Which two-handed weapon is best for a character to wield in 5e Dungeon & Dragons? Many nominally have the same or similar max damage but the dice used in rolling them have very different probability distributions, so even if the max damage is the same your average damage on hit is not.
Specifically this will be an analysis of heavy two handed weapons. A longsword wielded in two hands has a damage die of d10, the same a a glaive, but it can’t be used with Great Weapon Master (GWM) so it is harder to force higher concentration checks with it; not all of this will apply. I’m going to focus on expected damage per hit- a whole series of articles could be written on the trade offs between damage types and interactions with various feats.
Average Weapon Damage:
All die have a uniform distribution, assuming they are fair, their average being the median of possible rolls. 3.5 for a d6, 5.5 for a d10, and 6.5 for a d12, with E being the expected value or average of a successful hit with a weapon, not including attributes and other bonuses.
EX: E[1d6] = (1 + 2 + 3 + 4 + 5 + 6)/6 = 21/6 = 7/2 = 3.5 or
(1/2)*[6+1] = 3.5
Greatsword & Maul: E[2d6] = 3.5 + 3.5 = 7
Greataxe = E[1d12] = 6.5
Glaive, Pike & Halberd = E[1d10] = 5.5
Greatclub = E[1d8] = 4.5
The 2d6 weapons narrowly pull ahead of the Greataxe, though the axe offers a stronger interaction with the Half-Orc racial ability and a greater weight to the higher end of damage results, which can be useful for inflicting higher concentration checks on spellcasters. Polearms noticeably fall behind though they offer potential damage increases and battlefield control through the Polearm Master and Sentinel Feats, especially when combined with Tunnel Fighter Unearthed Arcana fighting style or the Knight UA fighter archetype.
The different damage types are relevant, notably skeletons are resistant to piercing damage, but again these are determined by the DM so I won’t try to quantify them. Also encumbrance is generous enough in this addition that a strength based character should be able to carry a backup weapon with a different damage type.
Averages with Great Weapon Fighting Style
The Great Weapon Fighting Style allows a 1 or 2 on the die to be rerolled- I will model the impact by replacing 1s and 2s in all distributions with the averages calculated above. All results will be rounded to the 10s place.
E[1d6|GWF] = (3.5+3.5+3+4+5+6)/6 = 25/6 = 4.2, so:
Greatsword & Maul = E[2d6|GWF]= 4.2 + 4.2 = 8.4, gain of 1.4
Greataxe = E[1d12|GWF]= (6.5+6.5+3+4+5+6+7+8+9+10+11+12)/12 = 88/12 = 7.3, gain of .8
Polearms = E[1d10|GWF]= (5.5+5.5+3+4+5+6+7+8+9+10)/10 = 63/10 = 6.3, a gain of .8
Greatclub = E[1d8|GWF]=(4.5+4.5+3+4+5+6+7+8)/8 = 42/8 = 5.3, a gain of .8
The 2d6 weapons not only have a higher average but disproportionately benefit from GWF since they have a much higher probability of rolling a 1 or a 2 with 2 smaller dice. On average they deal at least 1.1 more damage than a Greataxe, the equivalent of a +1 to every roll in a system that does not offer many damage increases. Absent other factors the Greatsword and Maul are a better choice for characters looking to wield two handed weapons.
Notably the Greatclub is outperformed by the 1d8 one handed weapons(E=6.5) such as the rapier, or even short sword(E=5.5) with the Duelist fighting style which also allows the wielder to have a shield or other utility item in the offhand, such as a spell focus. It objectively the worse than a 1 handed weapon for a character that gets to pick a fighting style and will not be considered further.
Polearm Master’s hilt attack also has an expected increase of .5 damage, but this conflicts with GWM bonus action attacks as well as with class features such as the Frenzy Barbarian’s extra attack or some Battlemaster maneuvers- the action economy is complex and I won’t try to quantify it.
Notably paladins with an extra 2d8(E[2d8] = 9) smite damage even with a level 1 slot will likely get less than 50% of their damage from their actual weapon- in these cases it may be better to take the Armored fighting style to conserve resources and help keep them up to use all their smites. Again the Tunnel Fighter style combined with the Sentinel feat in theory allows unlimited attacks per round against an enemy who isn’t careful in their movements. Depending on your campaign and DM it may be a better pick.
From here on out I will focus on the trade off of greataxe versus the 2d6 weapons in the case of critical hits.
Critical Hits with Savage Attacks (or Brutal Critical 1)
The Greataxe is better in the event it scores a critical hit and is wielded by a Half-Orc who may roll and additional weapon die in the event of a crit, from their Savage Attacks feature. In this case another d12 is rolled instead of only 1d6 for the Maul of Greatsword, offering an expected damage increase of:
E[1d12] – E[1d6] = 7.3 – 4.2 = 3.1 or 3 if GWF is not selected, with an increase of only about 2 if a pole arm is used.
