.00% normal hit (2.25 DPR) for a total DPR of 3.45

Conclusion

With your DMs proposed houserule, the expected DPR for any class is going to be decreased because of the fact that you're still only rolling 1 die, so the chance of a critical hit will not change. The biggest, well, advantage of rolling with advantage is it almost doubles your chance of a crit: 9.75% vs. 5.00% for a normal 20 crit and 19.00% vs. 10% for a champion fighter crit.

Indeed, that simple change reduces the overall expected damage output of the entire party, especially those classes that rely on burst damage in the form of more dice. As you gain in levels and get the extra attack feature, magic items/spells that add damage dice, and class features that change the damage dice done, the gap will only increase.

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    \$\begingroup\$ I may not have been very clear here. The +2 has no affect on the critical part. With the DM house rule I still have to roll a 19 or 20. A 17+2 will NOT count as a critical. \$\endgroup\$ – Guy May 11 '16 at 17:45
  • \$\begingroup\$ Oh, gotcha. I'll edit it when I have the chance \$\endgroup\$ – Christopher May 11 '16 at 17:45
  • \$\begingroup\$ Hell, I just noticed my math is off anyway. I'll have to do a full revamp when I have time. Gotta start work, though, so it'll be a while. \$\endgroup\$ – Christopher May 11 '16 at 18:09
  • \$\begingroup\$ @Guy updated with the right maths. Hope it helps! \$\endgroup\$ – Christopher May 11 '16 at 19:41
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    \$\begingroup\$ Disadvantage numbers added, @Tijnkwan, along with comparison to rogues who rely on more damage dice. \$\endgroup\$ – Christopher May 11 '16 at 23:04
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While there are certainly more robust answers on this page already, a friend told me that a simple Expected Value equation can illustrate the difference at a more "base level". This doesn't include any of the additional on hit effects (such as sneak attack, +x damage, etc.) that will certainly exacerbate the problem. It also doesn't incorporate AC which changes expected damage output dramatically.

Base Math

Average Critical Damage * Critical Chance % + Average Damage * Non-Critical Chance %

Using Advantage

9(.19) + 4.5(.81) = 5.355 expected damage on hit

Without Using Advantage

9(.1) + 4.5(.90) = 4.95 expected damage on hit

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Conclusion

For a fighter with critical hits on 19-20, attacking with Advantage gives a 40% increase in damage for AC 13 foes, 64% for AC 18 foes, 86% for AC 23 foes.

Attacking with the +2 bonus the GM is using in place of Advantage, these numbers are 14%, 22% and 48% respectively.

In other words, a +2 Attack Bonus is one third to one half as effective as the Advantage mechanic.

Personally, if my GM removed the Advantage mechanic, I'd ask for the Champion fighter feature "Crit on 19-20" be replaced with "+5 Attack Bonus".

Automatic Hits

Note that a critical hit is not only a doubling of the damage dice, it is also a guaranteed hit (Crawford, Crawford). If you are fighting a foe with high AC (something you need to roll a 19 or 20 to hit, regardless of bonuses), then removing the Advantage mechanic halves your DPR.

Attacks

Assuming: +5 Attack Bonus (level 3 fighter with 16 STR and no other bonuses), critical hit on 19-20.

Normal attacks:

  • Vs AC 13: 10.00% critical hit, 55.00% normal hit, 35.00% miss.
  • Vs AC 18: 10.00% critical hit, 30.00% normal hit, 60.00% miss.
  • Vs AC 23: 10.00% critical hit, 5.00% normal hit, 85.00% miss.

Normal attacks +2 (the GM's replacement for advantage):

  • Vs AC 13: 10.00% critical hit, 65.00% normal hit, 25.00% miss.

  • Vs AC 18: 10.00% critical hit, 40.00% normal hit, 50.00% miss.

  • Vs AC 23: 10.00% critical hit, 15.00% normal hit, 75.00% miss.

