Reconsidering Level Bonus: Hit Points

There was a lot of math involved in this.  Done by spreadsheet, but dear gods it was a lot of work.

At least it suggests that the change being proposed to Level Bonus calculations won’t lead to bad effects in Echelon.

See below for detail.

Old School (AD&D and BECMI) Hit Points

Both BECMI and AD&D used a model where a player would roll hit points at each level up to a certain point (ninth or tenth, depending on class) and add Constitution modifier (possibly variable — fighters in AD&D 2e could have higher hit point modifiers than other classes for Constitution 17 and 18) for each of those levels.  After that, a fixed number of hit points (again depending on class) were added per level and Constitution modifier no longer applied.

Unless (in AD&D only) you were a multiclass character, in which case you divided this value by the number of classes you have, every time you gain a level.

Yeah, it’s pretty obvious why they changed this in D&D 3e.

In all cases below I show only the mean.  For practical purposes it is close enough; once you roll 8-10 dice you’re going to be landing near the mean anyway, unless you use alternate rules such as “reroll anything less than half” or “reroll ones”.

Average Hit Points Per Class, BECMI

Looking up the details via the Rules Cyclopedia (which may vary slightly from the original box sets) I see for the core classes:

Class	Hit Die	After 9th	Hit Points at Various Levels and Constitution Scores
			1st			8th			9th			12th			20th
			3	10	18	3	10	18	3	10	18	3	10	18	3	10	18
Cleric	d6	+1	1	3	6	8	28	52	9	31	58	12	34	61	20	42	69
Fighter	d8	+2	1	4	7	12	36	60	13	40	67	19	46	73	35	62	89
Magic-User	d4	+1	1	2	5	8	20	44	9	22	49	12	25	52	20	33	60
Thief	d4	+2	1	2	5	8	20	44	9	22	49	15	28	55	31	44	71
Dwarf	d8	+3	1	4	7	8	36	60	13	40	67	22	49	76	N/A	N/A	N/A
Elf	d6	+1	1	3	6	8	28	52	9	31	58	N/A	N/A	N/A	N/A	N/A	N/A
Halfling	d6	N/A	1	3	6	8	28	52	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A	N/A
Mystic (optional)	d6	+2	1	3	6	8	28	52	9	31	58	15	37	64	N/A	N/A	N/A

(Ah yes, level limits.  Dwarves stop at 12th level, Elves at 10th, Halflings at 8th — because clearly they’re overpowering — and Mystics at 16th.  Technically the “Dwarf Con 3” is impossible because they must have a score of at least 9 — +0 modifier — but I’ll pretend he took Constitution damage.)

Average Hit Points Per Class, AD&D 2e

I dug up my AD&D 2e Player’s Handbook (and I regret my penchant for being thorough) and I see for the core classes:

Class	Hit Die	After 9th	Hit Points at Various Levels and Constitution Modifiers
			1st				9th				12th				20th
			-2	+0	+2	+4	-2	+0	+2	+4	-2	+0	+2	+4	-2	+0	+2	+4
Cleric	d8	+2	2	4	6	N/A	22	40	58	N/A	28	46	64	N/A	44	62	80	N/A
Thief	d6	+2	1	3	5	N/A	13	31	49	N/A	19	37	57	N/A	35	53	71	N/A
Warrior	d10	+3	3	5	7	9	31	49	67	85	40	58	76	94	64	82	100	118
Wizard	d4	+1	1	2	4	N/A	9	22	40	N/A	12	25	43	N/A	20	33	51	N/A

This isn’t strictly accurate because I skipped a couple of oddities.

• Fighters, if they have Constitution 17 or 18, can add +3 or +4 per Hit Die instead of only +2.
• Thieves and wizards get up to 10 Hit Dice instead of 9 (I didn’t bother with this, it makes only a very small difference, 1..3 hit points).

