Follow-Up: Testing Balanced Dice

When I wrote the previous blog on how to statistically test for balanced dice, I hadn't actually used the test on any of my own dice (as was requested in at least one comment). But shortly thereafter I started to get worried that I would be compelled by curiosity to (a) test all the dice in my house, and (b) wonder what testing dice-manufacturers use, if any, for quality assurance purposes.

For part (b), I'm broadly guessing that the answer may be "none" in any sense other than an eyeball-check. If anyone has other information I'd be interested in hearing it.

For part (a), I haven't checked all my dice, but I did check all my d20's, and the results were rather interesting. Again, the process outlined earlier is this -- roll 100 times, count the frequency of each face, subtract 5 and square each, then add them all up; at the end you have the SSE "sum-squared-error" (higher results indicating a more unbalanced die).

I started by testing all my "newer" sets (at least a couple years old), mostly translucent, pre-inked digits, with slightly softer edges (many by Chessex, I believe). The SSE results I got for them were 150, 148, 106, 100, and 98. Recall that we were going to reject a die as clearly unbalanced if the result was more than 150; the first two d20's I tested (which I was already suspicious about, because of prior in-game behavior) just barely squeaked through the test. The P-values for these dice would be 0.05, 0.06, 0.33, 0.39, and 0.42 (you can sort of think of these as the probability that each die is in fact balanced; that's not exactly correct, but a fair analogy for what P-values represent -- a higher value is better news for the "balanced die" hypothesis). For the first two dice, we've got at least moderate-level evidence that the die is unbalanced.

Now at the end, I tested what I presumed would be the weakest die in my collection: an older translucent red d20, with sharp edges, that I had to color in myself with a crayon. The other dice in this set still show the tab from where it was snapped off the molding sprue (although I can't see it on the d20 itself; these dice are probably from Gamescience). Well, unexpectedly to me, this d20 had the lowest error of the bunch: SSE = 80, significantly lower than anything else I had in the house, and clearly the fairest-rolling die of anything I tested (P-value = 0.66).

So my theory now would be that a die that has sharp edges is more likely to roll fairly than one that has rounded edges, even though I've been avoiding this "sharp-edged" set for years now because to my eye it looked less professional.

Back to item (b) above. Interested in seeing what, if anything, the dice manufacturers have to say about their QA process, I landed at the Gamescience website ( http://www.gamescience.com/ ). And the fascinating thing is that the home page has a couple of videos by the long-time owner Lou Zocchi, where he rants for a total of about 20 minutes on exactly this subject. Namely: (a) he intentionally produces dice with sharp edges, similar to requirements for casino gambling dice; (b) people complain about the cost, the lack of inked digits, and still seeing the sprue-tab, and (c) his competitors solve these problems by mass-painting the whole die and then throwing them into a tumbling sand grinder (removing excess paint, the sprue-tab, and rounding all the edges in an irregular fashion).

So I hadn't known about Zocchi's position before, and I hope I don't come off as a shill for Gamescience in this paragraph. (Zocchi doesn't use formal statistics in his presentations; he uses more user-friendly demonstrations, like stacking up competitors' dice and seeing that you get obviously different total heights from different directions). But at this point in time, I'm basically sold on that position; if I wanted dice that I really knew would roll in as fair a distribution as possible, I would go out of my way and get sharp-edged dice from Gamescience.

This is not to say there's no downsides. Frankly, their dice are more expensive. Moreover, I must say that the rounded-edged dice are a little more pleasant to look at and hold in the hand without the really sharp corners (and even more so to step on in stocking feet if you leave them on the floor; Gamescience dice are like deadly little caltrops).

But I'm glad I found Zocchi's videos, because an older gentleman devoting his life to fairly-rolling dice, and delivering a really angry rant about correct probability distributions, really warms and delights my heart.

I see on the Gamescience web site that Lou is selling the business at this time to Gamestation, Inc., which makes me a little sad, just at the time when I discover the advantages to his product. I'll salute Lou Zocchi for his years in the industry and hope that his dice stay available with the same level of quality in the future.

Growing Animals

Original D&D Vagaries; Advanced D&D Statistics; Proposal for Simplified Spellcasting 

In the prior post I was looking at the 5th-level D&D wizard spell, animal growth, throughout various editions of the game. One problem I pointed out is that the earliest editions commonly didn't give explicit stats for normal or giant animals (as required by different versions of the spell). For example, while the OD&D spell says that animals can "grow to giant-size with proportionate attack capabilities", the following is all you get in OD&D on the subject (from Vol-2, Monsters & Treasure):

That's from the table on p. 4. The first value is for Number Appearing (more for smaller types); the rest, including Armor Class, Move in Inches, Hit Dice, % in Lair, and Treasure Type, are all simply "Variable".

The text paragraphs are from p. 20. For large insects or animals ("giant" types), AC can be 2-8 (i.e., practically the entire range of AC's allowed in the game at this point), hit dice can be 2-20 (at the top end, significantly exceeding any other monster in the list), and damage can be from 2-4 dice (d6, of course; and again actually exceeding any other damage listing in the text of this book). So that more or less boils down to "anything you want, and feel free to surpass any of the other monsters we presented". Which to me seems like a big problem when numerous giant animals are included on the core wandering monster lists, quite heavily so at the starting levels (see Vol-3, p. 10). And an open-ended argument when a player casts animal growth in the game. (Also, you'll notice that reference to John Carter-type Martian beasts is included, and those types likewise show up in the desert wilderness random encounters on p. 18.) 


Now, in the Monster Manual (the first of the AD&D line, and almost entirely consistent with OD&D rules) we do get lots of explicit entries for giant animals. This solves the wandering-table issue, and would almost solve the animal growth spell issue (except that the spell is changed to double the normal animal's hit dice and now that's the thing you're generally lacking). But what are the general parameters for those animals (e.g., if we wish to use them as products of the OD&D spell)? Consider the table below which compiles all the AD&D Monster Manual giant animal types (ordered by increasing hit dice; optionally refer to PDF or spreadsheet ODS).

What do we see here? For starters, the hit dice range of 3-to-5 encompasses almost two-thirds of the giant animals (27/43 = 63%), with a fairly small number of monsters above that and below. In those few cases where hit dice for the "normal" animal type are given (see last column), the giant animal is never exactly double hit dice; some are close while others are much higher or lower. I'll also point out that there are some notable outliers to the high side, in the range of 7-15 hit dice, and they are almost entirely aquatic-type monsters (noted in bold at the bottom of the table -- perhaps emblematic of the rather terrifying "sea monsters" suggested in OD&D Vol-2, p. 15; in fact, the top-level hit dice match up quite nicely). Therefore, I've excluded those unusually huge aquatic monsters, and computed "Trimmed Average" statistics for all the other types, in the last row. Of course, since the majority of giant animals are in the 3-to-5 hit dice range, the average is 4 HD, etc.


So here's an idea: If we're playing under classic-type rules, and almost all of the land-based "giant animals" in the game are close to 4 HD anyway, why not just set that average as a standard profile in the description of the animal growth spell and run with that? That would mean: (1) We don't have to catalog, look up, or debate a list of either normal or giant animals when using the spell (as in any published edition of D&D). (2) We don't need to do a series of math calculations at the table when the spell gets used (most egregiously in 3E). (3) We can rein in the magic to some known parameters, and not run into trouble with some totally open-ended potential from the spell (like, say, casting it on a group of smilodons, killer whales, or triceratops; see also the polymorph problem -- link). Even using that simplifying assumption, we'd be within 1 HD of almost all the giant animals in the AD&D Monster Manual anyway, and with enormously less work.

So here's a proposal for a revised and simplified animal growth spell:

---

Animal Growth: (Range: 12 inches, Duration: 6 turns) Up to 6 normal animals in range grow to giant size. In most cases this provides: AC 6, MV 15, HD 4, Atk 1, Dam 2d6 (some creatures will exchange one die of damage for poison, paralysis, or blood-drain).  Optionally, use specific statistics from any monster book in use. This spell gives the character no special means of command or influence over the enlarged animals.

---

What do you think of that?

Exploding Dice Statistics

In January, after our Wandering DMs show on critical hits, someone in the patron's after-party chat (Joshua, but you knew that) suggested replacing Nat-20-triggered criticals in D&D with exploding damage dice all the time. It's not something I ever considered for D&D, but the statistical niceties of that immediately got stuck in my head. Let's look at that a little more closely. (Someone's probably already done that in the past, but as they say: "This is my blog; there are many like it, but this one is mine," or something.) 

A Formula for Exploding Dice

I wanted a simple, closed formula for the expected (average) value from rolling any variety of exploding die. Here you get some math so I can show my work. 

The essential trick/insight/tactic is this: Say we want the expected value E of an exploding d6. There's a 1/6 chance of getting any of the values 1, 2, 3, 4, 5, and then at the end, a like chance to get a value of (6 + E); that is, 6 plus the average result of the same process all over again (recursively, if you will). More generally, for an n-sided die, you have a 1/n chance for each value 1, 2, 3, ... , n − 1, (n + E). Some basic algebra solves for that E:

$$E = \frac 1 n (1) + \frac 1 n (2) + ... + \frac 1 n (n+ E)$$

$$E = \frac{1 + 2 + ... + n} n + \frac 1 n E$$

$$\frac{n - 1} n E = \frac {1 + 2 + ... + n} n$$

$$E = \frac {1 + 2 + ... +n} {n - 1}$$

Note in the 3rd step we subtracted \( \frac 1 n E \) from both sides, generating on the left-hand side a value of \( E - \frac 1 n E = \frac n n E - \frac 1 n E = \frac {n - 1} n E \).

