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Some class features, monster traits, and feats allow you to take a −5 to your attack roll in order to gain a +10 to your damage roll. At first glance this looks really good, but when will it deal more damage than not using it on average?
For instance if you have a total attack bonus of +8 and you deal 1d12 damage. You have to roll a 7 to hit an AC of 15. If you take the −5 that makes it where you have to roll a 12. That's going to reduce your average damage by a lot.
There are three main factors for influencing whether or not you should use the feats Great Weapon Master (GWM) or Sharpshooter (SS):
The main evaluation criteria for this should be the expected gains in damage after GWM/SS. That is, you have to answer this question: if I expect to do X damage before GWM/SS, and I expect to do Y damage after GWM/SS, then is Y-X positive?
Whenever the answer to that is yes, use GWM/SS.
Here's a graph of the expected gains from GWM/SS with base damage = 1.
Here's the same graph with base damage = 10.
Here it is again with base damage = 30.
As you can see, the lines represent the gains in using GWM/SS. If it is above the Zero Line, then use GWM. If it is below that, don't use it. If it is exactly at that line, GWM/SS is immaterial.
Expected Damage Gain After GWM/SS: This is the expected damage after GWM/SS minus the expected damage before GWM/SS
Target on Die: This is the number you need to roll on the die after modifiers are applied. So if the target's AC is 18 and you have a +8, your Target on Die is 10
As you can see, the higher the Target on Die gets, the less valuable GWM becomes -- or does it? There are several very interesting trend you notice right off the bat.
This is not surprising. If you can dish out 30 damage without GWM/SS, and you use GWM/SS, then you are also risking dealing no damage on a miss by taking a penalty.
The higher your base damage, the more you have to lose on a miss. Whereas, the additional 10 damage is always constant, so it is more significant if your base damage is small.
Sounds obvious. When you have advantage, there is a good chance for you to hit. If you use GWM/SS, you diminish that benefit.
This is when you want to be using Bless/Bane/Inspiration/Bend Luck to lower that Target on Die as much as possible.
While not very intuitive at first, this makes sense if you think about it.
When you are at a disadvantage and your Target on Die is small, you do not want to take a penalty because you risk missing your attack, and there is still a chance you might hit. But if your Target on Die is large enough, then you don't stand a chance to hit it anyway, with or without a penalty. If you take the penalty from GWM/SS, you can do bonus damage. So why not just take the penalty and hope for the best?
The maximum damage per hit where a damage trade-off is viable can be calculated as
Let \$P_1\$ = Percentage to hit before trade-off
Let \$P_2\$ = Percentage to hit after trade-off
Let \$D\$ = Increase in damage
$$\text{Maximum Damage Per Hit }= \frac{P_2 \times D}{P_1 - P_2}$$
The following spreadsheet shows the maximum damage where the -5 to hit for +10 to damage trade is worth it assuming no advantage/disadvantage.
If you have advantage we need to square your chance to miss so...
If you have disadvantage we need to square your chance to hit...
In general the higher your odds of hitting before making the trade the higher your base damage can be while keeping the trade beneficial. The inverse (low odds to hit need low base damage to be worthwhile) is also true and bottoms out with an abysmal worst case in disadvantage where if you are doing even 1 damage per hit normally it's a bad trade.
The edge cases here are when you're rolling for 20's or hitting even if you rolled -3. In either case since the -5 to hit doesn't change your chances you might as well go for it regardless of how much damage your base attack does.
This calculation ignores overkill. Enemies reduced to 0 hp are as dead as enemies reduced to -10 hp. If an enemy is likely to be killed without the extra 10 damage then you may prefer the greater chance to hit. Your preference here is going to be determined by a combination of your damage dice, your estimate of the enemies hp, your enemies expected damage in the coming round, your current hp, and so on. That calculation is highly situational and there's no one good answer to it.
Ceribia's answer, Markovchain's answer, and this forum post all describe very detailed what happens statistically if you use a -5/+10 feat. The calculations are done and are easy to follow, so I'm going to elaborate a bit on the consequences, provide a few examples.
That means +6 to hit and base average damage of 11.33. (source)
That means +10 to hit and base average damage of 13.33.
That means +6 to hit and base average damage of 27.08. (2d6 + 3 + 3d8[Divine Smite], GWFS applies to all dice here) (source, source)
That means +10 to hit and base average damage of 39.58. (2d6 + 5 + 4d8[Divine Smite] + 1d8[Improved Divine Smite], GWFS applies to all dice here)
It means that if you can increase your Base average damage (be it by additional dice from spells or feats, or by flat bonuses from magic weapons, feats or however else), the -5/+10 becomes less and less attractive. If you have less boni to damage and/or high boni to hit (for example from Bless or Precision Attack, the -5/+10 becomes more attractive.