This is still simpler than AD&D 1e, where we have the following exceptions:

• Druids roll 1d6 per level, up to 14th level (and nothing after that).
• Fighters, if they have Constitution 17 or 18, can add +3 or +4 per Hit Die instead of +2
• Rangers roll 2d8 at first level, then another 1d8 per level up to 10th level (then +2 per level after that).
• Magic-Users roll 1d4 per level, up to 11th level (then +1 per level after that).
• Illusionists roll 1d4 per level, up to 10th level (then +1 per level after that).
• Thieves roll 1d6 per level, up to 10th level (then +2 per level after that).
• Assassins (subclass of Thief) roll 1d6 per level, up to 15th level (then nothing after that).
• Monks roll 2d4 at first level, then another 1d4 per level up to 17th level (and nothing after that).

To be honest, it’s hard to call these ‘exceptions’ when only three classes fit the pattern shown in BECMI and AD&D 2e, and one of those itself has an exception in how the Constitution modifier applies.

D&D 3.x Hit Points

Compared to the hit point rules above, D&D 3.x makes it simple, at least on the surface.  For each level, roll the Hit Die for your class and add your Constitution modifier.  If the roll would be less than one, add one hit point instead.

Unlike previous editions this gets the complication that Constitution changes, both during play (damage, buffs, and so on) and when gaining levels.  Permanent Constitution buffs (or magic items that serve the same purpose) are very popular, for reasons that should become evident.

In D&D 3.x there are five Hit Die sizes only, and they get applied to all classes.  Classes no longer have the static hit points added after a particular level, so that is not important any more either.

Hit Die	Hit Points at Various Levels and Constitution Modifiers
	1st				9th				12th				20th
	-4	+0	+4	+7	-4	+0	+4	+7	-4	+0	+4	+7	-4	+0	+4	+7
d4	1	4	8	11	9	24	60	87	12	31	79	115	20	51	131	191
d6	2	6	10	13	10	34	70	97	13	44	92	128	21	72	152	212
d8	4	8	12	15	12	44	80	107	15	57	105	141	23	93	173	233
d10	6	10	14	17	18	54	90	117	22	70	118	154	34	114	194	254
d12	8	12	16	19	28	64	100	127	35	83	131	167	55	135	215	275

I showed hit point totals for Constitution modifiers of -4 (Con 2-3), +0 (Con 10-11), +4 (Con 18-19), and +7 (Con 24-25) because these are indicative values, not because they are likely to come up in play (I rarely see a player without a Constitution bonus, let alone one with a penalty).  Con 24-25 was chosen as the top modifier because in my experience few characters bother to push it any higher.

It should be evident that Constitution plays a much larger role in hit point totals in this edition than in previous editions.  Gaining full hit points at first level throws the math off a little, but in the long run that blip gets absorbed.

Hit points in D&D 3.x are much the same as in previous editions, for the same size of Hit Die, up to ninth level.  After that, things really take off, partly because the Hit Die rolled averages a point or two better than the static value previously gained, but mostly because the Constitution modifier — the ever-increasing, since at this point the PC can afford the magic needed, Constitution modifier) gets added at every level.

I think I’ll consider D&D 3.x hit points as an upper bound on the hit points I’d like to see characters with.  I’d really like to reduce the impact Constitution has on hit points.  At 20th level the difference in hit points between “minimum Constitution” and “maximum Constitution” characters ranges from about 5x (if the character has a d12 Hit Die for each level) and 9.5x (if the character has a d4 Hit Die).  The difference between “Maximum Hit Die” and “Minimum Hit Die” is about 2.75x (assuming the lowest Constitution scores) or 1.5x (assuming the highest Constitution scores).

D&D 4e Hit Points

I lack the specific knowledge needed to comment on D&D 4e hit points.  I haven’t played 4e and my usual online reference is evidently asleep or otherwise occupied.

Ah, someone was kind enough to direct me to a page at Wizards of the Coast that has the hit point calculations for the core classes.  You’ll have to scroll down a bit to get to it (look for “Quick Hit Points”), but it’s there.

