Falling off the RNG (Heavy Lifting)

In order to simplify my Falling off the RNG post, I removed a lot of the detailed calculations.  I felt they detracted from the post itself.  However, they may still be of interest (and I’d hate to throw them away after the time and effort I put into them).  I’m putting them here for those who like to see the work behind the conclusions.

Comparing Existing Editions

To start, I’m going to look at three different editions of D&D.  Since I have Nik on IM (hi Nik!  Thanks for your help) I’ll cover 4e first, then 3.x, then probably AD&D 2e (I have the books more or less handy).

The calculations and meandering thoughts below were used to find the values described in Falling off the RNG.

Dungeons & Dragons 4e

I don’t have personal experience with D&D 4e, nor even the books, so I’m looking to Nik to help me on this part.  By necessity I will simplify things slightly.  For example, in many cases you can use different ability scores to modify attack or defense scores; I will tend to use the common one.

D&D 4e went to great lengths to make things as balanced as possible between classes.  Attacks and powers can use different ability score modifiers (most weapons use Strength modifier, arcane attacks often use Intelligence) and can target different defenses (Armor Class and what used to be considered Saving Throws — Fortitude, Reflex, and Will).

Attacks are usually d20 + level bonus + ability modifier + enhancement bonus [+ proficiency bonus], against a defense that is usually 10 + level bonus + ability modifier + enhancement bonus [+ armor bonus; usually no ‘proficiency’ to be had].

I am told that a Fighter 10 can expect about +15 total attack bonus (+5 level bonus +5 Strength +2 magic weapon +3 proficiency), against Armor Class, and that a Wizard 10 can expect about +12 total attack bonus (+5 level bonus +5 Intelligence +2 magic implement) against Reflex (which tends to be a few points lower than Armor Class).

In other words, you can expect that the Fighter and the Wizard will hit, with their normal attack against appropriate targets, with about the same frequency.

Damage for at-will martial attacks is usually [W] + Strength modifier, and for at-will magical attacks usually 1d8 + Intelligence modifier.  Most one-handed weapons do 1d8 damage, but there are variations (larger weapons do more damage, lighter weapons do less)… for the point of this conversation, it is probably sufficient to equate these.  At-will powers at the epic tier tend to increase this by 1W or 1d8.

In short, it looks like everyone hits with about the same frequency and for comparable amounts of damage, at any particular level, with their good at-will powers.  It is possible to see some variance (a low-Strength fighter who doesn’t take steps to correct this), but overall it looks like the designers kept things very tightly constrained and are pretty solidly on the RNG.  In fact, I understand that the attack bonuses and defenses tend to grow at a rate somewhat less than one point per level.  Characters ten levels apart will have bonuses less than ten points apart, well within the RNG.

It’s pretty safe to say that within the powers examined here, classes are remarkably well-balanced with each other.  Encounter and daily powers appear to vary a great deal more, enough that for the purpose of this discussion I will ignore them, except to note that this is probably where you see most of the differences between the classes’ abilities.

Dungeons & Dragons 3.x

Okay, things are more complex here.

Ignoring multiclass characters, there are three progression rates for attack bonuses.  ‘Good’ Base Attack Bonus increases at one point per level, ‘Medium’ BAB increases at three quarters of a point per level, and ‘Poor’ BAB increases at one half of a point per level.

Assuming a character with Good BAB will try to maximize this, you can expect a Fighter to keep his Strength at a maximum (improve it every four levels, maximize the Strength buff item, find and use wish spells to build it up).  It is trivial to see how to get a Medium-size human Fighter to an effective Strength of 34 at 20th level

Honestly, in my campaign this would be either 29 or 23, depending on era.  I long ago ruled that the level-gained ability score bumps were inherent bonuses and thus did not stack with those from wish spells (I never have seen this definitively ruled), and a while after that I dropped enhancement bonuses from my campaign.  However, I digress; rules as written there are enhancement bonuses, and I believe the wish gains stack with level bumps.  Of course, this isn’t including racial bonuses — a half-orc here would have a Strength score two points higher.

Let’s assume the Fighter optimizes.  At 20th level he can expect to have an attack bonus around +36 (+20 BAB, +9 Strength, +5 magic weapon, +2 Weapon Focus) [or +39 if level and wish gains stack].

A Cleric (+3/4 BAB) would probably start with a lower Strength score and probably wouldn’t develop it as much.  At 20th level a +6 belt of giant strength is chump change, but he probably hasn’t put level gains into his Strength score (but have spent wish spells on it… but I’ll assume not).  This character effectively has Strength 20, and would thus have an attack bonus somewhere around +26 (+15 BAB, +5 Strength, +5 weapon, +1 Weapon Focus) [maybe +28 if he used four wish spells to bring his Strength up a bit].

A Rogue (+3/4 BAB) would almost certainly take Weapon Finesse (use Dex in place of Str, with light weapons and rapiers).  At 20th level he would have an attack bonus somewhere around +30 (+15 BAB, +9 Dexterity, +5 weapon, +1 Weapon Focus) [+33 if level and wish gains stack].

A Wizard (+1/2 BAB) probably started with an even lower Strength score, but might have found +2 gauntlets of ogre power or a +4 belt of giant strength).  For simplicity, let’s say he effectively has Strength 12.  He may have an effective attack bonus of +14 or so (+10 BAB, +1 Strength, +3 magic weapon, no weapon focus).

Also there are iterative attacks — the Fighter can get four, at +36/+31/+26/+21, the Cleric can get three, at +26/+21/+16, and the Wizard can get two, at +14/+9.