What other ways of writing this are there? That top sum 1 + 2 + ... + n has a name, specifically the (nth) triangle number, which could be denoted simply \(T_n\). It's like the factorial function, but with adding instead of multiplying; and it's sequence A000217 in the Online Encyclopedia of Integer Sequences. So we could just write:

$$E = \frac {T_n} {n-1}$$

If you don't like that (and how dare you), we could use what I refer to as the Gaussian formula to replace it with more primitive operations; \(T_n = \frac {n(n+1)} 2\). Substituting that into our formula for E, we get this version of our exploding-die expected value formula:

$$E = \frac {n(n+1)} {2(n - 1)}$$

Note that the expected value for multiple dice is this number times the number of dice; e.g., for 3d6, just compute E for n = 6 and then triple it (expectation always being a linear operator).

A Table of Exploding Dice Values

Here you have a table of expected values for our standard mostly-Platonic dice, in both the normal-roll and exploding-roll forms. Consider the ratio column at the end, which shows the effective multiplier that exploding is giving you on average. The benefit is best for smaller dice; e.g., for a d4 you're multiplying your expectation by 1⅓, and then it goes down for larger dice. And yet this benefit doesn't flip the ordering of any dice; the benefit on d4 (which I kind of like, say, making daggers a bit more fearful) doesn't even reach the expectation for a standard (non-exploding) d6.

That said, watch out that things get weird in the edge-cases if you go off the top or bottom of this table. If you have 2- or 3-sided dice mechanics, those exploding expectations are actually the same: E(d2exp) = E(d3exp) = 3. And theoretically an exploding 1-sided would generate infinite damage! (E.g., consider the "ordinary rat" by Gygax in the AD&D MMII with that damage specifier; the standard 1st-level adventure just got a lot more dangerous.) On the other end, larger die values have a benefit ratio decreasing towards 1; in the limit for an infinite-sided die, there would be no benefit at all (uh... not that you'd need any).

Note that the ratio of exploding-to-normal damage expectation has a nice, short formula for it. (Thanks to Drew on Twitter for noticing this from the decimal values above.) This is:

$${n(n + 1) \over 2(n - 1)} \div {n+1 \over 2} = {n(n+1) \over 2(n-1)} \times {2 \over n+1}$$

$$ = {n \over n-1} = {n - 1 + 1 \over n-1} = 1 + {1 \over n-1}$$

Graphs of Exploding Dice Values

Here's the other thing that I think is really nifty about the exploding-die mechanic; the distribution becomes right-skewed (instead of uniform, or bell-shaped for multiple dice). Here's the probability distribution for an exploding 1d6 (from Troll Dice, turned on its side for more familiar presentation):

And possibly more illustrative, here's the chart for exploding 2d6 (like the damage for giants in Original D&D): 

Note how that slopes off gently on the extended right-hand side. My understanding is that right-skewed distributions like this are far more common (really, the only thing) for natural or biological processes -- you have a hard lower limit (often 0 or 1), but in theory no hard upper limit (on the right), and therefore the population/sample-space spreads out in exactly this way.

In fact: The famed evolutionary biologist, Stephen Jay Gould, wrote an entire book dedicated to exactly this observation: it's called Full House: The Spread of Excellence from Plato to Darwin.

Are those compelling results? (And did you already know that?)  Would you consider using always-exploding weapon damage dice in your D&D games, instead of a Nat-20 trigger?

Wandering Monster Tables

Today we'll look at the Wandering Monster tables ("Monster Level Tables") in OD&D Vol-3, p. 10-11. I won't recreate all of the tables here; I'll just present some summary analysis and comments.

These tables come in Monster Levels 1 to 6 (like most things in OD&D), and you'll be consulting some random one based on a roll on the Monster Level Matrix before it (see previous post). The 1st-level table has 8 monsters, levels 2-4 have 10 monsters each, and levels 5-6 have 12 monsters each. (Compare to the OD&D spell lists which also have increasing numbers of entries at the higher levels.) The specific monster you get is determined by a simple roll of the equivalent die (i.e., a uniform probability distribution and a one-to-one correspondence).

Now, analysis is somewhat complicated because many of the lower-level monsters (mostly giant varieties of normal insects and animals) are not given specific statistics in OD&D. You get some very general guidelines in Vol-2, p. 20, but basically the individual DM has to intuit appropriate stats. I'll write more about this in a later post -- but for now let's agree to fill in the missing stats from the AD&D Monster Manual and use that as an approximation.

Here's a summary of the Hit Dice at each level (after discarding a small number of outliers). Analysis follows:

1. Range: Up to 1+1. Average: 1.
2. Range: 1+1 to 2+1. Average: 2.
3. Range: 3 to 5+1. Average: 4.
4. Range: 4 to 6. Average: 5.
5. Range: 4 to 8+2. Average: 6.
6. Range: 5 to 12. Average: 9.

Monster levels are based directly on Hit Dice. It's pretty easy to see that monster levels are based simply on Hit Dice. 1 HD creatures are all in the 1st level chart. 2 HD creatures are all at 2nd level, all 3 HD creatures at the 3rd level, etc. At higher levels, you get a wider range, but the bounds are always increasing (or at least, nondecreasing). On the one hand, there's no provision for a low-HD but powerful-special-ability monster to get bumped up to a higher level (that would be a development for later editions of the game), but at the same time, OD&D special abilities are more-or-less scaled to the Hit Dice of the monster anyway.

NPC classes are also based on Hit Dice. A number of entries specify NPC class types, and their positioning is again dictated purely by OD&D-style Hit Dice (always d6-based). For example, consider the magic-users: the 2nd-level chart includes Conjurers and Theurgists (HD 2 and 2+1 in Vol-1); the 3rd-level chart includes Thaumaturgists and Magicians (HD 3 and 3+1); and so on and so forth. This is one case where the danger level of magic-users is probably underestimated by looking solely at Hit Dice. (Note also that different levels of a given class generate multiple entries in the charts, so, lots of NPC entries; I think I'd prefer if these were merged to a single entry in each table, i.e., more room for non-human types.)

Clerics are largely missing. Guess what? Whereas there are entries for almost every level of fighter or magic-user from 1-9 or so, clerics are uniquely missing from the majority of the tables. In fact, they only appear in two places: Evil Priests at level 4 (cleric level 3, HD 3), and Evil High Priests at level 6 (cleric level 8, HD 7). I'd argue that this is yet another example of the game resisting a full commitment to integrating Catholic-style clerics into the milieu, partly due to the clash in thematics that they represent.

Inclusion if and only if the monster is hostile. Look at the OD&D Monster Reference Table, which spans two pages at the beginning of Vol-2 (pages 3-4). The first page (p. 3) covers all the basically hostile-to-men, dungeon-dwelling monster types (humanoids, undead, and classical chimera-types), and everything on this page is included in the OD&D Wandering Monster charts. The second page (p. 4) covers all the benign-to-men and/or fey/enchanted-woods-types, and almost nothing on this page is included in the dungeon Wandering Monster charts (with two exceptions: Minotaurs and the Ochre Jelly). Later D&D products include entries such as for dwarves and elves in the dungeon wandering monsters, but they do not appear here for OD&D -- and I personally find this to be highly laudable. Consider how this neatly implies the "always attack by default" rule for the OD&D Reaction Table (discussed previously here).

Frequency of special abilities increase with level. While it appears that monster level determination is based largely on Hit Dice (see above), the tables also have the very satisfying quality that the prevalence of monsters with powerful special abilities increases monotonically by level. If we count up monsters with major special abilities (such as poison, paralysis, regeneration, hit only by silver or magic, spell-casting etc.) we obtain this:

1. 2/8 = 25%
2. 4/10 = 40%
3. 6/10 = 60%
4. 6/10 = 60%
5. 8/12 = 67%
6. 9/12 = 75%

(I suppose if I gave full reign to my tiny bit of number-OCD, I'd want to see that statistic at the 3rd level be 50%; perhaps by making the snakes there non-poisonous, or something. A longer rant on those snakes in particular awaits for the later post.)

Idiosycratic animal inclusions. The insect/animal types included on these tables (mostly missing stats as noted above) are not exactly what I would think to initially associate with D&D. First of all, while most are noted as "Giant" types (rats at level 1; hogs, ants, snakes, weasels, beetles, and scorpions at levels 3-4), several are not. At levels 1-2 you've got Centipedes, Spiders, and Lizards listed without that identifier. While later AD&D shows Giant versions of each of these, it's unclear exactly what's meant in OD&D; poison is not suggested for them in the LBBs, and Vol-2 suggests "any hit will kill the smaller" in its guidelines for types like these (p. 20). Secondly, the common inclusion of above-ground mammals like Weasels and Hogs (not even boars at this point) on deep-level dungeon charts really rubs me the wrong way -- noting, of course, that among the many levels of Greyhawk Castle, "The sixth was a repeating maze with dozens of wild hogs (3 dice) in inconvenient spots, naturally backed up by appropriate numbers of Wereboars." (Gygax writing in Europa Magazine #6-8, April 1975 -- thanks to Allan Grohe for documenting that here, 1st paragraph).

Mostly missing the clean-up crew. As noted above, the Ochre Jelly appears at level 3 (uniquely, but for the Hydra, listed in the singular), yet none of the other "clean-up crew" amoeba/slime types do. This is particularly odd because elsewhere it's implied that these types should be found only as wandering monsters. ("Note that Ochre Jellies, Black Puddings, Green Slime, etc., are generally distributed randomly, usually in passages, without treasure."; Vol-3 p. 7).