They use a fairly straightforward formula

level * class-based-number + another-class-based-number + Constitution

Simple enough.  Let’s see what it looks like

Class	Formula	Hit Points at Various Levels and Constitution Modifiers
		1st				9th				12th				20th
		3	10	18	24	3	10	18	24	3	10	18	24	3	10	18	24
Cleric, Ranger, Rogue, Warlock, Warlord	level*5+7+Con	15	22	30	36	55	62	70	76	70	77	85	91	110	117	125	131
Fighter, Paladin	level*6+9+Con	18	25	33	39	66	73	81	87	84	91	99	105	132	139	147	153
Wizard	level*4+6+Con	13	20	28	34	45	52	60	66	57	64	72	78	89	96	104	110

Three different hit point formulas.  All are entirely linear.  Constitution has large effect at low level and smaller effect at high level, proportionately.

After the conclusion I came to last night (that could have saved me a lot of effort had I thought the problem through first), that the hit point curve is linear tells me it isn’t what I’m looking for for Echelon.  Still, it is nice and simple, and over time does reduce the impact of Constitution.

Echelon Hit Points

Here is where I need to start examining the impact of the proposed change.  First I’ll evaluate the existing model, then see what this change does to it.

Current Hit Point Model

Echelon doesn’t have Hit Dice, but a previous iteration (the one that’s currently online that might go away) used a combination of level, tier, Base Attack Bonus, and Constitution score to determine hit points using a formula.  The formula changed over time as I experimented, but the last version was (level + BAB + Con) * tier.  This resulted in an unsmooth curve that I actually liked because it helped make the transition between tiers more noticeable.  The curve could be smoothed by using (level/4) instead of tier, and using the ‘ceiling’ (i.e. round up all fractions, like the opposite of truncating), but I didn’t do that.

Note that I’m using “D&D-equivalent levels” in the tables below.  Echelon characters are basically four levels higher than their D&D equivalents (a starting wizard is ‘fifth level’ instead of ‘first’) because I have expanded the lower-than-first level into a full tier in Echelon.  Also, I show Constitution 12, which is frankly outside the expected range, just to show the biggest values under consideration.  Normally creatures are likely to be constrained to scores between one and nine inclusive.

Hit Die	Hit Points at Various Levels and Constitution Scores
	1st				9th				12th				20th
	1	5	9	12	1	5	9	12	1	5	9	12	1	5	9	12
Good BAB	22	30	38	44	108	124	140	152	132	148	164	176	294	318	342	360
Medium BAB	18	26	34	40	92	108	124	136	116	132	148	160	258	282	306	324
Poor BAB	16	24	32	38	80	96	112	124	100	116	132	144	222	246	270	288

In this model, the ratio between BAB progressions is fairly consistent.  In the values shown above,

• Good:Medium averages 1.13:1 (low 1.10:1, high 1.22:1)
• Medium:Poor averages 1.12:1 (low 1.05:1, high 1.16:1)
• Good:Poor averages 1.27:1 (low 1.16:1, high 1.38:1)

The influence of Constitution becomes smaller at higher levels because the level and Base Attack Bonus both get larger, while Constitution is not expected to change.  At the lowest levels, when Constitution dominates the equation, the hit point ratio between highest and lowest allowed scores only reaches 2:1, and that for the Poor BAB creatures.  At this point it might be argued that Constitution, the ‘meat’ of the creature, is the key element of the hit points.  After that hit points become more and more learned ability, luck of the gods, and what have you (in other words, what D&D has always tried to say hit points are).  At higher levels the ratios drop quite a bit, until the hit point ratio between ‘highest Constitution’ and ‘lowest Constitution is comparable to that between Good BAB and Poor BAB.

This suits me fine.  The numbers are a little big, but the relationships look generally good.

Proposed Hit Point Model

Here I will examine a few options and formulas for hit point calculation.  I no longer have Base Attack Bonus in the same form I did (I might if I find a way to do it, but for now I’ll assume I won’t).

I’ll present a few formulas, then evaluate them below.

Formula	Hit Points at Various Levels and Constitution Scores
	1st				9th				12th				20th
	1	5	9	12	1	5	9	12	1	5	9	12	1	5	9	12
(level+Con)*tier	12	20	28	34	56	72	88	100	68	84	100	112	150	174	198	216
(level+Con)*tier/2	6	10	14	17	28	36	44	50	34	42	50	56	75	87	99	108
(level+Con)*tier*2	24	40	56	68	112	144	176	200	136	168	200	224	300	348	396	432

I ended up deciding that I could not readily model D&D hit point totals in any edition I am familiar with (I can’t say about D&D 4e).