(In many cases they are actually decreasing DPR)
If you want that, just take the Ability Score Increase. Great Weapon Master and Sharpshooter are not meant for every situation, as proven by the calculations in other answers. It is more of a way to convert excess hit chance to damage.
Even if your calculations show that GWM is beneficial, because you are fighting a prone, naked elf, you should consider that it wont be any more dead from 20 damage than from 10, when it has only 1 HP.
If you impose any conditions on a hit (prone, poisoned, frightened, etc), it might be more benefitial to attack without the -5.
The Archery fighting style provides a +2 bonus to attack, so it can happen quite frequently that -5/+10 is worth it for the Ranger but not the Barbarian.
On the other hand, proning is probably the easiest way to provide Advantage, but only for melee characters.
The top 2 answers explain perfectly when you are supposed to use it, but once you have it, it is too late.
If your allies have a way to reliably provide Advantage for attacks, you just take the feat.
If not:
1st level: Human Variants could take it, but unless you are playing the Zombie Apocalypse, don't. With your low hit chance against the usual opponents, you will not have the opportunity to use it very often. Polearm Master / Crossbow Expert increases your DPR much more often.
4th level: A Battlemaster Fighter can quite reliably prone with Trip Attack. From level 5 it is possible to attack normally and prone, and use the second attack with -5/+10. Otherwise an Ability Score Improvement provides better average DPR increase.
8th level: Same as level 4.
12th level: This is where you should take it, and only because you can not increase the ability above 20.
Good adveturing groups have ways to provide Advantage for heavy hitters, the above advice is only meant for the less fortunate.
Level 4, Greatsword, Great Weapon Fighting Style, Strength 16. Is GWM better than ASI?
Base:
Attack: +5 (Str 16, Proficiency +2)
Base damage: 11.33 (Great Weapon Style, Str 16)
DPR vs AC 12: 7.93 (0.70 x 11.33)
DPR vs AC 15: 6.23 (0.55 x 11.33)
DPR vs AC 18: 4.53 (0.40 x 11.33)
ASI:
Attack: +6 (Str 18, Proficiency +2)
Base damage: 12.33 (Great Weapon Style, Str 18)
DPR vs AC 12: 9.25 (0.75 x 12.33)
DPR vs AC 15: 7.39 (0.60 x 12.33)
DPR vs AC 18: 5.55 (0.45 x 12.33)
GWM:
Attack: +0 (Str 16, Proficiency +2, -5)
Base damage: 21.33 (Great Weapon Style, Str 16, +10)
DPR vs AC 12: 9.59 (0.45 x 21.33)
DPR vs AC 15: 6.40 (0.30 x 21.33)
DPR vs AC 18: 3.20 (0.15 x 21.33)
Crit: 0.42 (0.05 * 8.33)
In most cases ASI gives you a better DPR than GWM.
There are several other answers here, but, IMO, none of them actually answer the real underlying question:
$$Maximum\,AC=\left\lfloor\frac{\left(2\,\times\,Attack\,Bonus\right)\,-Average\,Damage\,+\,32}{2}\right\rfloor$$
(The \$\lfloor\,\,\rfloor\$ indicate the mathematical floor operation or, here, rounding down.)
So, if you have a +8 attack bonus and deal 1d12 (i.e., 6.5) damage, your result is:
$$\left\lfloor\frac{\left(2\times8\right)-6.5+32}{2}\right\rfloor=20$$
So, with a +8 to attack, deal 1d12 damage on a hit, and the target's AC is 20 or lower, then it is mathematically correct to use -5/+10.
Let's say you have a level 9 Fighter with 20 Str, Great Weapon Fighting, Great Weapon Master, and a +1 greatsword. Your attack bonus is \$4+5+1=10\$. Your damage on a hit is \$2d6\,(8.33)+6=14.33\$. (The average of 2d6 here is 8.33 and not 7 due to Great Weapon Fighting.)
$$\left\lfloor\frac{\left(2\times10\right)-14.33+32}{2}\right\rfloor=18$$
So, mathematically, you should use -5/+10 on any target with an AC of 18 or less.
The above is assuming that you do not have advantage or disadvantage. I will cover that below, but the reality is that using -5/+10 is essentially always correct when you have advantage, and conversely it is essentially always incorrect when you have disadvantage.
Yes, you do have an outliers such as when you require a natural 20 to hit the target, or when you've got Dex 10 and a Blowgun with the Sharpshooter feat, or sometimes when you have a very large number of sneak attack dice, but none of those cases very common.
Further, critical hits also have no bearing on the calculation at all because critical hits add the same amount of damage to both types of attacks and occur at the same rate on both attacks. A natural 20 always hits, and essentially every high level Champion is already always going to hit any target's AC on a natural 18 or better even with the -5 no matter how your DM interprets a critical hit.
