## Saturday, November 4, 2017

### A Rule for Shields from Descending Armor Class, Applied to Ascending Armor Class

I've been playing 'D&D 'ever since I was a young boy and found my dad's Dungeon Master's Guide on his bookshelf (the original, by Gygax).  I was still too young to understand a lot of it (Gygax opens with a discussion of uniform vs binomial probability distributions...), but I spent hours of days flipping through the pages, looking at the pictures and reading the descriptions of a fantastic world.  It inspired me to make my own games about exploring through planned dungeons with friends and having them use ability scores to get through.

By the time I was old enough to try to really figure out the rules, version 3.0 was out, which made a lot of changes.  One of the biggest was replacing the original Descending AC and THAC0 system with the slightly more intuitive Ascending AC system.  This system always just made sense to me, whereas DAC and THAC0 always seemed weird and complicated.  I never understood how DAC worked, and never really saw a good explanation of it, and so just never used it.  It was only recently, when getting into old school tabletop RPGs, that I decided to look up an explanation of how DAC and THAC0 used to work.

In the process, I found there are all kinds of debates about which is better, which frankly is stupid because unless you're using the original B/X table-lookup method (which does not correspond to any simple linear equation), then THAC0 and DAC is exactly identical to Attack Bonus and AAC.  Algebraically, one is just a different way of writing the same equation.

Let DAC = Descending Armor Class, AAC = Ascending Armor Class, AB = Attack Bonus.

In 1e/2e rules, where unarmored DAC is 10, to determine if you hit, you roll 1d20 and check if
1d20 >= THAC0 - DAC.
If it is, then you hit.

In 3.0/3.5 rules, where unarmored AAC is 10, to determine if you hit, you roll 1d20 and check if
1d20 + AB >= AAC.

Notice, if you use the conversion
AAC = 20 - DAC
AB = 20 - THAC0
then this latter method just gives
1d20 + 20 - THAC0 >= 20 - DAC
which, after you cancel the 20's and move the THAC0 over is just
1d20 >= THAC0 - DAC.
So both give the exact same thing.  (That's actually the condition you use to find this conversion.)

I guess the real argument is over which is easier to use.

I have always thought AAC is the most intuitive, because better should be bigger!  An AC of 200 sounds super tough, while an AC of 0 sounds like you have no armor.  That's what made the most sense to 8-year-old me, anyway; whereas I only understood the Descending system after growing up and getting a bachelor's degree in mathematics.

That doesn't mean DAC doesn't have some benefits, though.

The idea of the DAC system is that your armor class represents the chance to hit you, as opposed to a score needed to hit you.  Smaller AC means a smaller chance to hit you.

DAC works by effectively treating the wearer's armor class as a modifier to the attack roll, and has a constant DC of 20 for every attack -- so an attack roll could also be expressed as
1d20 + AB + DAC >= 20.
This, I think, makes a bit more sense of the Descending system.  The smaller DAC,  the harder it is to hit.  This is also a pretty intuitive way to phrase it.

It is also possible to easily phrase things in terms of "chance to hit" with the DAC system.  Calculate Attack Bonus as
AB = 21 - THAC0 + modifiers
Your chances to hit the target are
(DAC+AB)/20
so if you have an AB of 3 and are trying to hit someone with DAC 4, then you will succeed 7/20 times, or 35% of the time.  This leads to a different way to run attack rolls, which is a hit on
1d20 <= DAC + AB,
so roll under or equal to the sum of DAC and Attack Bonus, consider a 1 (an "ace") to be a crit; a fumble is now a 20.  Of course, this requires calculating Attack Bonus differently (consistent with OE and B/X rules of DAC 9 for unarmored), which means the normal Attack Bonus from standard d20 rules is increased by 1.

At first glance it seems crazy, but after some thought it actually makes a bit more sense than the old method, since it correlates more directly with probabilities to hit.

One other cool flavor feature of DAC is that it makes a clear demarcation between natural and supernatural, as even full plate armor with a shield and high DEX only brings you to 0 DAC.  To get anything below 0 DAC, you have to have magical enchantments.  If a creature has a DAC of -4, it is immediately put into the category of otherworldly and you know to expect a difficult battle.  Whereas the equivalent AAC of 24 just isn't as neat.  It's just 4 higher than 20.  There isn't any clear, non-arbitrary line between what is naturally achievable vs. what is the result of powerful magic.

None of this is actually the point I wanted to address, though.

What I actually wanted to talk about is an interesting house rule I came across for shields that depends on the DAC system.  I saw this rule on stackexchange when looking for info on DAC.

The rule figures that shields get a lot of disrespect, only offering a single measly point of AC bonus.  In reality, shields were a crucial part of armed combat.  To make them a bit more awesome, this user made the house rule that a shield halves your DAC.

So if you are unarmored (DAC 10), your DAC with a shield is 5.  If you're wearing chainmail (DAC 4), your DAC with a shield is 2.   If you're in plate (DAC 2) your DAC with a shield goes down to 1.

It's a pretty neat rule idea.  You can see some of the limits, though.  For one, it seems way too powerful at higher DAC.  Someone with a DAC of 18 (representing I guess someone stark naked, bound at the ankles, with a target painted on them and a magic curse) gets lowered to DAC 9, the same as wearing padded armor.  For two, at higher AC, it stops working, and then starts hurting.  Someone with a DAC of 0 gets no benefit from a shield.  Someone with magical armor granting them DAC -2 is actually hurt by the shield, up to DAC -1.

Despite all that, though, it's not a bad rule.  It's simple, and makes shields a bit more special.

The original poster said this was one of the benefits of DAC, because there was no simple way to do this with AAC.  Which of course is wrong.  Doing the math on paper, this rule amounts to: halve your AC, then add 10; that's your new AC with a shield.