With advantage:

  • Vs AC 13: 19.00% critical hit, 68.75% normal hit, 12.25% miss.

  • Vs AC 18: 19.00% critical hit, 45.00% normal hit, 36.00% miss.

  • Vs AC 23: 19.00% critical hit, 8.75% normal hit, 72.25% miss.

Expected Damage

Assuming an average damage 4.5 (martial weapon) +3 (STR) (which means 11 damage on a critical hit and 7.5 damage on a normal hit).

Normal attacks:

  • vs AC 13: 532.50 damage per 100 hits.

  • vs AC 18: 345.00 damage per 100 hits.

  • vs AC 23: 157.50 damage per 100 hits.

Normal attacks +2 (the GM's replacement for advantage):

  • vs AC 13: 607.50 damage per 100 hits.

  • vs AC 18: 420.00 damage per 100 hits.

  • vs AC 23: 232.50 damage per 100 hits.

With Advantage:

  • vs AC 13: 743.63 damage per 100 hits.

  • vs AC 18: 565.50 damage per 100 hits.

  • vs AC 23: 293.63 damage per 100 hits.

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  • \$\begingroup\$ Minor detail, but important: a crit isn't an auto hit, but an auto hit is a crit. PH 19: "If the d20 roll for an attack is a 20, the attack hits regardless of any modifiers or the target's AC. In addition, the attack is a criticai hit, as explained later in this chapter." A crit on a 19 isn't an auto-hit if you needed to roll a 20+ to hit, and thus a miss. Page 191 further says, "When you score a criticai hit, you get to roll extra dice for the attack's damage against the target." Not that a crit automatically hits. \$\endgroup\$ – Christopher May 12 '16 at 2:57
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    \$\begingroup\$ @Christopher Crawford has clarified otherwise here and here. \$\endgroup\$ – CTWind May 12 '16 at 4:17
  • \$\begingroup\$ That's an…interesting misinterpretation of the rules. \$\endgroup\$ – Christopher May 12 '16 at 13:16
  • \$\begingroup\$ Makes sense to me. Critical hits are a type of hit, and Improved Critical says you score a critical hit on a 19 or 20. If the word 'critical' wasn't there and it just said 'your weapon attacks score a hit on a 19 or 20' I would certainly read that as not being able to miss on either of those rolls. \$\endgroup\$ – CTWind May 12 '16 at 16:52
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I realized, as I was going online to relearn probability to the point where I could be useful to you, that I wasn't sure if your DM does things the 3.5 way and only lets you crit if you roll a 19 or 20 naturally.

If so, then your crit rate (success AND failure) remains unchanged, no matter the stats, and all you get is bonus to-hit.

Average roll on a d8 is 4.5, so if you want to know your average damage per hit, there you go. Crits do an average of 9 damage. This guide assumes that, like Advantage and Disadvantage, you get no inherent bonuses to damage for having it, only what you get for landing a crit.

+2 = 10% chance of crit, 5% chance of flop.

-2 = 10% chance of crit, 5% chance of flop.

However, the thing about Advantage and Disadvantage is that, by increasing the number of times you roll, you get more chances to hit it big or fail horrendously, so that would suggest that these bonuses should factor into success or failure.

If the system he's using doesn't factor this in, point this out to your DM and see what he says about it. Go with what he says regardless of his answer as long as he understands the statement, but just make sure he knows.

If the bonuses are factored into the roll (read "treated as the natural roll") as opposed to being tacked on at the end, then it will look very different...

+2 = 20% chance of crit (crits on 17, 18, 19, 20), ?% chance of flop (1+2=3, so either advantage means you can't bungle something in this system, or it's calculated normally, the bonus added after you see the result. BRING THIS UP WITH YOUR DM).

