I've always liked playing a tanky character. I don't know why, exactly - maybe just a general anxiety about injury or death, or perhaps it's the sense of heroism at being able to take the hits for your friends. My main character in World of Warcraft for nearly twenty years (dear lord) is a Protection Paladin, and my first really long-term D&D campaign character was an Eldritch Knight Fighter who focused primarily on getting his AC up to absurd levels.
So, given that, I figured I'd talk broadly and try to figure out what the real cost of using a shield rather than going with a more aggressive equipment loadout winds up doing for you.
The Basics:
At the most basic level, a mundane shield gives you an additional 2 AC. That means, in absolute terms, a 10% lower chance of being hit with attacks. It's not really that, though, exactly, because of how statistics can warp depending on the way we're looking at them. For example, if you have Chain Mail on and are fighting a creature with a +4 to hit, they'll have a 45% chance to hit you if you don't have a shield versus a 35% chance to hit you if you do have one, but what that means is that their chance to hit you is actually just about 78% of its previous hit chance, meaning a reduction in hit chance of 22% (See how insane this is? I think that's why we have the adage "There are three kinds of lies: lies, damned lies, and statistics").
So, obviously, there's better survivability with a Shield than without one. But we're also giving up damage. Again, at the most basic level, barring something like a higher-level Shillelagh cantrip, one-handed melee weapons cap out at a d8 damage die. A two-handed weapon, though, can get up to 2d6 (as usual, we're setting aside modern and futuristic firearms). That's a difference between an average of 4.5 and 7, so about 2.5 damage.
Complicating Things:
The thing is, while 4.5 is only about 64% of 7, it's not in a vacuum. We add a modifier to this damage (likely Strength if we're using melee weapons,) which smooths out the difference a bit. If we have +3 to Strength (likely at level 1 if we're a Strength-focused character) we'll be hitting for 7.5 versus 10 on average with mundane weapons, which means that, say, a Longsword is doing 75% of the damage of a Greatsword. All the various bonuses we might get to these weapons, be it a +2 bonus to damage from a +2 weapon, or maybe 2d6 extra damage from a Vicious Weapon, or even adding 1d6 to each hit with a spell like Hunter's Mark, will further smooth out the damage distinction. If we're a Paladin hitting with a +2 Longsword or Greatsword and doing a 2nd level Divine Smite against a fiend and have +5 to Strength, we're talking about 5d8+7 versus 2d6+4d8+7, or 29.5 versus 32, which gives the Longsword 92% of the damage of the equivalent Greatsword - pretty darn close.
But Also Complicating Things the Other Way:
That being said, there are some other elements that can widen that gap. No one-handed weapon has the Graze mastery, for example, which can (and I think most often) significantly increase the damage-per-attack of a weapon (even with relatively low ACs, though you hit diminishing returns when you start to need to get a Nat 1 to miss). Similarly, there are feats like Great Weapon Master that can add a good chunk of damage to a heavy, two-handed weapon.
In the above example with the +2 weapons, Great Weapon Master might have added an additional 4 (if we're thinking the character is, like, level 10) to the Greatsword's damage. And depending on the hit chance (or, more accurately, the miss chance,) Graze will also effectively add more to the damage-per-attack because of the cases where the Longsword would have done nothing.
But What About Defense?
It's clear that going with something other than a Shield will increase your damage output, but what are the benefits to the Shield?
Well, Shields can also come in +1/+2/+3 varieties. But it's also impossible to evaluate specifically how much damage they prevent unless we have more context. If we've got +2 Plate, the Defensive Fighting Style, and a +3 Shield, giving us an AC of 27, the monster with a +5 to hit is literally no worse at attacking us than someone with +6 or +7 to hit, and no better than someone with +4, because all of these would only ever be able to hit us with a natural 20. (Is it likely we'd face such a foe at a point in a campaign where we had acquired such gear? Probably not.)