In an equation,
New AC = 10 + 0.5*(Old AC)

The hard part (dividing by 2) is present in both, whereas adding ten just amounts to mentally placing a "1" digit in front of 1/2 your AC.  It's barely "math" and more like squinting.  If your AAC is 12, then half is 6, goes to 16.  If you AAC is 18, then half is 9, goes to 19.  It's basically the same difficulty as the original method.

Working this out, you will see corresponding limitations.  After AAC 20 (corresponding to DAC 0), a shield stops being useful and becomes detrimental, whereas someone with the miserable AAC of 2 (DAC 18) gets brought up to a whopping AAC 11 (DAC 9).

But, it still gives a neat take on the shield.  Most neat of all is that it suggests a way to grade shields.  This works for DAC or AAC, but I will just use AAC to illustrate.

Rather than all shields halving your DAC, you can instead say that different shields give a different fractional modifier.  The three that seemed to work best (numerically) were 2/5, 1/2, and 3/5 shields, which alter ACs as
New AC = 10 + 0.4*(Old AC)
New AC = 10 + 0.5*(Old AC)
New AC = 10 + 0.6*(Old AC)
I'm not exactly sure what to do with rounding (truncate, round up, round the way you learned in chemistry class?) and the original poster didn't specify and only used even numbr examples.  (The corresponding benefit to DAC, if frac = fractional benefit, is frac_DAC = (1 - frac_AAC), i.e. 2/5→ 3/5, 2/5 → 3/5).

A chart of how these work is as below (I use rounding down)
 Old AC 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 2/5 10 10 10 11 11 12 12 12 13 13 14 14 14 15 15 16 16 16 17 17 18 18 18 19 19 20 1/2 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 3/5 10 10 11 11 12 13 13 14 14 15 16 16 17 17 18 19 19 20 20 21 22 22 23 23 24 25
This adds that bad shields (crappy wooden ones, say) not only give you less of a bonus, but they stop being useful sooner; whereas good shields (magical mirror shields, say) not only give you more of a bonus, but continue being useful even for higher ACs.

You could also decide that above the limit of usefulness, the  shields just give a static +1 bonus.

The limit of usefulness is calculated as 10/(1 - frac) rounded down.

The major problem is this really is not easy to do in your head.

The math is much easier when the multipliers are 1/something, like 1/4, 1/3, 1/2.  However, these cut off in usefulness far too soon (13 for 1/4 and 15 for 1/3 respectivly); though if you resolve to use a constant bonus thereafter, it may be worthwhile.  The table is
 Old AC 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 1/5 10 10 10 10 10 11 11 11 11 11 12 12 12 12 1/4 10 10 10 10 11 11 11 11 12 12 12 12 13 13 13 1/3 10 10 10 11 11 11 12 12 12 13 13 13 14 14 14 15 15 1/2 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20

which carries up to just past where the shield "bonus" becomes detrimental.  Notice the bonus increases by 1 in integer steps or 5, 4, 3, and 2.

The cutoff of usefulness to certain shield classes makes some roleplaying sense.  If you are wearing rune-enchanted plate made of star-iron from the realm of Fey, but also carrying a shield made of tanned hide stretchd over a bone frame that you got back in the first dungeon off a liazardman -- then really, why bother with it?  It's time to throw that shield away and get a better one.  But if you have a shield made of the same rune-enchanted star-iron, then it also makes sense that this shield is awesome and contributes to protect your character even into supernatural AAC levels.

At the same time, if you've got nothing but a big solid moveable wall to hold up in front of yourself, you're going to rely on it an awful lot.  You can kind of see this, in that Greek hoplites wearing leather armor carried giant shields to battle, whereas the classic tourney knight in full plate wasn't above grabbing his opponents sword with his hand midswing.  And even if you're naked and bound at the ankles and magically cursed to absorb all thrown projectiles, ducking behind a piece of wood will still help you deflect a lot of the blows.

So I think an idea that works, using different shield classes, is to have the shield cut off when it cuts off, and provides no benefit to AC after that; not even a constant +1 benefit.

To put everything together: rounding down and cutting off, the table works out to:
 Old AC 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 1/5 10 10 10 10 10 11 11 11 11 11 12 12 12 13 14 15 16 17 18 19 20 21 22 23 1/4 10 10 10 10 11 11 11 11 12 12 12 12 13 13 14 15 16 17 18 19 20 21 22 23 1/3 10 10 10 11 11 11 12 12 12 13 13 13 14 14 14 15 16 17 18 19 20 21 22 23 2/5 10 10 10 11 11 12 12 12 13 13 14 14 14 15 15 16 16 17 18 19 20 21 22 23 1/2 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 21 22 23 3/5 10 10 11 11 12 13 13 14 14 15 16 16 17 17 18 19 19 20 20 21 22 22 23 23

Beyond AAC 23, you need a really incredible shield for it to be worth anything compared to your crazy-powerful armor.

A 1/5 shield seems kind of worthless -- maybe this is an old oaken branch your PC picks up in desperation.  A 1/4 shield might be leather stretchd over a wooden frame or made lashed bones.  A 1/3 shield might be a simple wooden shield, made of a solid flat piece of lumber.  A 2/5 shield might be a banded wooden shield made of planks of wood fixed together by a metal frame.  A 1/2 shield can be a shield made of steel or other metal.  A 3/5 shield may be magical or of superb craftsmanship.

There are other ways to give shields back the respect their deserve.  You might consider damage reduction, and in particular the "Shields Shall be Splintered" rule.  These have the benefit of being a bit simpler to implement.

Or if you're not above some 6th grade Algebra at the game table (or look-up), then consider the option above.