Outliers. I mentioned above that for Hit Dice analysis purposes, I discarded a small number of outliers, so I should detail what they are. At the 2nd level you've got Thouls, which are entirely missing from OD&D and AD&D (more in a future post). You also have Lizards at the 2nd level; if this is the same as AD&D's Giant Lizards, then they have higher Hit Dice than anything else at that level (HD 3+1), and they're also the only huge-sized creature until levels 5-6 (15' long; noting that I think the size listings introduced for AD&D are generally over-inflated, and could be cut in half). At level 4, those troublesome Evil Priests have significantly lower hit dice than anything else (3), and at a level 6 the Purple Worms have much higher dice (15), so each of these I've precluded from the Hit Dice statistics above.

Conclusions. In general, I think the OD&D wandering monster tables are highly praiseworthy in being meaty, well-themed for D&D, highly playable, and sufficiently limited in scope as to be easy to use and highly manageable and comprehensible. They present mostly mythic, recognizable-to-the-layman creatures. There are just enough monsters present to believe that they could possibly all exist in some particular megadungeon somewhere (i.e., Greyhawk). They compare favorably to later iterations such as the AD&D DMG tables which are overly large and fiddly, reduce interesting creatures like Skeletons and Zombies to near-absence in favor of a truckload of rats/beetles/shriekers, and are subject to the complaint that they don't make any sense as a presentation of a particular dungeon ecology. In the few OD&D cases where some particular entry presents a monster that is missing, duplicated, undesirable, or overly idiosyncratic, then it's a trivial operation to use the slot for something more common in your own personal dungeon or campaign (notably Thieves, for example).

I really like the simplicity of these tables for the most common basic D&D monsters, and I wouldn't think that anything more complicated is really desirable (e.g., see AD&D DMG, FF, or MM2). If monsters more rare than those on these lists require hand-placement by the DM, then I think that's a good thing, as well.

Monte-Carlo Measures of Monster Levels, Pt. 1

Abstract: Recently I've been expanding my "Arena" simulator program for OD&D combat analysis (current version 1.07). In particular, I implemented special abilities for all of the monsters in the OD&D dungeon wandering monster lists (with a few exceptions). As I did, this, it became possible to run simulations against all of those monster types to get a measure of the "true" power level for all of those monsters. This, in turn, then gives some confidence for exactly how much XP should be handed out for each monster in the OD&D Vol-1 context (recalling that at that time, there was no awareness of the need to modify XP for monster special abilities). See the chart below for the output, and then I may spend some number of blog posts discussing this model, it limitations, and the results. The second column, "Equivalent Hit Dice", indicates overall power level, and is what I would generally expect to use for monster XP awards (specifically: this number × 100). Spreadsheet version here: ODS file.

The Model: The Arena program simulates one-on-one gladiatorial fights between single fighters and some other opponent in original D&D (which could be either men or monsters). The rules implemented are essentially OD&D with 5%-step smoothing applied to attacks and saves.

After trying a few different models for assessing monster power levels, here's what seemed to work the best. NPC fighters are faced off against each monster in sequence: these fighters are assumed to be in chain mail & shield (base AC 4), with a +1 magic sword (to allow attacks on gargoyles, etc.), and ability scores rolled straight-3d6 (no bonus on average, except the 1 pip from the magic sword); but they may be of any level. No other improvements for equipment, improved magic, or higher ability scores are made for advanced levels. Fighters do not ever get multiple attacks (as they did not in OD&D; even for under-1-HD types, they are likely only being compared to 1st level fighters anyway, and we want a consistent basis for comparisons here in any event). This choice seems to allow for a fair measurement of the "raw" power of each monster, gauged against an opponent of fixed AC and damage output (i.e., the fighters only vary in HD alone, so "Equivalent Hit Dice" is in fact the output).

For each monster and any suggested fighter level, we can generate a random monster and fighter of that type, and have them battle to the death in the Arena. This is repeated 10,000 times for a given suggested level, and a proportion of times that the fighter wins produced (a Monte-Carlo method). Then we run a binary search on all possible levels from 0 to 50 to find the fighter level that gives the win percent closest to 50%; and we call this the "Equivalent Hit Dice" (EHD) of the monster.

Some of the confidence that this model gives us are that the EHD's generated for all the NPC fighters in the monster list are in fact identical to their actual HD (level). Also, the brute humanoid types like orcs, gnolls, ogres, and giants, have EHD in approximate linear progression to the Hit Dice, with trolls specially being valued at about double-HD -- the same as shown in our prior analysis of monster values in Chainmail, Vol-3 Mercenary costs, and Book of War (link).

Note that these results in EHD are not the same as the level of PC fighter that would be fairly matched against a given monster. If a PC is assumed to have better equipment, more magical arms and armor, and also receives a survival benefit (filtering effect) to ability scores, then those fighters will be much more proficient in battling monsters than shown here (like, say, able to contend with a monster of EHD +50% over their own level, as a zeroth-order estimate).

The Monsters: I implemented almost all the monsters in the OD&D random dungeon monster tables, and almost all of their special abilities. Sticking with OD&D rules, Hit Dice and damage dice are still d6's in almost all cases (damage 1d6 for all attacks by default, with exceptions noted in the Vol-2 text: e.g., ogres 1d6+2, giants 2d6, earth elementals 3d6, and a maximum 4d6 suggested for the largest of sea monsters or a Tyrannosaurus).

By necessity, many interpretations must be made regarding the monster statistics. In particular, OD&D included no statistics for any of the giant animal types referenced in the random tables (giant spiders, centipedes, ants, snakes, scorpions, etc.) In some cases we can look to Sup-I which included alternate attacks and damage for some of these types, although translating back to d6-based dice was done. In other cases I looked to the AD&D Monster Manual for the intention behind giant animal Hit Dice, armor class, attacks, and special abilities (the AD&D MM is compatible with OD&D in almost every way, and was chiefly a compilation of prior material).

I'll leave that as the end of Part 1 of this analysis. Next time: Commentary on individual monsters and special ability effects. Below, two downloads: First, a data file of the exact monsters and statistics used in this simulation, if you want to inspect that. Second, the current Arena code (Java) if you want to audit or compile that yourself.

• Arena Monster data file (CSV).
• Arena v. 1.07 source code (ZIP).

P.S.: Tell me if you would desire the Arena program being packaged up so you could run it from a command-line with needing to compile it yourself (say: to input and measure your own unique monsters, or monsters with adjusted stats, or experiments with totally different assumptions)? If more than a couple people express interest in that, I could do it.

Missing Monsters

So here's a bit about the monsters that appear in the OD&D Wandering Monster lists, but don't actually have explicit statistics given in those rules. A rundown by category, from most common to least:

Giant Animals. We're all familiar with these. Things listed as "Giant X" where X is some normal animal type include: Rats, Hogs, Ants, Snakes, Weasels, Beetles, and Scorpions. Most of these are in monster levels 3 or 4 (although Giant Rats are level 1). All of these have some analogous entry in the AD&D Monster Manual, so post-1977 you could use that resource to fill things in (recognizing that some monsters got boosted in the OD&D --> AD&D switch; see the last post). But what would the DM do running with just OD&D itself? Some guidelines in Vol-2:

LARGE INSECTS OR ANIMALS: This category includes giant ants and prehistoric monsters. Armor Class can be anything from 8 to 2. Hit Dice should range from 2 to anywhere near 20, let us say, for a Tyrannasaurus Rex. Also included in this group are the optionally usable "Martian" animals such as Apts, Banths, Thoats, etc. If the referee is not personally familiar with the various monsters included in this category the participants of the campaign can be polled to decide all characteristics. Damage caused by hits should range between 2-4 dice (2-24 points). [OD&D Vol-2, p. 20]

A few comments here: Taken literally, the rule in the text above (min 2HD) would make for really heavyweight Giant Rats, which are of course level 1 and given only 1/2 hit die in AD&D. Note also that the damage parameters are rather terrifying -- in OD&D, 2 dice is the same as a Hill Giant, while 3-4 dice is the same as a double- or triple-size Purple Worm (i.e., Sea Monster). Personally, looking at the Wandering Monster charts, my first guess would have been to assign a number of Hit Dice according to whatever monster level is indicated -- and you'd be accurate to the AD&D Hit Dice (or one less) in each case.

Normal Animals. Oddly, there are a few animals pointedly listed without the "Giant" identifier. These are: Centipedes, Spiders (level 1 each), and Lizards (level 2). What to make of these? Obviously, in AD&D they're each listed as a Giant type with attacks like weak poison, but there's no indication of that in OD&D. They seem to be referenced in this entry from the monsters book:

INSECTS OR SMALL ANIMALS: These can be any of a huge variety of creatures such as wolves, centipedes, snakes and spiders. Any hit will kill the smaller, while larger beasts (such as wolves) will receive one Hit Die. Generally speaking they will be Armor Class 8. [OD&D Vol-2, p. 20]

So perhaps at one point the Centipedes and Spiders listed in OD&D effectively had just 1hp? Were we meant to intuit a poison attack, if anything? It's hard to tell here. (One other thing that I point out elsewhere is how ridiculously huge the 3HD Giant Lizard is in AD&D -- pretty much like every monster given a specific size in feet there -- 15' long, whereas all other monsters in OD&D from levels 1-4 are basically human-scaled.)