My first instinct was simply to treat crits as a 1/20 chance but we are considering only the pool of hits- in which case the proportion of critical hits rises substantially.
For example, at level 1 an optimized strength based martial character using the stat array should have a +5 to hit modifier, +3 from strength and +2 from proficiency. A goblin has an AC of 15. You must roll at least a 10 to hit the goblin, an 11/20 chance- so the probability of hitting is 11/20 with one of those hits being a crit.
Therefore the expected increased damage from a crit with, which occurs in one of 11 hits while fighting a goblin, is E[crit]/11, or in the case of a Half-Orc wielding a Greataxe, 3/11 = .3, compared to an expected increase of 1.1 damage from rolling the greatsword damage again compared to a greataxe.
Even with only a 55% chance to hit the extra critical damage is much less than the roughly +1 bonus to damage from wielding a maul or greatsword. For the greataxe/half orc combo to be worth while we need:
1 = 3/(21 – x)
with x being the minimum roll d20 roll to hit a given enemy, and rounding the 2d6 advantage down to 1 for simplicity. The less likely you are to hit the more valuable crits become. However, we can solve this equation to find the minimum to hit roll needed to make the greataxe better than the maul or greatsword on average. We multiply both sides by the denominator:
21-x = 3 => x = 18
The crit combo give less expected damage unless you hit only on an 18 or higher; an armor class of 23 for an optimized level 1 character. For context the Tarrasque, a CR 30 creature, has an AC of 25. It seems the greatsword or maul is a better choice even for half orcs unless all the enemies you fight have Shield prepared. For martial characters of other races the 2d6 weapons have more expected damage per hit, including for crits, and as such are optimal even for crit fishing builds.
Expanded Crit Range with Savage Attacks
What if our half orc is a champion fighter with an expanded crit range? For the 19-20, we have two crits in our pool of hits, or an expected gain of 6 over the greatsword, and an expected gain of 9 for a 15th level champion fighter who crits on a 18-20. The minimum to-hit rolls needed to make them viable are:
1 = 6/(21-x) => 21-x = 6 => x = 15
1 = 9/(21-x) => 21-x = 9 => x = 12
With Champion Fighter we start to get some plausible to-hit rolls to make the half-orc greataxe build outperform the 2d6 weapons. For a level 3 fighter the AC that makes this perform better is a 20, plausible if you are fighting an enemy with the Shield spell or an actual shield. For the 15th level champion with a +10 to hit, enemies with buff spells and magic armor are quite likely to have an AC of 22 with magic equipment, but it’s a long time for this to pay off and stop being situational.
Brutal Critical (Barbarian)
Barbarian has a class feature, Brutal Critical, similar to the half-orc racial except that it adds up to 3 weapon die. We have effectively examined up to the 3 die case with a tripled chance of critical, with a trippling of the crit chance being equivalent to rolling 3 die instead of one on a crit, at least from the standpoint of averages. So a 17th level Barbarian will find it worthwhile to wield a greataxe in the same circumstances as a 15th level half-orc champion fighter, the loss of expected damage per swing being offset by the extra crit damage against heavily armored enemies.
However this class feature stacks with the half-orc racial, the better to encourage that classic race/class combination, allowing up to 4 extra weapon die to be rolled in the event of a crit, in addition to the usual doubling of damage dice.
A critical with 4 extra weapon die (A 17th-level half-orc barbarian):
1 = 12(21 -x) => 12 = 21 -x => x = 9
or at least a minimum target AC of 19 for a barbarian capable of rolling this many dice, assuming maximized strength.
Expanded Crit Range and Brutal Critical
A character with two extra weapon die on a crit and an expanded crit range of 19-20, say a half-orc champion fighter 3/barbarian 9 is statistically identical to a level 17 half orc barbarian with 4 extra weapon die on a crit, with the greataxe being superior with a minimum roll to hit of 9 or more without advantage. Bonuses acrue quickly from here, for 3 weapon die from 17 levels in barbarian or 13 levels and the half orc racial:
1 = 18(21-x) => x = 3
so that the greataxe deals more damage per hit against almost any enemy except those not wearing any armor. A 20th level half-orc with Champion Fighter 3/ Barbarian 17 will always have a higher damage per hit, the cutoff to-hit roll for the greataxe being negative.
1 = 24(21 -x) => 24 = 21 -x => x = -3
However this is unlikely to be worth giving up the increased to hit chance, damage, AC, and hp from the barbarian capstone.