A good while ago, I did some math on how AC affects our total damage taken, and I was surprised at how much less effective it was than damage-reduction features like Rage or Deflect Attacks (I didn't do Uncanny Dodge, but presumably that's comparable to the latter, though I'd guess not quite as good in most cases). That said, I didn't investigate how much really big, significant boosts to AC affect things. A magic shield at most is going to give you essentially a permanent Shield spell (minus the immunity to Magic Missile,) though being able to stack it with that spell (as in the case of an Eldritch Knight) can push your AC to the stratosphere. Still, the existence of crits does create a kind of hard cap on how useful high AC can be - you'll never be able to reduce the enemy's hit chance below 5% (ok, technically you can push it to .25% if you can impose disadvantage).
But it's not like it does nothing. If we figure a monster would hit you 55% of the time for 1d6+2, they'll do an average of 5.5 55% of the time and an additional 3.5 5% of the time, giving them an average damage of 3.025 plus .175, or 3.2 damage per attack. Adding a mundane shield to that means they only hit you 45% of the time, so it becomes 2.475 plus that crit bonus (which hasn't changed) so it becomes 2.65 total per attack, which is about 83% of the damage you would have been taking.
And in a higher-level scenario, with a, say, +2 Shield and, say, Plate armor, if a monster with a +10 to hit is going after you, without the shield they'd be hitting you 65% of the time, but the shield reduces that to 45% of the time. If they're doing 4d6+6 damage (we're looking at a Dao in this case,) that's 20 damage on average on a hit and an additional 14 on a crit. So, 65% of 20 is 13 while 45% of 20 is 9, and then crit bonuses are .7 (boy this is easier when it's a nice round number) so the damage goes from 13.7 per attack to 9.7 per attack, which is only about 71% of the damage we had been taking.
Other Elements:
Another thing to consider is that Shields do enable some other things - the Shield Master feat is one example. I think we also need to address the Dueling fighting style.
Dueling adds 2 damage to our one-handed weapons if that's all we are attacking with, which is typically what we'd be doing if we're using a Shield. And while that might not look like a ton, the equivalent damage-boosting Fighting Styles for other weapons might not do as much. Great Weapon Fighting, which changes rolls of 1 or 2 on damage dice to 3s, is actually a minuscule boost in damage (it turns a Greatsword from an average of 7 to an average of 8, and I believe that and the Maul are the weapons that benefit the most from it). This effectively turns a d8 weapon into a d12 weapon in terms of average damage (though not quite, as they do less on a crit).
Thus, in a vacuum, you might think that this is a no-brainer, but again, what we're missing out on are things like the Graze or Cleave masteries that don't show up on one-handed weapons, as well as the Great Weapon Master feat, which will pull these types of weapons ahead.
Another kind of easy-to-forget element here is Somatic spell components. Technically speaking, without something like War Caster, you can't perform the somatic components of spells if you don't have a hand free. This means that an Eldritch Knight who is going either Sword-and-Board or Dual-Wield will really want to pick up that feat, or they have to do what I did, which is just drop your weapon every time you cast Shield, and then either use an item interaction or your bonus action War Bond to get the weapon back. Because you only need both hands during the attack with a two-handed weapon, you'll have the hand free otherwise when wielding it.
Note though that there's also some ambiguity around Spell Focuses - these can replace material components that aren't consumed or expensive, and you can perform the somatic components of a spell in a hand that is holding a material component. But there's this weird little edge case: if a spell doesn't have a material component, can you perform its somatic components while holding a spell focus? I think a very strict reading of the rules would say no, but I also think any DM who isn't an insane sadist would say yes. Thus, an Eldritch Knight who fights with a quarterstaff could theoretically let that staff be a wizard's staff and thus get around the restriction (I wish that the 5.5 EK had been able to treat any simple or martial weapon as a spell focus. Hell, make that what our War Bond does, and make the feature more relevant!)
But All This Nuance Aside...
Ok, let's say that we want to just get an example of how two characters with similar builds, except for their choice of weapon loadout, will do compared to one another.
I'm going to break from my "level 10 character versus Death Knight" math just because I actually think it winds up being a bit skewed due to the high AC and legendary resistances at play.