Thouls. So, Thouls are listed in the OD&D monster level 2 chart, but they were never given statistics in any OD&D or AD&D publication. Looking only there, I might think to give them 2HD (by level) and some abilities like Ghouls (of whom they seem cognate, and they come immediately prior to in the encounter table). Of course, Moldvay Basic D&D later gave them 3HD, paralysis, regeneration like a troll, and the appearance of a hobgoblin. Perhaps more interestingly, we have evidence that the original Greyhawk Castle notes show the monster with 4HD and similar abilities -- which would make it distressingly powerful compared to other creatures on the level 2 list. (This information from Gygax, via Gene Weigel, via T. Foster over at the Dragonsfoot boards.)

White Apes. These occur in the level 4 listing. Granted all the "John Carter from Mars" references, it's pretty obvious that these refer to the White Apes of Barsoom, who appear alongside such types as Apts, Banths, etc. (mentioned in the first quote above). These are huge, 4-armed, semi-intelligent gorillas lacking any direct statistics in D&D. I assume that AD&D's "Ape, Carnivorous" (5HD) is the replacement here, although it only has standard limbs. 3E D&D has the "Girallon" monster which is quite obviously the White Ape under different trade dress (7HD, huge, white-furred, and 4-armed; stats here and picture here).

Other Considerations. The preceding account for all of the OD&D Wandering Monsters that lack statistics in that game. Here's a few other thoughts:

Let's say you think to replace the missing monster stats with stuff from the AD&D Monster Manual. You've got to be careful for a few reasons. First, note that some animals may not have been intended as "giant", but that's all you have available in the MM. Second (and related), note that some amount of Hit Die inflation occurred between OD&D and AD&D, and that might be true for the missing monsters as they appear in the MM. Third, you've got to make some decisions about exactly which AD&D monster of several sub-types should fill the slot (my suggestions, generally looking for appropriate Hit Dice and size: Spiders-->Large Spiders, Lizards-->Giant Lizard, Giant Hogs-->Wild Boar, Giant Beetles-->Boring Beetles, Giant Snakes-->Poisonous or Constrictor Snakes?) Fourth, and again related, some of these types will skew more dangerous than others on the same list (e.g., poisonous large spiders HD1+1 at level 1, giant lizards HD3+1 at level 2). In particular, using either of AD&D's Poisonous or Constrictor Giant Snakes at level 3 is really dangerous (respectively HD 4+2 or 6+1 by special attacks each; compare to Ogres listed at level 4 without any instant-death ability) -- I recommend a 3HD Constrictor type at that level.

One final peculiarity is that, for some reason, most of these types have damage entries listed in Sup-I Greyhawk, but are still missing the more basic stats (AC, MV, HD, etc.) To see this, remember that throughout OD&D, basic monster stats were in a separate table/location from attacks and damage (those being introduced in Sup-I -- and really this whole issue is what made the Monster Manual product so highly desirable and the first publication in the AD&D line). So if you look at the new, large Sup-I attacks/damage table, although they're all strung one-after-the-other in a single unified listing, you see three identifiable sections: 1st, all the original monsters from Vol-2 (Sup-I p. 16-18); 2nd, "missing monsters" of the type we discuss here (midway down p. 18); and 3rd, all the monsters being introduced elsewhere in Sup-I (p. 18-19).

Just looking for a moment at these orphaned damage entries, we see these types -- Giant (Sumatran) Rat, Wolf, Dire Wolf, Lion, Sabre-tooth Tiger, Giant Weasel, Mastodon, Giant Spider, Giant Lizard, Giant Toad, Giant Snake, Giant Crab, Giant Beetle, Giant Scorpion, Crocodile, Tyrannosaurus Rex. Note that the spider and lizard are now explicitly "Giant". While many of these damage stats match what you see in the Monster Manual, that's not the case for any of the monsters with multiple sub-types in AD&D, so you can't use that to synchronize them (e.g., snakes, spiders, beetles). For example, the entry for the Giant Snake indicates Attacks: 1 bite/1 constriction, Damage: 1-6 bite (poison noted), and 2-8/turn of constricting; now, if I allege above that the poisonous or constrictor snakes from AD&D are particularly scary, imagine an OD&D snake of the same hit dice that's both poisonous AND constricting. Holy smoke! (Recommend bumping that to a higher monster level if you use it.)

But irrespective of these Sup-I damage listings, there are still no basic stats anywhere in OD&D for our "missing monsters". Were they originally intended to be in Sup-I Greyhawk and got edited out at some point? Interesting to think about.

OED Monster Database

One of the places that OD&D can be most successfully criticized is in its presentation of the monster listing. In many places in the original work key details are missing, contradictory, or left to the DM (e.g., all of the normal and giant animals that appear in the encounter charts). As of the first supplement, one had to look in at least four different places for all of a monster's information: (1) the main table (HD, AC, MV, etc.), (2) the alternative attacks/damage listing, (3) the alignment category, and (4) the main text description, each of which appeared in different far-flung sections, even different books for one monster. The overall situation is what motivated release of the Monster Manual as the first hardcover book, which compiled all the statistics from OD&D monsters in one place, before any other volumes of the AD&D rules.

Below you'll find the OED Monster Database, a compilation I've made over the years for OD&D monster statistics, in line with my OED games and house rules. This gives me a convenient one-stop resource for OD&D monster statistics when I'm writing other material. One might ask: Why not just use the Monster Manual? One reason is that I very much like to stick with the original d6-based hit dice, attacks, and damage, as found in the LBBs (i.e., we do not recognize the alternative monster hits starting in Supplement I). Moreover, here are other reasons why I think this exercise was worthwhile:

1. Provides a consolidated listing of OD&D-style monster statistics.
2. Creates automatic summary stat blocks for insertion to adventures (see sheet 2).
3. Software can provide data-integrity checks for monster records.
4. Software also assesses "equivalent hit dice" valuations for encounter balancing and XP awards.
5. Forced me to think through any ambiguous adjudication cases in code (this prompted many "rules archeology" investigations and margin notes that you see on this blog).

Generally I've compiled everything I could find from OD&D Vol-2, Sup-I, land creatures from Sup-II, and TSR (The Strategic Review) No. 1 and 2 -- for a total of 147 monsters. Not included are aquatic monsters from Sup-II, demons from Sup-III, or deities from Sup-IV. For things like giant animals I turned to the Monster Manual and back-ported the information there, translating variant damage into units of d6's as we would normally expect/prefer.

Among the things you'll see is that any kind of special ability is given a keyword (and optionally one numerical parameter) for readability by my "Monster Metrics" program (more on that this Saturday); hopefully a knowledgeable DM can parse what those notes mean. The third-to-last column shows the EHD (equivalent hit dice) as determined by that program. A number of monsters are fundamentally outside the ability of my model to determine EHD, and so currently indicated by a question mark (?). These would include monsters with expansive wizard-type spell ability or need spells to defeat -- for example: the lich, lammasu, gold dragon, beholder, titan, and oozes with total weapon immunity.

XP Awards by EHD

Let's consider XP awards for a minute. My preference is to award XP by simply multiplying HD by 100 (and this is supported by evidence of a fundamentally linear relationship between risk and HD). Of course, this method from OD&D Vol-1 overlooks the value of special abilities. Sup-I introduced a variant XP table, and a secondary column to award bonuses for special abilities (which was of course carried forward into later editions like Holmes, B/X, AD&D, etc.). But in most of these works the DM still needs to make a subjective decision about what abilities warrant this bonus -- and I would argue in many cases it vastly undervalues some very nasty abilities (esp. on creatures with low HD).

There's a much easier way to do this, without any new tablature required or DM subjectivity, by just assigning a revised hit die value for XP purposes, which I've taken to calling "equivalent hit dice" (EHD). I gauge this with my Monster Metrics program by running several thousand fighters of different levels at each monster until we can determine a "fair" fight in each case. But the core of this idea (as simple as it is) predates the Sup-I and later variant XP charts, appearing first (briefly) in The Strategic Review, Vol. 1, No. 2 (Summer 1975), p. 4:

For purposes of experience determination the level of the monster is equivalent to its hit dice, and additional abilities add to the level in this case. A gorgon is certainly worth about 10 level factors, a balrog nut [sic] less than 12, the largest red dragon not less than 16 or 17, and so on. The referee's judgement must be used to determine such matters, but with the foregoing examples it should prove to be no difficulty.

This seems like a much more elegant way to assess the risk/reward of exotic monsters, and it's also the simplest measurement model I could construct in my software, so this is what I now include for each monster (where appropriate) in the Monster Database. Let's just briefly compare the sample evaluations from TSR #2 to what comes out of my program for EHD:

So we can see that, compared to our model, Gygax seemed to to undervalue the more exotic special abilities of monsters (in this case: petrification, immolation, and fire breath), which is consistent with his low valuation for specials in the variant Sup-I XP tables. Perhaps on closer inspection we could generously grant that Gygax's numbers correctly assess the minimum fighter level which could possibly match the creature man-to-man, but this is not an ideal metric of the danger to parties of several men of lower or higher levels.

Finally, let's look at the distribution of the EHD values currently recorded in the monster database:

Here we see that the distribution is a nice logarithmic curve, with the largest number of monsters at the lowest level (EHD 1-5), and decreasing regularly after that. (If we zoom in a bit more we'd see that the highest number of monsters are actually the level of EHD 2, outnumbering EHD 1 by a small margin.) This seems to be useful for a fantasy campaign, and nicely echoes the curves we'd expect to see in demographic and economic statistics from real-world data.