Let's see how we arrive at that formula.
I think it's easy to see that answer will depend on three factors:
The first two you can know pretty easily before the game even begins. The target's AC, however, is a value that varies for every combatant. Therefore, it will be most useful to determine what AC is the most effective.
So, what we want to know is:
$$Expected\,damage\,from\,normal\,attack<Expected\,damage\,from\,-5/+10\,attack$$
Expected damage from a normal attack is, in most cases, best understood as the mean average damage on a hit multiplied by the chance to hit.
$$Expected\,damage\,from\,normal\,attack=Average\,damage\times\frac{21+Attack\,Bonus-Target\,AC}{20}$$
Expected damage from a -5/+10 attack is the same, but we need to write it using the same terms as above. So, we get:
$$Expected\,damage\,from\,-5/+10\,attack$$
$$=(Average\,damage + 10)\times\frac{21+Attack\,Bonus-5-Target\,AC}{20}$$
$$=(Average\,damage + 10)\times\frac{16+Attack\,Bonus-Target\,AC}{20}$$
So, that gives us this inequality:
$$Average\,damage\times\frac{21+Attack\,Bonus-Target\,AC}{20}$$
$$<$$
$$(Average\,damage + 10)\times\frac{16+Attack\,Bonus-Target\,AC}{20}$$
Now we just need to solve for Target AC. However, I'm lazy, so I made Wolfram Alpha do it. I used \$a\$ for the average damage, \$b\$ for the attack bonus, and \$x\$ for the target AC. I get the solution:
$$x<\frac{1}{2} (-a + 2 b + 32)$$
Which is the same as:
$$Target\,AC<\frac{\left(2\,\times\,Attack\,Bonus\right)\,-Average\,Damage\,+\,32}{2}$$
When the above inequality is true, it is mathematically correct to use -5/+10 on your attack.
You can repeat the above method to determine functions for advantage and disadvantage by substituting the different equations for calculating to-hit into the above inequality. However, you quickly run into pretty monster equation solutions for both advantage and disadvantage due to square roots.
However, here's the solution for advantage:
$$\frac{-\sqrt{a^2+10a+1600}-a+2b-8}{2}<Target AC<\frac{\sqrt{a^2+10a+1600}-a+2b-8}{2}$$
Here's the solution for disadvantage:
$$Target\,AC<\frac{-a-\sqrt{a^2+10a}+2b+32}{2}$$ $$Target\,AC>\frac{-a+\sqrt{a^2+10a}+2b+32}{2}$$
Again, where \$a\$ is the average damage and \$b\$ is the attack bonus.
Let's take the same level 9 Fighter as above with +10 attack bonus and 14.33 average damage on a hit.
Advantage:
$$-23.24 < Target\,AC < 20.91$$
Disadvantage:
$$Target\,AC < 9.50$$ $$Target\,AC > 28.17$$
And you can always test a given AC by calculating the damage per attack for that specific AC.
Note that with disadvantage you're getting results that are basically off the die, since you can't roll a 9 on d20+10.
Let's take an extreme example to test the rule of thumb. Let's take a level 19 Rogue/Level 1 Fighter with Sharpshooter, Crossbow Expert, Dex 20, a +3 Hand Crossbow, Bracers of Archery (+2 damage), and Archery Weapon Style (+2 to hit). We've got \$6 + 5 + 3 + 2 = 16\$ to hit. We do \$1d6 (3.5) + 5 + 3 + 2 + 10d6 (35) = 48.5\$ average damage.
Normal:
$$Target\,AC < 7.75$$
Unsurprisingly, the -5 from Sharpshooter it too much here. The +10 damage doesn't pay off when we're dealing with nearly 50 damage on average.
Advantage:
$$-45.56 < Target\,AC < 21.06$$
It's still preferred for essentially all targets here (the number of enemies with AC greater than 21 can be counted on one hand, I believe).
Disadvantage:
$$Target\,AC < -18.89$$ $$Target\,AC > 34$$
And, unsurprisingly, we're off the die here.
One final note, but we're pretty explicitly ignoring the idea of overkill here. If you're fighting particularly weak enemies, then +10 damage might be entirely overkill damage. That is, you'd have killed the target without the +10 damage anyways. That's technically within the scope of the discussion above, but I've only answered when it's correct to use the ability based on maximizing the damage value. IMO, overkill is just something that isn't much of a concern.
It's pretty trivial to know the AC of a target but fairly difficult to know it's HP. Further, many situations where you might overkill are obvious (e.g., kobold scouts).
However, if you want to consider overkill, then what you need to do is change how you calculate average damage. However, this really sucks because your average damage is different between both sides of the inequality.