Instead, what we're going to look at are two 8th level Eldritch Knight Fighters, both Strength-focused. One has taken the Great Weapon Fighting Style and uses a Greatsword, while the other has taken the Dueling Fighting Style and uses a Rapier (yes, I'm dooming myself to using Vex). The idea here is that, while the sword-and-board character is going for higher armor, they're still trying to output as much damage as possible, and a Vex weapon is probably the best option there for a one-handed weapon (and while we usually think of Rapiers as being used with Dex-based characters, there's nothing preventing you from using Strength with it).
Both will be weaving in Booming Blade as their cantrip, but because they don't yet have Tactical Master, they're not going to be doing any shenanigans with the Push mastery to trigger Booming Blade's secondary damage. Given that they have second level spells, both will also use Enlarge/Reduce to give themselves an extra d4 of damage on a hit, and then Action Surge to get an Attack action on the same turn.
The Greatsword build will take Great Weapon Master, Mage Slayer, and War Caster (admittedly less necessary on a two-hander build) while the Rapier build will swap out Great Weapon Master for Shield Master.
For our target, I think a pretty reasonable monster for an 8th level character to be fighting might be a Troll. (Ok, maybe we should be using Green Flame Blade, but this will do the same damage in a single-target situation). At CR 5, we're at a point where an 8th level party might be fighting two or three of them, depending on how hard the fight is, but we'll assume that they're spread out.
And hey, let's even give them +1 weapons, which is pretty reasonable to assume at level 8. This actually effectively simulates being capped at 20 Strength, as with those feats, they'll actually just be at 19 Strength because War Caster will be used to boost Intelligence instead.
So, let's break it down:
Greatsword Build:
After casting Enlarge on ourselves, we action surge to get in our regular attacks, making two swings with the Greatsword. We have a +8 to hit, and the Troll has an AC of 15, meaning we hit on a roll of 7 or higher, or a 70% hit chance.
We'll start off with Green-Flame Blade (thanks to War Magic at level 7, we can cast Green-Flame Blade as one of our attacks,) which will hit for 2d6 (enhanced by our fighting style)+1d8+1d4 (this from Enlarge) + 5 (a mix of Strength and the weapon's magic bonus). The 2d6 is actually 8 in this case thanks to Great Weapon Fighting. Now, admittedly, I think you could interpret this as also boosting the d4 and d8 from Enlarge and GFB, respectively, as the fighting style says "when you roll damage for an attack you make with two hands," and doesn't specifically call out the weapon damage. I don't know if this was errata, because I could have sworn they narrowed that to only the weapon dice, but let's apply it to the other dice as well to make this pretty bad fighting style a little better. Thus, a d4's average goes from 2.5 to 3.25 and a d8 goes from 4.5 to 4.875.
With Great Weapon Master, we'll now be landing our Green-Flame Blade for 8+3.25+4.875+8, or 24.125, and our crits will be adding 16.125. 4 of that damage is guaranteed thanks to Graze, so we'll pull that out and add it in later. Thus, we're looking at 20.125x70%, or about 14.1, plus 16.125x5%, or about .8, for 14.9, then adding back in that guaranteed 4 to give us 18.9 average damage on this first attack.
Next, the second attack loses the d8 from Green-Flame Blade, so we just do 2d6+1d4+8. As before, we cut out the guaranteed 4 from Graze, so it's now 15.25 on a hit and adding 11.25 on a crit. 15.25x70% is about 10.7, and 11.25x5% is about .6, so we're doing 11.3, then adding back in that 4 from Strength to give us 15.3 damage.
Thus, with these two attacks, we're doing 34.2 damage.
Finally, we need to figure out how likely it is we get a Hew attack. With two chances for a crit, there's a 9.75% chance that one or both of them do give us a crit. But unlike the first two attacks, we don't add our Proficiency bonus. Thus, it's just 2d6+1d4+5, or 16.25 on a hit, or 11.25 on a crit, but Graze still applies, so we can cut out that 4 and add it in later, meaning we can use 12.25x70%, or about 8.6, plus 11.25x5%, which we already know is .6, for 9.2 average damage if we get this attack, though that only happens 9.75% of the time, so it's really only adding roughly .9 extra damage per turn.