That being said, here's the link to the current OED Monster Database, at the OED Games website (ODS format):

OED Games: Monster Database

Spells Through The Ages – Animal Growth

This week we're looking at the 5th-level D&D wizard spell, animal growth. No need for artificial growth hormone with this bit of magic in your book. But how did the spell itself grow over the years?

Original D&D

Growth of Animals: A spell which will cause from 1-6 normal-sized animals (not merely mammals) to grow to giant-size with proportionate attack capabilities. Duration: 12 turns. Range 12".

Originally Gygax called the spell growth of animals. It can affect 1-6 animals; but is that a random roll, or at the election of the caster? Of course, the biggest problem is that while the spell says the targets "grow to giant-size", OD&D nowhere provides concrete statistics for any giant animals. There is one line in the table of Vol-2 that simply says "Variable" for all statistics; and a short block of text that essentially says they could be, well, anything at all. So with this spell the DM and players are utterly on their own about negotiating what benefit the caster gets. (Note that this spell was one of those excluded from Cook's D&D Expert rules, so next we'll go directly to the AD&D line.)

AD&D 1st Ed.

Animal Growth (Alteration) Reversible
Level: 5
Range: 8"
Components: V, S, M
Casting Time: 7 segments
Duration: 2 rounds/level
Area of Effect: Up to 8 animals in a 2" square area
Saving Throw: None

Explanation/Description: When this spell is cast, the druid causes all animals, up to a maximum of 8, within a 2" square area to grow to twice their normal size. The effects of this growth are doubled hit dice (with resultant improvement in attack potential) and doubled damage in combat. The spell lasts for 2 melee rounds for each level of experience of the druid casting the spell. Note that the spell is particularly useful in conjunction with a charm person or animal or a speak with animals spell. The reverse reduces animal size by one half, and likewise reduces hit dice, attack damage, etc.

Above, we're looking at the druid-spell entry, to which the magic-user section back-references. The name is now animal growth, as it will be in all later editions. The number affected has increased from 1-6 to a maximum of 8 (technically by the wording "all animals" not modifiable by the caster). The effect has changed in kind of a weird way: in OD&D, it said that animals "grow to giant size", even though OD&D lacked explicit giant animal stats. Here in AD&D, you now do have a catalog of specific giant animals available (via the Monster Manual), but the spell effect does not refer you to them, instead saying "effects of this growth are doubled hit dice" -- which is now a whole new problem, because what the MM generally does not have are the stats for the normal version of the animals which are getting doubled here (think: badgers, weasels, spiders, centipedes, etc.). That is: the spell zigged while the monsters zagged. Later on, the Monster Manual 2 gave lots of entries for "normal" animals that I suppose you could use as a basis for the wording of this spell. To me it seems like the best solution would be to extrapolate away from the exact language above and use those giant entries in the MM. But fundamentally this "doubling" rule will be retained consistently in 2E & 3E.

AD&D 2nd Ed.

Animal Growth
(Alteration)
Reversible
Range: 60 yds.
Duration: 1 rd./level
Area of Effect: Up to 8 animals in a 20-ft. cube
 
When this spell is cast, the wizard causes all designated animals, up to a maximum of eight, within a 20-foot-square area to grow to twice their normal size. The effects of this growth are doubled Hit Dice (with improvement in attack rolls) and doubled damage in combat. The spell lasts for one round for each level of experience of the wizard casting the spell. Only natural animals, including giant forms, can be affected by this spell.
 
The reverse, shrink animal, reduces animal size by half and likewise reduces Hit Dice, attack damage, etc.

The component of both versions of the spell is a pinch of powdered bone.

This is almost a direct copy-and-paste from 1E with some very minor edits (changing area in scale inches to feet, etc.). The "double Hit Dice" rule is kept intact. Cook does change the wording from "all animals" in the area to "all designated animals", so the caster can now clearly choose how many to gigantify.

D&D 3rd Ed.

Animal Growth
Transmutation
Level: Drd 5, Sor/Wiz 5
Components: V, S
Casting Time: 1 action
Range: Medium (100 ft. + 10 ft./level)
Targets: Up to one animal/two levels, no two of which can be more than 30 ft. apart
Duration: 1 minute/level
Saving Throw: None
Spell Resistance: Yes

A number of animals grow to twice their normal size. This doubles each animal's height, length, and width, increasing its weight by a factor of eight. This increase in size has a number of effects:

Hit Dice: The creature's HD double, doubling the creature's base attack bonus and increasing its saves accordingly.

Size: The creature's size increases one step. This increase reduces its AC (according to the new size), reduces its attack bonus (according to the new size), affects its ability to grapple, and so on. The creature gains an enlargement bonus to Strength and Constitution scores, and its damage with natural attacks increases. This spell does not affect Colossal creatures.

When the spell ends, the creature's hit points return to normal, and all damage the creature has taken while enlarged is divided by 2.

The spell gives the character no special means of command or influence over the enlarged animals.

This version still maintains the kernel of the spell since 1E -- doubling the normal creature's hit dice (and 3E did provide a fairly extensive list of normal animals in its core rules: see 3E Monster Manual Appendix 1.) The number grown has changed to a variable one-per-two-caster-levels. And consistent with 3E's elaborate rules for monster creation and size changes, you now have a whole gauntlet of math calculations that you're supposed to make from the spell: first increasing hit dice, base attack bonus, and saving throws (due to the hit dice basis). Then reducing AC and attack bonus (due to size penalties). Then increasing Strength and Constitution and subsequent attack bonus and damage (due to size bonuses). No matter how you slice it, if you do this extemporaneously during a session at the table, play will be halted for some extended amount of time while you assess these changes, which I personally find to be highly unacceptable. If I recall correctly, a lot of published 3E material would pick one "favored" type or companion for NPCs and present the long stat block for when that animal gets animal growth cast on it. In some sense you'd have to admit that the original simplicity of "double stats on the fly" (to the extent there was some source animal to double in the first place) is in fundamental conflict with the strictly-constructed 3E system for creating and calculating monsters.

Questions

In early editions (0-1-2E), how did you come up with stats for the grown animals? Did you actually take normal stats and double them, or did you turn to the Monster Manual and use the appropriate giant type? Were you ever troubled by the lack of an upper bound on the hit dice possibly created (for example: 15HD giant sharks and sea turtles in the AD&D MM)? Were you able to use the 3E spell on the fly as written at the table, or did that grind your game to a halt?

Testing Balanced Dice

If you're like me, at this point you've seen a whole bunch of fellow players keeping track of their "lucky" dice, trying to "train" dice by storing them best-side-up, stuff like that. I got to wondering recently what it would take to practically test whether you've got a fairly balanced die or not, and fortunately I've read enough statistics at this point to finally track down what it would take. (I was a bit torn about whether to put this post here or in my math blog; I do think it belongs here a bit more, because it's a specific application of a pretty well-known "Pearson's chi-square" hypothesis testing procedure.)

Testing a d6: Let's say you've got a d6. Roll it 30 times. Keep a tally of how many times each face comes up, from 1 to 6. (Note that we expect the number of appearances from each face to be about 5; 30/6 = 5). At the end, go through the counts and subtract 5 from each, square them all, and then add them all up. For a fair die, the total at the end should be no more than 55.

Testing a d20: Alternatively, say you're looking at a d20. Roll it 100 times. Again, keep a tally of how many times each face comes up, from 1 to 20. For each of the counts, go through and subtract 5, then square them, then add them all up at the end. For a fair die, the total at the end of this process should be at most 150.  

Comments: The process specified above uses the minimum number of rolls you can get away with (5 times the number of sides; more on that below). It has a significance level of 5%; that is, there's a 5% chance for a die that's actually perfectly balanced to fail this test (Type I error). There's also some chance for a crooked die to accidentally pass the test, but that probability is a sliding function of how crooked the die is (Type II error). A graph could be shown for that possibility, but I've omitted it here (usually referred to as the "power curve" for the test). 

How does this work? As mentioned above, it's an application of the well-known "Pearson's chi-square test" in statistics. The process results in a random number whose probability follows a "chi-square" curve after some large number of rolls. Formally, the different faces of the die are called the potential "outcomes" of rolling the die; the count of how many times each one comes up is called the "frequency" of that outcome; and the number of times you expect to see each one is called the "expected frequency". For the "chi-square" function to be applicable, you've got to have an expected frequency of 5 or more for each possible outcome (hence the requirement for a number of rolls of at least 5 times the number of sides). 

The numerical process after you're done rolling can be referred to as the (very, very common) method of finding the "sum squared error" (abbreviated SSE). The "error" is how far off each frequency was from the expected value of 5 (hence count - 5); the "squared error" is when you square that value (thus making everything positive, among other niceties); and the "sum squared error" (SSE) is when you add all of those up. If the die showed every face exactly 5 times, the SSE would be exactly zero; the more crooked it is, the more error, and hence the larger the SSE value at the end. 

Normally in Pearson's procedure, you'd take each "squared error" and divide by the expected frequency, then add those, then check a table of chi-squared values to see how likely that result was (compared to your initial expectation). But since we expect every side of our dice to be equally likely, there's a simplification that I've done above. For example, for a d6 test at a 5% significance level (degrees of freedom one less than sides on the die, so df = 5), I go to a table of chi-square values and look up X²(5, 0.05) and see 11.070. That means I would normally reject the fair-die hypothesis (i.e., the null-hypothesis Hₒ, "there is no difference from a fair die") if X² > 11.070. Under Pearson's procedure this would imply the following (letting "O" indicate the observed frequency of each outcome, and "E" indicated the expected frequency of each side, e.g., 5 in each case here): 

$$\Sigma (O-E)^2 / E > 11.070$$

$$\Sigma (O-E)^2 / 5 > 11.070$$

$$\Sigma (O-E)^2 > 11.070 \times 5$$

$$SSE > 55.35$$

A similar simplification is done for the d20 process. 