Critical Hits with Advantage and Savage Attacks
Adding advantage makes things complicated- crits become more likely but a hit in general is also more likely. We are going to have to get into higher statistics for this- the Cumulative Distribution Function of a uniform distribution is:
CDF: (x – a – 1)/(b – a) with a being the start of the distribution and b being the highest draw possible. The CDF is the probability of drawing a number of x or less from the distribution, or in this case rolling x or less. For a d20 a=1 and b=20, and we subtract an additional 1 from x as we hit on a tie, so we have:
CDFd20: (x- 2)/(19) for 0 < x < 20
So our probability of hitting on a given roll is 1 – CDFd20, again with x being our number needed to hit. Our probability of hitting with advantage is our miss chance squared as we must roll under with both rolls, or:
Chance to hit = 1 – CDFd20^2 = 1 – [(x-2)/(19)]^2
assuming away an AC of 1 as no such creature exists in the Monster Manual and it breaks the math. Our expected gain from a crit with a half orc wielding a great axe is:
3*(probability of a critical hit given a hit) = 3*[(crit chance)/(hit chance)]
The probability of a critical hit given advantage is:
1- (19/20)^2 = 1- .9025 = .0975
So the cutoff to hit roll for a non-champion fighter half orc wielding a greataxe to outperform a 2d6 weapon is:
1 = 3*[.0975/(1 – [(x-2)/(19)]^2)]
The solution to which is x = slightly less than 18 according to Wolfram Alpha, barely a budge from the attack without advantage. So even with advantage, say from barbarian’s Reckless Attack, this math doesn’t work out with 1 die.
However, gains are larger with two extra weapon die on a crit:
1 = 6*[.0975/(1 – [(x-2)/(19)]^2)]
with a solution of about 14 for Wolfram Alpha, compared to 15 without advantage. The gap widens significantly for 3 weapon die, with a minimum to hit roll of 9 or higher needed to make the greataxe have a higher average per hit, compared to a 12 without advantage, a condition quite likely to be met at any level.
With Brutal Critical a half orc barbarian can add up to 4 weapon die on a crit, giving us:
1 = 12*[.0975/(1-[(x-2)/19]^2)]
Which has an imaginary result for x- meaning it is always better for a 17th level Half-Orc barbarian to wield the greataxe and Reckless attack regardless of the minimum AC to hit, at least for the purposes of maximizing damage.
Expanded Crit Range with Advantage & Savage Attacks
Unlike the barbarian weapon die, the champion fighter expanded crit range interacts differently with advantage than it does with normal hit probabilities- we must recalculate the crit probability in each instance. Notably this only applies to half-orc champion fighters or those multiclassed into barbarian, or both, and an unmulticlassed champion fighter will need their allies to give them advantage.
Probability of Crit on 19-20: 1 – (9/10)^2 = .19
Probability of a Crit on 18-20: 1-(17/20)^2 = .2775
So the minimum to-hit roll needed to make greataxe superior for a Champion Fighter/Half orc with advantage or a Chamption Fighter 3/Barbarian 9 is:
1 = 3*[.19/(1 – [(x-2)/(19)]^2)]
or x = 15 since rolling a 14.4 is not possible. The result for a Barbarian 3/Champion Fighter 15 or Half-Orc Chamption fighter 15 is:
1 = 3*[.2775/(1 – [(x-2)/(19)]^2)]
or x = 10, a plausible requirement; an AC of 20 on enemies will not be uncommon at this level.
Expanded Crit Range with Advantage and Brutal Critical
For two weapon die and an expanded crit range the cutoff is:
1 = 6*[.19/(1 – [(x-2)/(19)]^2)]
which again gives an imaginary result so with at least two extra weapon die on a crit, expanded crit range, and advantage, the greataxe is the two-handed weapon with the highest expected damage per hit for any possible minimum roll to hit. This can be achieved by a half orc Champion Fighter 3/Barbarian 9.
With a 17th level barbarian and 3rd level champion fighter half-orc 4 weapon die on a crit with expanded crit range is possible, but the greataxe is already proven superior without the expanded crit range in this case so it will not be considered.
15th level Champion Fighters will need to be at least level 24 to benefit from Brutal Critical, so multiple weapon die and a triple crit range will not be considered.
TL,DR: The greatsword and maul almost always have a higher expected damage per hit, the greataxe being superior only for high level half-orc barbarians or champion fighters, or characters with levels in both, going up against heavily armored enemies. Most campaigns will not reach the point where the greataxe is superior, the crit damage just doesn’t scale fast enough, and the fall off is worse for polearms.
Given that a character with disadvantage has a 1/400 chance of critting I will not consider damage trade offs when attacking with disadvantage, it seems unlikely a greataxe crit build will ever perform well in that case.
I wanted to look at how the increased weight of polearms and the greataxe at the upper tail of their damage distribution increases the likelihood of forcing a failure on a concentration check but the topic is extensive enough that it deserves its own article.