Thus, our Greatsword build is doing 35.1 damage per turn.
Rapier Build:
Now, I've taken on the Vexing conundrum of the Vex mastery, which is all well and good in play but a pain to calculate. For the purpose of damaging our foe, though, Shield Mastery is kind of irrelevant, as we'll get advantage on our second attack if we hit with our first anyway, thanks to Vex, regardless of whether we knock the Troll prone. It's still worth a shot because it can help our allies get advantage and also potentially slow them down if we need to open up distance, but it won't affect our damage calculations.
Things also get a little complicated if we try to extrapolate this out to subsequent turns - because one hit makes the next more likely, our chance to hit in absolute terms goes up the longer the fight goes on, though in practice, any miss will effectively reset the counter. With only two attacks currently, our first turn will be relatively simple, but we'll touch on what it looks like on later turns even if we don't do the full calculation.
Our attacks, thanks to Enlarge and the Dueling fighting style and our magic weapon, will hit for 1d8+1d4+7, meaning 14 damage on a hit and an extra 7 damage on a crit.
Our first attack will have the same 70% hit chance we had with the Greatsword, so that will be 14x70%, or 9.8 plus 7x5%, or .35 for crits, giving us 10.15 damage on this first attack.
Now, for the second attack, 70% of the time we'll have advantage, but 30% of the time we won't. That 30% will look like the damage of the first attack. But at advantage, 70% becomes 91%, and our crit chance becomes 9.75%. Thus, in those cases, we'll get 14x91%, or 12.74, plus 7x9.75%, or .6825, for an average of about 13.42. Therefore, to get our actual average damage on this second attack, we're talking 13.42x70%, or about 9.4, plus 10.15x30%, or basically 3, giving us 12.4 average damage on this second attack.
Oh duh, I forgot Green-Flame Blade. That's kind of important.
Now, given Vex, we might actually save Green-Flame Blade for our second attack, given that it increases our hit and crit chance if we had hit on our first attack. I think we can just calculate this separately and add it, and because it's pure dice, we can add the hit and crit chances together. 1d8 is 4.5, so for the cases where we don't have advantage, we're talking 70%+5%, or 75%, and that times 4.5 is 3.375. This happens just 30% of the time, so it's giving us roughly 1 extra damage. Then, if we do have advantage, we're adding 91% plus 9.75% (yes, that'll be over 100%, but that's fine) for 100.75%, giving us, well, basically 4.5. That's going to be 70% of the time, though, so it'll come to 3.15, and thus we add to that second attack 4.15, meaning it's doing 16.4 average damage.
We then add both attacks together to give us about 26.6 damage per turn.
As promised, we'll consider how this pans out in future terms. Our first attack gives us a 70% chance to get advantage on our second attack, but Vex carries over to subsequent turns (and reaction attacks,) so given we have a 70% chance to have advantage on our second attack, that means that there's an absolute chance to hit with our second attack of 91%x70% plus 70%x30%, or about 64% plus 21%, or 85% chance to get advantage on the first attack of the next turn. Then, for attack two on turn two, it becomes 85%x91% plus 15%x70%, or 77% plus about 10%, meaning we're now at an 87% chance to have advantage, and so on and so forth. So, the longer a fight goes, the better your damage is going to be with a Vex weapon, but only in a kind of Schoedinger's superposition of possibilities and absolute probabilities, because if you actually miss, it all resets.
Damage Comparison:
As you can see (and if I actually got my math right) the Rapier doing 26.6 damage per turn and the Greatsword doing 35.1 damage per turn means that for your shield, you're doing about 76% of the damage.
And I will say, this is a scenario with a lot of factors that are smoothing out the difference between the two.
Again, there might be reasons that survivability is a bigger concern for you. But I think that you do wind up getting a better return on investment for prioritizing damage over AC.
That's why, if I ever get to return to my Eldritch Knight character (it'd be great to have that party escape the Nine Hells, where the campaign fell off) and get license to rebuild him, I'd probably go with a Great Weapon build instead of his supreme tankiness.
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