Now, if you wanted to improve the accuracy of the test you could obviously roll the die more times. You would then be able to reduce the chance for either a Type I or Type II error, but never totally avoid either possibility. (In practice, we normally keep the Type I error "significance level" fixed, and work to reduce the Type II error, thus improving the "power" of the test). 

I'll end this with how you can form your own test, for any die, to whatever power level you desire. I'll assume you keep the significance level at 5%, and let E be the expected number of times each face should appear in your experiment (E = rolls/sides; having E ≥ 5 is a requirement!). Then you will reject the Hₒ "balanced die hypothesis" if the SSE you get at the end is greater than X ∙ E, where X is the table lookup for X²(sides-1, 0.05). For a d4, X = 7.815; d6, X = 11.070; d8, X = 14.067; d10, X = 16.919; d12, X = 19.675; and for the d20, X = 30.143. Have fun.

Pearson's chi-squared test at Wikipedia

Edit: Someone in the comments asked for a look at an actual working example -- included below in a photo of my scratch paper from the last time I tested a d20. 

See also:

• Follow-Up: Testing Balanced Dice
• Balanced Dice in Dragon Magazine
• Testing Balanced Dice Power

Compleat Character Creation Catalog: AD&D

This is Part 2 of looking at statistics for how likely various start-generation methods are to qualify you for desired classes in early D&D (partly inspired by our recent chat on Wandering DMs). Last time we looked at stat-generation in OD&D (3d6-in-order, maybe with a few point-swaps allowed) and the various sub-classes presented in supplemental material (ultimately resulting in the same list of classes that appear in AD&D). Here we'll look at the AD&D game and its very different matrix of possibilities for generating ability scores, and the ways it overhauled those classes and requirements.

A few reminders of ground rules: We're just looking at entry possibilities for single-classed humans, not considering demi-humans with racial adjustments, multi-class combinations, etc. Statistics were generated by Monte Carlo-type simulations from a C++ program (and spot-hand-checked in a few cases and against other resources). Today's entry is only dealing with rules as presented in the 1E AD&D PHB/DMG -- rules changes from 1E Unearthed Arcana (6 years later), will be considered in a follow-up post.

Ability Generation Methods

A wrinkle in the publication history of AD&D is that PHB came out in 1978, and it gave no rule for how players should generate ability scores -- it's one of several things locked away in the DMG ("The referee has several methods of how this random number generation should be accomplished to him or her in the DUNGEON MASTERS GUIDE"; PHB p. 9). And the DMG didn't come out until a year later, in 1979. So I'm assuming in the intervening year players were still using the OD&D 3d6-in-order method by default?

When the AD&D DMG did finally appear, Gygax specifically calls out using straight-3d6 as a bad idea in these rules (p. 11):

While it is possible to generate some fairly playable characters by rolling 3d6, there is often an extended period of attempts at finding a suitable one due to quirks of the dice. Furthermore, these rather marginal characters tend to have short life expectancy -- which tends to discourage new players, as does having to make do with some character of a race and/or class which he or she really can't or won't identify with.

He then offers four alternative methods that can be used for the AD&D game, as shown below:

I'll rephrase those slightly. For comparison sake, I'll also include the OD&D method as "Method 0" to see how it stacks up against the AD&D methods. And note that while the book identifies them with Roman numerals, I'll use Arabic identifiers here, to more easily synch with my C++ code and its customary output:

• Method 0: Roll 3d6 in order for the six abilities. (The OD&D rule, of course, not actually in AD&D for player characters.)
• Method 1: Roll 4d6 and drop the lowest die, six times, and order to taste. (This is the default rule from 1E all the way to 5E -- and most likely 6E as well, which won't be changing that element, according to a draft Character Origins document released by WOTC the other week.)
• Method 2: Roll 3d6 twelve times, take the best six, and order to taste.
• Method 3: Roll 3d6 six times for each ability, and take the best of the six in each case. 
• Method 4: Roll up twelve separate characters using 3d6 in order (i.e., as per the OD&D Method 0). Pick your favorite of the twelve characters.
  

(Yes, Virginia, there is a "Method 5" variant from the later Unearthed Arcana, but we'll address that next time.)

One thing to note is that the AD&D methods require exponentially more dice-rolling as one proceeds to the later methods:

And, here are the base-level descriptive statistics differentiating the various methods:

We see that Method 3 produces notably higher scores than the other methods -- it has an average of 14, while the other AD&D methods all hover around 12 to 13. Method 3 is also the most reliable in that it has the lowest standard deviation (e.g., the gold-standard for modern portfolio theory). Yet, we'll see later that, due to its lack of re-ordering to taste, Method 3 is not necessarily the best method in regards to the question of qualifying for certain classes.

I'll also note that for Method 4, I assumed that from the dozen characters generated, you pick the one with the highest sum (or average) of abilities -- although that too may not be the way it actually gets played, depending on the which character class one most desires. Again, we'll see the results of that further below.

Classes and Requirements

Recall that OD&D had no ability score requirements for the core classes in the LBBs, but did start using that as a mechanic to gatekeep the various sub-classes added in supplemental works. AD&D generally raised all of the requirements, in particular, instituting that any class had to have a score of at least 9 in their prime requisite (explicitly counter to the OD&D table of ability modifiers). Higher scores were required for sub-classes and even nonhuman racial choices.

And there's a further wrinkle in the fine print of the ability score rules that's easy to overlook: each of the character ability tables has a note for the 3-5 score range that restricts a character to only one particular class. For example, in the Strength table (PHB p. 9), it says, "Here or lower the character can only be a magic-user". Low Intelligence restricts you to Fighter; low Wisdom to Thief; and low Dexterity to Cleric -- while low Constitution mandates the Illusionist subclass, and terminally low Charisma restricts the character to being an Assassin.

So this results in an implicit "hopeless character" rule that requires re-rolling an AD&D character in some cases -- if you get more than one ability score in the 3-5 range, then the conflicting class mandates are not mutually satisfiable, and the character cannot pick any class at all. This gets echoed years later in the rules for D&D Basic; for example, Mentzer writes (Basic D&D Players Manual, p. 48), "If two or more Ability Scores are less than 6, the character may have problems later on. This type of character should also be discarded, unless the DM says otherwise." This is just a rephrasing of the same implied criteria in the AD&D PHB. (Also, I'm guessing that the 5E "no hidden rules!" dogma was motivated precisely in response to Gygaxian mechanics like this.)

This is a place where the otherwise excellent analysis of AD&D class access at Athenopolis.net made a misstep, I think -- they show the default ability minima to be 3 unless otherwise stated, when it's actually 6 by the AD&D ability tables (excepting precisely one 3 allowed for any given class). Fortunately, we'll see that in most practical cases this makes for only a small difference in our results.

A further interpretive assumption I'll make is that these very-low-score restrictions permit entry into an associated sub-class, unless otherwise stated. For example, if you have a 3 in Strength, such that the table explicitly says you can only be a Magic-User, then I'll assume you could alternatively be an Illusionist (if your other scores qualify). But that won't make a detectable difference either way you read this corner-case issue.

In summary, here are the requirements for the various classes in the 1E AD&D PHB as I see them:

Qualification Chances

So here's what our simulator reports for qualification probabilities across the various AD&D stat-generation methods. Numbers are all percentages, rounded to the nearest whole number. At the bottom there's an extra row denoted "No Class" for those unfortunates trapped by having more than one very low ability score, as described above. Classes are presented here in the same order as tables in the AD&D PHB:

Note that the 0's and 100's in this table are all the result of rounding; nothing is guaranteed, but you get very close in many cases. In light of the inflated AD&D requirements, we see that we really had to progress to something other than the OD&D Method 0; many of the classes would be effectively impossible by that method (less than a half-percent chance to be accepted), our primary four classes now only have about 60% pass rate, and 13% of all characters are "hopeless" and can't enter any class.

Note that those primary classes, rolled under the OD&D method but with AD&D requirements, are really the only places where our numbers notably differ from those of Athenopolis.net (they show acceptance values about 10% higher than what we have here). These are the classes where the 6-not-3 default requirement affects the most abilities. Of course, this whole column is purely theoretical, because Method 0 is nowhere officially supported by AD&D -- unless we consider the one-year window between PHB and DMG when that may have been still practiced? All the other entries are usually less than a 1% difference in our mutual results.

By using any of the official AD&D methods, we see that the primary classes return back to near certainties (for example, you're 99.8% likely to qualify for a cleric, fighter, magic-user, or thief, using Method 1). It's actually Method 1 that generally has the best chances to qualify for a given subclass, with Method 2 generally a close runner-up -- these are the two methods that allow re-ordering abilities to the player's taste. If you're stuck with abilities in a predetermined order, as in Methods 3 and 4, regardless of how much they otherwise boost the numbers -- remember, it's Method 3 that generates the highest scores on average -- it's fairly unlikely you'll have the exact profile you need, and qualification rates are a lot lower as a result. Method 4 in particular, using a stack of a dozen Method 0-OD&D characters, is very bad in this regard (showing 0, 1, or 2 percent for a number of classes).

Let's focus mostly on that prominent Method 1 (there's a reason it was carried forward as the starting point for all later editions, while the other methods were mostly abandoned). It boosts previously very rare subclasses to more common occurrences: for example, about one-quarter of all characters qualify to be a Paladin or Ranger (compare to less than 2% in OD&D). Druids go from 6% to 78%. Illusionist was 9%, now it's 36%. Assassins scaled up from 5% to 94% (and, again, are even required in the case of very low Charisma).