In D&D 3.x the curve is quadratic in that Constitution usually goes up as a function of level (approximately “original score + 2*floor(level/4)” when you consider items, wish, and other inherent improvement)… but this is still strongly influenced by the original score.  In D&D 3.x, the Constitution has greater effect as the character gains levels, in Echelon the Constitution has less effect as the character gains levels.  The curves can be expected to cross, but they won’t really be congruent at all.  That said, the first formula does a passable job of emulating 3.x hit points for fighters.

In AD&D and BECMI, Constitution has some effect early and and doesn’t change much, until it drops out of the equation entirely at high level.  The curve is generally bi-linear (two different linear curves, changing slope around 9th level).  Given that Echelon is still quadratic by level, and the AD&D/BECMI curve gets shallower at high level, these are irreconcilable and will not be congruent.

This lack of congruency between Echelon and previous editions of D&D may not be an issue, since I actually don’t like the other hit point curves.

Formula	Hit Points at Various Levels and Constitution Scores
	1st				9th				12th				20th
	1	5	9	12	1	5	9	12	1	5	9	12	1	5	9	12
(level+Con)*tier	12	20	28	34	56	72	88	100	68	84	100	112	150	174	198	216
d10 Hit Die	6	10	14	17	18	54	90	117	22	70	118	154	34	114	194	254

Happily — but not coincidentally — the Constitution values ‘1, 5, 9, 12’ would have modifiers ‘-4, +0, +4, +7’ in D&D 3.x.

The hit point curve is much shallower in Echelon’s formula than it is in D&D 3.x, but it is also quite a bit gentler within the level for differences in Constitution.  The fit between the highest-Constitution hit point totals at each level isn’t too bad.  The ‘higher-than-highest’ Constitution hit point totals starts with D&D 3.x somewhat below Echelon’s totals and ends somewhat higher.  ‘Average’ and ‘low’ Constitution hit point totals are quite a bit more divergent.

I’d be prepared to use this total for D&D 3.x style of play.

Formula	Hit Points at Various Levels and Constitution Scores
	1st				9th				12th				20th
	1	5	9	12	1	5	9	12	1	5	9	12	1	5	9	12
(level+Con)*tier/2	6	10	14	17	28	36	44	50	34	42	50	56	75	87	99	108
AD&D Warrior	3	5	7	9	31	49	67	85	40	58	76	94	64	82	100	118

This progression looks like it fits the AD&D Warrior (Fighter) pretty well, even though the Constitution modifiers in AD&D are only half those seen here.  The AD&D Warrior actually ends up above the Echelon hit point curve in the middle levels (Master tier and into Champion, it looks like) but moves back in line afterward.  It doesn’t fit the D&D 3.5 d8 Hit Die, let alone the others.

I think this formula would work well for an old school style of play.

The last formula I showed was a consideration of how ‘toughness’, unusually tough characters, might be modeled by increasing the tier multiplier (that is, treat as if the character had a talent that increased his tier multiplier by one for each tier).  I won’t bother trying to fit it to any previous edition of D&D.

Closing Comments

As much as the formulas I examined won’t quite model the hit point curves available in any previous edition of D&D, I could see using them without too much worry.  D&D hit point curves have always been an issue with me, especially the effect Constitution has on hit points D&D 3.x.

While I admit to preferring that ‘martial characters’ be generally tougher than ‘nonmartial characters’, I am coming to the conclusion that I might prefer to move away from this trope.  I can imagine settings in which it might be desirable that this trope be used (almost anything Sword & Sorcery), but there are enough examples of exceedingly tough to kill wizards out there that I can live without it.  Perhaps the effect of Constitution alone is enough.  Perhaps I might use the ‘old school’ formula and give martial characters a bump up (either through modifying the formula, or just straight hit point bonuses from talents taken).

In any case, while I have not chosen a specific solution or formula here, I am satisfied that the change to Level Bonus will not particularly affect hit point calculations in a bad way.  I may end up doing a little more experimentation and modeling before I find the formula I want, but since I have never really been satisfied with the hit point curve in previous editions, it’s something I want to fix anyway.