The different classes do damage roughly in proportion to their total attack bonuses.  Between higher Strength and Weapon Specialization and access to better weapons and Power Attack, the Fighter can lay down a lot of hurt.  The Wizard, not so much (milder weapon, lower Strength, no Specialization, lower enhancement bonus on the weapon).  The Cleric lands somewhere in the middle.  The Rogue can actually do a fair bit of hurt as well, that +10d6 sneak attack damage can leave a mark.

No matter how subtle the wizard, a knife between the shoulderblades will seriously cramp his style.

— Vlad Taltos

So far, sounds pretty unbalanced.  The Fighter and Wizard are clearly off the RNG, and Rogue and Wizard are just hanging on.  Let’s look at Armor Class.

• The Fighter tanks up and gets to AC 43 pretty easily (+5 mithril full plate, +5 large shield, +5 ring of protection, +5 amulet of natural armor, +3 Dexterity).  However, it counts only as AC 18 against touch attacks, AC 40 when flatfooted.
• The Cleric follows the same path for AC 43 as well.  However, it counts only as AC 18 against touch attacks, AC 40 when flatfooted.
• The Rogue goes with lighter armor but still manages AC 38 (+5 mithril chain shirt, no shield, +5 ring of protection, +5 amulet of natural armor, +9 Dexterity).  AC 29 against touch attacks, AC 29 flatfooted.
• The Wizard can’t wear heavy armor or a shield, practically, and ends up around AC 33 (bracers of armor +8, +5 ring of protection, +5 amulet of natural armor, +5 Dexterity, he can afford to push that a little more).  AC 20 against touch attacks, AC 28 when flatfooted.

At full AC we’re still on the RNG, but that can be stripped pretty hard (the heavy armor is useless against touch attacks).  Let’s check saving throws.

• The Fighter has Fort/Ref/Will of +23/+14/+13 (assuming Con 22, Dex 16, Wis 14 and a +5 cloak of resistance)
• The Cleric has Fort/Ref/Will of +22/+14/+26 (assuming Con 20, Dex 16, Wis 29 and a +5 cloak of resistance)
• The Rogue has Fort/Ref/Will of +16/+26/+13 (assuming Con 20, Dex 29, Wis 14 and a +5 cloak of resistance)
• The Wizard has Fort/Ref/Will of +16/+16/+23 (assuming Con 20, Dex 20, Wiz 14 and a +5 cloak of resistance)

Okay, best and worst are within 13 points of each other, though the numbers above assume a certain amount of optimization (all four using +5 cloaks of resistance, for one thing).  It is trivial to come up with degenerate cases — a Ftr20 with Wis 8 who doesn’t do something to improve that and doesn’t get a +5 cloak of resistance could have a Will save of +5 and fall off the RNG (the Cleric has Will +26).  Because of the consequences of failed saves, he will hit very hard and Darwin the Merciless will scrape up his remains and cart them away.

It is pretty clear that D&D 3.x does not exhibit the same kind of balance as D&D 4e.

Advanced Dungeons & Dragons Second Edition

This part should go much more quickly than either of the two above.  I haven’t played D&D 4e and had to get the information second-hand, I have played D&D 3.x but there are a lot of options.  I have also played AD&D 2e and it’s much simpler.  I will convert to bonuses rather than work with the AD&D values (decreasing THAC0 and decreasing saves).

• A Fighter 20 has an attack bonus of +29 (THAC0 1 = +19, +4 from Strength 21, +5 sword, +1 specialization).
• A Cleric 20 has an attack bonus of +19 (THAC0 8 = +13, +3 from Strength (gauntlets of ogre power), +3 mace).
• A Rogue 20 has an attack bonus of +15 (THAC0 11 = +10, +0 from Strength (gloves of dexterity), +5 short sword).
• A Wizard 20 has an attack bonus of +10 (THAC0 14 = +7, +0 from Strength, +3 dagger)

Somewhat lower than D&D 3.x, but the Fighter and Wizard are hanging onto the RNG, very barely (the Wizard can tie the Fighter if he’s lucky — 1/400 chance).

Armor Class is a little closer, perhaps.

• A Fighter 20 can expect AC 30 (+5 full plate, +5 large shield, probably not Dex > 14 — and a cap of AC -10 anyway)
• A Cleric 20 can expect AC 30 (+5 full plate, +5 large shield, probably not Dex > 14 — and a cap of AC -10 anyway)
• A Rogue 20 can expect AC 22 (+5 leather, -5 Dex (Dex 21, being generous) — could be as much as 29 if he uses defenses such as the Wizard, but magic leather armor is archetypal)
• A Wizard 20 can expect AC 24 (bracers of defense AC 2, ring of protection +6, probably not Dex > 14)

AD&D 2e had no consideration of touch attacks that avoid armor, surprise might be able to cost you Dexterity bonus to AC but it’s usually so small I’ll ignore it.  We’re well within the RNG here.

Saving throws are pretty close, compared to later editions.

Saving Throw	Fighter 17+	Cleric 19+	Rogue 21+	Wizard 21+
Paralyzation, Poison Death Magic	3	2	8	8
Rod, Staff, or Wand	5	6	4	3
Petrification or Polymorph	4	5	7	5
Breath Weapon	4	8	11	7
Spell	6	7	5	4

There are suggestions about using ability score modifiers to adjust saving throws, but since they don’t do much and infrequently apply even when the rule is applied (you need Dex 16 or better to get a +1 to dodging-type saves) I’ll ignore them.  The saves are all on the RNG (smallest difference for a single save type is three between best and worst, biggest difference for a single save type is seven between best and worst).