One thing that jumps out at many of us when we first see this is that the Monk is actually even harder to qualify for than the much-treasured Paladin class -- it's about half the rate of Paladin success in both Methods 1 and 2. But the truth is that getting 3 scores of 15+ is harder than getting a single score of 17+, which are the critical requirements in both OD&D and AD&D. It's probability, kids!

Then you have the totally overhauled Bard class (also removed to PHB Appendix II), which is the only class in the book with explicit minimum requirements for all 6 ability scores (needing at least four 15's, one 12, and a 10). This is an overwhelmingly challenging prospect, with success rates of only 2% for Methods 1 and 2 (but oddly 4% with Method 3, making Bards the only class that Method 3 is better for). Contrast this with the gentle requirements in the OD&D version which gave it the highest acceptance rates for any subclass there.

Forgive me for a rabbit-hole on further weirdness with the Bard regarding the topic at hand: As any AD&D player knows, you can't start as a Bard (unlike the OD&D version); you have to starts as a Fighter, get to at least 5th level, switch to a Thief, progress to at least 6th level, and then switch to the actual Bard super-class (in the direction of what 3E would later identify as "prestige classes" in immense profusion). Note that among the several 15's the text says a Bard needs for their abilities, one of those is Dexterity. But: the core rule for humans switching classes ("The Character With Two Classes", PHB p. 33) dictates that one needs a 17 or 18 in the prime requisite of the class you're switching into -- in this case, Dexterity, when you make your first switch to Thief. So how do those rules interact? I can see arguments for (1) the core switch-class mechanic still apples, so you have a "hidden rule" that a Bard really needs Dexterity of 17+, or (2) the specific description of the Bard class overrides that, and Bards-to-be have a special allowance in their class switching. With that latter rule I can imagine a case where some rules-lawyer says they're aiming for Bard, double-classes as Fighter and Thief with only Dex 15, but then later reneges (a very niche case, to be sure). Nonetheless, for this analysis I've gone with option (2) and assumed the Bard truly needs only Dex 15. (If you instead rule that they really require Dex 17, then the chance to create one drops to approximately 1% for Methods 1 to 3, and less than 0.01% for Method 4.)

Finally, as we see in the last row, the "hopeless" no-class possibility that we outlined above is really a nonissue using any of the AD&D generation methods. With Method I the chance of this happening is only 0.06% (6 times in every 10,000 characters rolled), and with other methods it's even lower (less than 0.01% each).

What surprising things do you see in those results? Would this analysis make you re-think what method(s) you'd want to use in a new AD&D game? Next time: in what ways Unearthed Arcana stirred stuff up a few years later.

The Case for Level Zero

What evidence is there in classic version of D&D for the status level-0 characters? This could be important for a few reasons: (1) It would make a difference in the statistics for mass-combat armies, whether the basic troop type was level-0 or level-1 or something else; and (2) It could provide justification for common 1st-level boosts (as in ability scores or hit points), under the statistical survivability analysis that we've done via the Arena simulation program (link).

Original D&D

In Original D&D, 1st-level characters are distinguished from Normal Men in at least three ways.

1. Starting-level fighters are given the level title "Veteran", implying that some significant action had to take place prior to reaching the 0-XP level (Vol-1, p. 16).
2. Veterans are given Hit Dice of "1+1" (that is: 1d6+1; Vol-1, p. 17), whereas normal men such as Bandits, et. al., are given "Hit Dice: 1 die/man" (that is: 1d6 with no addition; Vol-2, p. 6). 
3. Veterans are also given fighting capability in the Chainmail-based system of "Man + 1", while normal men would presumably be simply "Man" (with no bonus; Vol-1, p. 17). 

That said, in the d20-based "Alternative Combat System" combat and saving throw tables, there is no distinction in abilities; a footnote reads, "Normal men equal 1st level fighters" (Vol-1, p. 19). But other than that, we see that even in the earliest work there was a clear distinction drawn between 1st-level fighters and other Normal Men (even those who appear in large armed groups, making a living via combat and raiding, with high-level leadership, as per the Vol-2 listing for Men such as bandits, brigands, pirates, etc.).

D&D Sup-I: Greyhawk

Here's a puzzle: what hit dice do Normal Men have under D&D Supplement-I, Greyhawk, using the alternate hit dice rule (p. 10: e.g, d8's for fighter levels)? The text says, "Use of this system is highly recommended, but if it is used all monsters should be based on the 8-sided die system." Does this include Normal Men, or merely every other being in the monster listing? If the former, then they are here equal to 1st-level fighters and unique compared to any other ruleset (see below); but if the latter then they have just become less strong than Orcs, for example.

Personally, I think this switch to d8-dice for monster hits was one of the major, early missteps in the evolution of D&D. So much simpler to roll batches of d6's for this purpose, and I see no compelling need for expert fighter dice to match monster dice. 

Swords & Spells

Gygax's Swords & Spells supplement for mass warfare in Original D&D appears to have the first actual use of the phrase "0 level". The combat tables for average damage per figure, unlike the OD&D core books, have a separate row for level "0", with the footnote: "0 Level is normal man-type (i.e., 1 die or less, and not able to progress upwards in levels.)" (p. 25). The two-page spread of weapon-based combat tables in Appendix A are in fact entirely based on Level 0 combatants (p. 31-32), with modifiers for other levels applied from a prior page (p. 27).

These rules include different "Troop Classifications" in six tiers: Peasants, Levies, Regulars, Elite, Guards, and the combination Elite Guards. The latter are quite rare: "As a rule, guards should be limited to 10% of an army, and elite guards would surround the personage of the commander only, for example" (p. 5). But the following all-italic paragraph establishes that any 1st-level fighters qualify for this special category: "NOTE: Scale figures representing human/humanoid (and highly intelligent) creatures of 1st level or above or with 1+1 hit dice are always considered as having elite guard status. For example, the following types of troops are classified as elite guard status: Veterans (1st level fighters), Elves, Hobgoblins..." (p. 6). This status gives bonuses to formation, movement, morale, and melee. So in summary: 1st level is shown to be something very special indeed.

Advanced D&D

In Gygax's AD&D, there are even more copious distinctions made between 1st-level characters and those less than that. The Player's Handbook states, "It is important to keep in mind that most humans and demi-humans are '0 level'." (p. 106). The Dungeon Master's Guide notes in the section on hiring mercenary soldiers, "Note that regular soldiers are 0 level men-at-arms with 4-7 hit points each" (p. 30: again highlighting that even trained career combatants are 0-level by default; see also class followers on p. 16). The combat and saving throw tables here give a separate, reduced column for level 0 fighters (p. 74, 79). A footnote reads, "Dwarves, elves and gnomes are never lower than 1st  level (unlike halflings and humans, which may be of 0 level)" (p. 74: possibly correcting PHB p. 106 above). A table on the campaign's "Typical Inhabitants" specifies combat ability of "0 level", or even as low as -1 to -3 from that mark for sedentary types, with hit points ranging from 1-3 to 2-7 (p. 88). Of peasant revolts it is noted, "Troops will be 0 level" (p. 94). For energy draining by undead, the stated rule is: "If this brings the character below 1st level of experience, then the individual is a 0 level person never capable of gaining experience again. If a 0 level individual is drained an energy level, he or she is dead (possibly to become an undead monster)" (p. 119). City and merchant guards are also 0 level (p. 191-192).

In the Monster Manual section on Men, the first thing stated is, "Normal men have from 1-6 hit points each." (p. 66), and this is reiterated in the following entries "Hit Dice: 1-6 hit points" for Bandits (Brigands), Buccaneers (Pirates), Dervishes (Nomads), Merchants, and Pilgrims. So in this regard the hit points are identical to those for Men in OD&D Vol-2 (actually downgraded for Dervishes), and it argues that the likely intent in OD&D Sup-I was also to maintain them at the level of 1d6 hits, less than 1st-level fighters or 1-hit-die monsters.

The DMG also has the following statement in the section on henchmen (p. 35):

Number of Prospective Henchmen: Human and half-orc characters suitable for level advancement are found at a ratio of 1 in 100. Other races have an incidence of 1 in 50. However, as most of these characters will be other than low level adventurers and already in a situation they are satisfied with  -  and humans more so than other races, unless the development of the area is primarily other than human - about 1 in 1,000 population will be interested in offers of employment as a henchman. NOTE: This figure must be adjusted by the DM according to the locale, for if  it  is an active adventuring area, the incidence of prospective henchmen might be as great as 1 in 200, while if it  is  a settled and staid area, incidence might be as low as 1 in 5,000.

Note the opening line, "Human and half-orc characters suitable for level advancement are found at a ratio of 1 in 100." If taken at face value, then this could have interesting demographic implications; however, it seems to broadly contradict the fact elsewhere that for about every 100 normal men, there is at least one leader of 8th-10th level, along with a host of other lower-level fighters (see Men: Bandits in the Monster Manual p. 66 and other places). And surely the claim that "Other races have an incidence of 1 in 50 [for level advancement]" is in even starker contrast with the footnote on DMG p. 74 that "Dwarves, elves and gnomes are never lower than 1st  level". So coming up with some synthesized interpretation for these different statements seems very challenging; they seem as though if coming from completely different campaign systems. And yet -- there is no doubt that level 0 men are the norm; the only question is what their relative frequency is (high or extremely high).