Say we're looking at a fighter with duelist style dealing \$1d12 + 7\$ damage and attacking a creature with 13 hit points. Average damage caps at \$13\$ then. You can't ever deal more than 13 damage. That means instead of \$6.5 + 7\$, we calculate the average of \$8, 9, 10, 11, 12, 13, 13, 13, 13, 13, 13, 13\$ which is \$11.75\$. Then we calculate average damage with the +10, which is \$13\$ since rolling a 1 still deals more damage than the creature has. Doing this calculation with weapons dealing \$2d6\$ damage is much more annoying because there are \$36\$ outcomes. If you're looking at calculating with Sharpshooter and \$5d6\$ Sneak Attack damage with a \$1d6\$ hand crossbow, good luck because that's \$6^{6}\,=\,46,656\$ outcomes to average. Not difficult, but not trivial.
Now we use our inequality method, but we can't write it in the same terms anymore so it's a much bigger pain.
Again I solved for \$x\$ with Wolfram Alpha. I used \$a\$ for the average non-GWM damage, \$b\$ for the attack bonus, \$d\$ for the average GWM damage, and \$x\$ for the target AC. You want to use the solution where \$a\,\ge\,0\$ and \$a\,\gt\,d\$ since both of those should always be true. When I have time I'll come back and translate it and update the question again.
Let's take a Fighter or Ranger with the Archery Fighting style into consideration.
We'll start with level 3, max dexterity, using a Long Bow (1d8 piercing damage) for simplification.
Your normal attack modifier would be: 5 (20 dex) + 2 (level 3 proficiency bonus) + 2 (Archery Fighting Style) = +9 to hit. Your attack roll is 1d20 + 9, and 1d8 + 5 damage.
This build is already hitting 83.3% of most shots against any creature a party of 4 level 3 players are going to fight except a few that present a Deadly Challenge.
Adding in Sharpshooter (The ranged equivalent of the melee attack version) gives you a -5 penalty to hit for +10 damage on top of whatever damage you are rolling for the weapon itself. Taking the same example, the attack modifier then becomes +4 . Which is: 5 + 2 + 2 = 9 - 5 = 4 . Your attack roll becomes 1d20 + 4 and 1d8 + 5 + 10 for damage. So if this attack hits, that's a guaranteed 15 damage before we even roll the 1d8 damage dice for the Long Bow.
Most average characters with a strength or dex of 14 / 16 are hitting at + 4 / +5 at level 3 anyway. a +4 to hit against level 3 encounters that are below Deadly Encounter CR ratings are going to hit 68% of the time.
The Risk / Reward of Sharpshooter / Great Weapon Master is quite mathematically sound and I have seen it used to brutal effectiveness given a build made around balancing out the -5 penalty. Where it becomes hard to determine mathematical soundness is when you add in homebrew creatures or Encounters that are unorthodox or a group of more than 4 players where challenge ratings are calculated different and thus harder creatures with Higher AC will show up.
Are there other variables that factor in to just how and when taking -5 to hit for +10 damage will be effective? Yes, quite a few in fact, but these are base percentages for a level 3 character taking in to consideration average conditions.
Add in Bless (for an extra 1d4 to the attack bonus) or a Bards Inspiration Dice and this penalty is non existent and can let you tackle even higher AC encounters.
There are far too many variables to give this a definitive yes or no. In other words, the trade off is completely situational and depends on dozens of factors.
There's no real way to tell you if it's mathematically sound when there's many major variables (and countless other minor ones) in the equation.
Examples:
X = modifier being used to attack with (STR or DEX)
Y = your proficiency level
Z = target's AC
So basically if you're making an attack, your major variables amount to:
(D20 roll) + X + Y >= Z in order to hit.
If the target is a low AC creature, sacrificing -5 to hit might be worth it because you're making up for it by gambling on a 70% chance to hit anyways. Whereas the minor variables start weighing in when you want to get an almost guaranteed hit.
Possible minor variables include but are not limited to;
Health of party members, protecting an NPC, number of enemies, your own current health, protecting an innocent, and any other factor that would rely on you either making sure you hit, or potentially gambling on taking a target out with a single shot.
Basically, there's too many variables to take into consideration to give a real definitive, "Yes, this is worth the swap," because in some situations it's demonstrably untrue (fighting an AC 23 creature), and in other situations you should be using the +10 damage (fighting creatures with much lesser AC.)
Then you have to start factoring in advantage and disadvantage, which sways the numbers even more. And if you're playing a Rogue does the chance of missing your sneak attack for +10 flat damage suddenly make it worth it...... just..... way too many variables for a definitive answer.
From personal experience: Mobs of low AC creatures means spam the +10 on hits. Large, single target creatures with high AC, don't try for the extra +10. But that's really, really simplified.