-2: 0% chance of crit (20-2 = 18 aka not a crit. Wow, that's rough), 15% chance of flop (3-2=1 (>_<))

In this version, where the bonuses are trying to mimic the tendencies shown in the normal version, saying the buffs and debuffs are too strong is an understatement. Worse yet, it's incredibly restricting and unrealistic, as having advantage means you can't fumble, and having disadvantage means you can't crit, which makes anything that grants these bonuses to you and these debuffs to the enemy are OP AF, which makes halflings, the Lucky feat, and rogues in general OP.

And I don't even want to CONSIDER half-orc ferocity + greataxe + barbarian rage + advantage... Q-Q

However, without the use of the new "natural" number, the bonuses don't pack enough of a punch to really matter, which shoots the whole concept of Advantage and Disadvantage in the foot. The whole point of 5e was to get away from tiny bonuses and long addition in favor of quicker, easier mechanics, hence the double dice-rolling.

A character with Disadvantage could potentially roll 20/20 and crit with a blowgun. A character with Advantage could roll Snake Eyes and bungle a back-stab and sneeze at the most inopportune moment. It's simple, easy to keep track of, and does its job.

Advantage and Disadvantage are set up in such a way that there are hundreds of possible resolutions. 400, to be precise.

With Advantage, your chances of getting a crit are equal to the number of results where you roll a 19 or 20, which means that the other die can be any number that is equal or lower and not change the outcome.

If one die is a 20, the other die can be any number from 1 to 20, so we have 20 possible win conditions for that die, however, the other die could be a 20, and that would be a win condition for that die too, so when we add them, we count the result of 20/20 only once. 19 + 19 + 1 = 39

Rolling a 19 for a crit means any number from 1 to 19 counts as a new win condition, but, like before, 19/19 is only counted once. 18 + 18 + 1 = 37

We add them together to get 76/400 possible resolutions, or 19% chance of a crit with advantage using a war pick. Not bad!

With Advantage, the only way to get a critical fumble and not default to a number higher than 1 is to roll two 1's, which is only one possible state out of 400. 1/400 = .25%, so that's slightly more balanced than flat-out immunity to critical fumbles.

With Disadvantage, it's different. The only way to get a crit is to roll a 19 or higher on both dice, so that's 4 possible win conditions. (19/19, 19/20, 20/19, 20/20). 4/400 = 1/100 = 1% chance for a crit with disadvantage. Sure beats 0% chance amirite?

The chance of a critical fumble is equal to the number of scenarios where a single one is rolled. The other die can be anything from 1 to 20, so that's 20 results for each individual die, save for 1/1, which is in both categories and will only be counted once because of that. 19 + 19 + 1 (sound familiar?) = 39/400 = 9.75% chance, which is less severe than 15%.

In Conclusion The exact workings of the bonuses are unclear. Make sure you understand them by communicating with your DM. Once you have an answer, refer to the section dealing with this. Adding the bonus to the end of your roll after seeing the natural result ala 3.5 doesn't pack the punch required to substitute for the Advantage system. It's better than nothing, but less helpful than Advantage and less of a risk than Disadvantage. Trying to make the bonuses stronger by giving them the ability to make a roll of 17 a "natural" 19 and therefore a crit, or a roll of 3 a "natural" 1 and therefore a critical fumble, is too powerful as a buff or a debuff, both for and against the players. Advantage and Disadvantage should be given a chance. If it helps, roll the two d20s simultaneously to save time. Show this to your DM by copy/pasting it into a Word document and then sending it to him. Either email it or print it and hand it to him.

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    \$\begingroup\$ I'm not sure where you're going with the first section of this answer, especially when you mention criticals "the 3.5 way". At least in D&D, a critical always refers to the NATURAL dice roll, before any modifiers. I don't see any evidence as to why his DM would try to apply the modifier to the natural roll. \$\endgroup\$ – Discord May 11 '16 at 19:12
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    \$\begingroup\$ The question says that the roll must still be a natural 19 or 20. You can edit your answer to cut out the material that's based on not knowing that, which would tighten it up considerably and make it a better answer. \$\endgroup\$ – SevenSidedDie May 11 '16 at 19:31

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