Unearthed Arcana

In the AD&D Unearthed Arcana work, Gygax implements a mechanic for gaining XP and advancing through 0-levels in two places: for the Cavalier and the Magic-User. For the former (p. 14):

A cavalier character must be of proper social class, and is usually of noble or aristocratic origin. Only those characters of Upper Class  social status may immediately enter into the cavalier class. Those of lower social standing are generally excluded from becoming cavaliers, but certain members of lower social classes may be so honored. Such a character must be sponsored by a higher authority  of  greater status, and begins first as a 0-level Horseman (a retainer for a Knight), then a 0-level Lancer, and finally becomes a 1st-level Armiger of the cavalier class. The 0-level Horseman starts at -1500  experience points and has 1d4 + 1 initial hit points. The Horseman becomes a Lancer at  -500 experience points and gains another d4 roll for cumulative hit points. The Lancer becomes a 1st-level cavalier at 0 experience points, and gains another d4 in hit points. In contrast, a character whose social standing qualifies him or her for immediate entrance into the cavalier class begins as a 1st-level Armiger with 1d10 + 3 hit points. The character’s hit-point bonus for high constitution (if applicable) is first received at either Horseman or Armiger level, and is then applied to each additional hit die from second level on as normal. The special abilities of the cavalier class are only gained when the character attains Armiger status.

These statistics appear likewise in the class table for cavaliers (p. 15). For the Magic-User, 0-levels are the mechanic by which one can use Cantrips even before the character can cast a 1st-level spell (p. 45):

Cantrips are the magic spells learned and used by apprentice magic-users and illusionists during their long, rigorous, and tedious training for the craft of magic-use. An aspiring magic-user or illusionist may use 1 cantrip per day as a 0-level neophyte (-2000 x.p. to -1001 x.p.), 2 cantrips per day as a 0-level initiate  (-1000  to -501), and 3 cantrips per day as a 0-level apprentice (-500 to -1). Cantrips must be memorized just as higher-level spells are.

So these below-zero-XP rules are broadly consistent in their application. In each case there are actually more than one 0-level in question (two for the cavalier; three for the magic-user). Somewhat oddly in my view, the increasing XP-per level is reversed for the 0-levels; the earliest levels actually require more XP, and the level closest to 0 the least (in both cases, just 500 XP to get to 1st level from the immediate predecessor).

(Side note: Post-Gygax, there was an AD&D-branded Greyhawk hardcover book with an appendix on running 0-level characters. However, those rules do not use standard XP, instead using a totally different AP point-buy system instead -- e.g., get an AP point and use it to boost an ability or buy a skill by one point -- and so we need not consider it further here.)

Holmes D&D

While the status of "0-level/normal men" is fairly consistent throughout all the Gygaxian works, the Basic D&D line veers off in another direction -- one which has caused a few mistakes and incorrect recollections on my part in the past. First, J. Eric Holmes started to dispose of them. Under "Non-Play Characters", he writes: "Generally, only the lowest level of character types can be hired, i.e. first level" (p. 8); that is, he seemed to collapse the distinction between 0-level hirelings and 1st-level henchmen in mainline D&D. Secondly, at the start of the monster list, he writes "Monsters' hit dice are 8-sided", and then three lines later in the lead-off Bandit entry, they get "Hit Dice: 1" (p. 22). So it's hard to read that in any way other than giving Bandits a full 1d8 hit points; a conceivable reading of Supplement-I Greyhawk (see above), but definitely at odds with the listing in the AD&D Monster Manual. On the other hand, the combat tables do include a row for "Normal Man" (p. 18) which didn't exist pre-Swords & Spells, but other than that I can't find reference or use of the term.

Edit: Zenopus Archives has pointed out on his site that in Holmes' original manuscript submitted to TSR, his combat and saving throw tables did not include any separate row for "Normal Man". Rather, they were inserted in an editorial pass by Gygax himself before publication of the final product. But apparently Gygax did not pick up on those more subtle areas where the concept was submerged (hirelings and hit dice). Thanks to Zenopus for that analysis (links: one, two).

Moldvay/Cook D&D

The Moldvay Basic and Cook Expert D&D rules take this a step further, by attempting to make almost every man in an adventure a member of some class. Now, Moldvay does state that, "A retainer may be of any level (0, 1, 2, 3, or higher) and of any class (normal man or a character class)" (p. B21), and he does include rows for "Normal Man" in the combat and save tables (p. B26-27). But that level is used mostly for minor non-human monsters (e.g., bats, rats, centipedes, insect swarms, goblins, kobolds, and mules). The Bandits which previously represented the canonical normal men are here declared to be "NPC thieves" who save as "Thief: 1" (p. B30); but while 1st-level thieves get 1d4 hit points, once again the monster entry simply says "Hit Dice: 1" -- which means 1d8 by the rules of the monster section -- so these seem out-of-joint. Also, a new monster type of "Noble" is given, and for these, "In the D&D BASIC rules, a noble will always be a 3rd level fighter. However, the DM may choose to make a noble any class and level... A noble will always be accompanied by a squire (a 2nd level fighter). A noble might also be accompanied by as many as 10 retainers or hirelings (usually 1st level fighters)" (p. B39-40).

Edit: Thanks to JB in the comments for noticing that I'd overlooked a dedicated monster entry for the unclassed "Normal Human" (p. B40); I missed this because it's the only place in this or any other ruleset where the descriptor is other than "Normal Man". The stat block gives them 1d4 hit points. The description says this: "Most humans are 'normal' humans, though people with certain professions (such as merchant, soldier, lord, scout, and so forth) help in some adventures"; my reading is that those "certain professions" are all classed by default (c.f., "Noble" above and all the soldiers in any B/X adventure). Finally, Moldvay gives a unique rule for advancement, far more generous than Gygax: "As soon as a human gets experience points through an adventure, that person must choose a character class."

In the Cook Expert rules, the only place the phrase "0 level" or "normal man" appears is in the matching combat/save tables, the rules for the legacy potions of heroism and control human, and the stats for seamen and the normal hawk. Mercenaries are not explicated one way or another (p. X22). However, in the catch-all monster entry for Men, they are all given a full 1 Hit Die and saves as "Fighter: 1"; including all of the types of Brigands, Buccaneers, Pirates, Dervishes, Merchants, and Nomads (p. X35) -- very much at variance with the OD&D and AD&D game lines. Likewise, all of the adventures in the X-series of modules specified base soldiers in any army as being universally at least 1st-level fighters: see modules X4, X5, the comprehensive army rosters for all of the Known World in module X10, and so forth.

Conclusions

While there is some attraction to the simplicity of the Cook/Moldvay method (all soldiers are at least 1st-level fighters), I do think that the core D&D system makes the most sense if we interpret things in the sense of OD&D/AD&D with a "0-level/normal man" class prior to 1st-level characters. This justifies 1st-level fighters having the title "Veteran", their improved attack/defense capability, the status that trainee wizards might have prior to casting their first spell, etc.

As usual, I frown upon special-case discontinuities such as that "0 Level is... not able to progress upwards in levels" (per Swords & Spells). In this sense, I look favorably upon Gygax's addition in Unearthed Arcana of some kind of rules for pre-1st-level experience progression. Even if we never expect to use that during gameplay for PC's, it gives us something to hang our hat on for the purpose of campaign-wide NPC demographics, and how the matriculation from 0-level to 1st-level might occur. (And even for active play, I've had at least one player in the last year take over an NPC lantern boy who started racking up a surprising number of kills and a developed persona -- moreso than his original dead PC -- and surely that should be rewarded.)

Here's what I think I would do for the unwashed masses in my "Original Edition Delta" games: Start at 0-level. Roll 3d6 in order for abilities. Roll 1d6 for hit points. Earn 1,000 XP to get to first level. On achieving 1st level, add 1 hp if becoming a fighter, or subtract 1 hp if a becoming wizard (minimum 1 hp before Constitution bonus). A slight wrinkle here is that 1st-level fighter graduates in this way could only have 2-7 hit points, but that actually matches the Original D&D Vol-1 text, and is on average the same as a 1d8 roll.

Note that I veer off from Gygax a bit here in the exact XP needed to graduate from 0-level. For both the cavalier and magic-user, he set the level immediately before 1st at 500 XP; although in both cases, there were other "negative" levels that one needed to progress through initially.  Looking at the standard geometric XP tables, with fighters starting at a 2,000 XP step for 1st level, the clear extrapolation is that the level before that should be 1,000 XP (i.e., half). In addition: under certain other demographic assumptions, having a 500 XP step may actually produce a greater number of 1st-level types than normal men, a result that we should certainly avoid.

A few notes for Book of War: statistically, these distinctions are all below the level of granularity recognized in that game. For example, say a target is wearing chain & shield (AC 4 in any edition above). If the attacker is 0-level, with d6 damage, against a d8 hit die, then the chance to score a kill with one hit is 11%; but if the attacker is 1st-level, with d8 damage, against a d6 hit die, then the chance is 21%. Any other permutation is somewhere within these bounds; but using the BOW d6-mechanic, all of these round off to the same 1-in-6 chance to score a kill (16.6...%). So that's why it's really nice that BOW is a statistically accurate representation of whichever edition of the game you normally play with, regardless of these small details; and at all times I can thus assume that common 1 HD human soldiers are, for simplicity, effectively the same as common 1 HD orcs (at least in reasonable lighting).

So we do like giving the 0-levels a chance to be heroes someday. But what implications does this have for campaign demographics or 1st-level abilities and hit points, having survived the 0-level to get there? More on that next time.