A Model of Archery for D&D

Problem Statement; Evidence; The Model; Conclusions for the Game; Open Questions

Problem Statement

As you may know, I've written about archery mechanics before (particularly in terms of maximum ballistic range in a low-ceiling dungeon; links one, two, three, four). What I've gone back-and-forth about a few times is the best way to modify chances to hit at different ranges. What I would like to develop here is a best-fit archery model, partly out of raw curiosity. This model may or may not be directly usable for D&D; but if not, then hopefully we can simplify, modify, or massage useful game mechanics from this basis.

Evidence

First of all, it's critical that we note the difference between using a bow at very long range against army formations (which are easily hittable even at maximum bowshots of around 200 yards; see clout archery competitions), versus use of a bow in a man-to-man context (which may be impossible even at close range if the target is aware and moving, or maybe a 50% shot at 100 yards against an immobile target by the world's greatest marksmen; see Longman and Walrond, Archery, discussed more below). Conflation between these markedly different situations and success rates has caused much confusion in the past (starting with Chainmail's use of identical ranges for both, and continuing even up to this year with Len Lakofka's updated article on archery in &-Magazine #7, which retains the same core system.)

Based on my prior work (link), if used indoors with 10' high ceiling, then no bowfire can be used past about 150' distance (as opposed to outdoors where a longbow may certainly be fired 210 yards, etc.). For simplicity in may game, I set the standard indoors missile fire ranges at 30/60/120' -- i.e., 10/20/40 yards, or 6/12/24" on the tabletop. (Compare this to 3E DMG, p. 65: Table 3-3: Direct Fire Range.) As we will see, at these smaller ranges, bowfire can be quite accurate, at least against a motionless target.

For example, modern bow-hunting sites generally expect shots to be sure-hits out to about 20 yards, with questionable hits to possibly 40 yards (link). Targets here are presumed to be unaware and immobile (careful shooting from ambush), but still the desired target is very small: possibly just an 8" diameter area in the chest (i.e., the goal is a one-shot kill through the vitals; this might argue for some added mechanic for called criticals at short range, left for future research). Furthermore, the consensus is that shots against immobile targets are near-certain, while shots against moving targets are nearly useless. Of course, this is with modern equipment: range-finders, sighting pins, high-quality bows, etc. Even 3E D&D staffer Dave Noonan agrees, in his 2006 "Proud Nails" article: "I did enough bow hunting in high school to know that a 110-foot range increment for a composite longbow is bogus. A shot beyond 30 yards or so is rarely worth taking... whenever one of my players makes a 400-foot bow shot, I grind my teeth" (link).

Another example comes from SCA (Society for Creative Anachronism) experience with archery in mock combat situations. An SCA document of recommendations on bow-fighting (link) calls this same up-to-40-yards distance "point blank range" (i.e., effectively no arc in the arrow flight), and calls for practice against a man-shaped target, with aimed shots against unshielded portions of targets like the legs, neck, and face. Again, the claim in this document is that shooting an unaware man is easy, but one aware and moving around can easily dodge or block an arrow with a shield. Shooting in & among friends in melee is expected and successful behavior; in so doing, shots are aimed at specific individual parts, such as a visor looking over a shield, etc. (not random targets, very different from the rule in AD&D DMG p. 63). This is with intentionally low-weight draws and very soft, blunt arrows. For example:

Before I continue, as a bit of a side note, I should say that SCA combat videos are otherwise very educational (and fascinating) as a suggestion of the overall pacing & tactics that might get used for mass fights with a few hundred individuals per side. In particular, there are long minutes of general inaction, with face-offs across lines of opposed spears, each side looking for an opening;  and then some aggressive push with lots of blows struck rapidly in a very short time. Of course, this would be a resource that we didn't have in the 70's (and I find video to be invaluable for the rather narrow case of studying some particular action's timing and movement in space). Likewise, small-action contexts may be very similar to D&D dungeon action (with a few men on each side carrying mixed weapons, and very fast action). As usual, this is very distinct from the Gygaxian oversight/assertion that shots and blows are made only once per minute. Here are two other videos that I found quite interesting:

But now let's consider evidence on longer-range shooting; for example, expert bowmen competing in the top English archery tournaments over the course of a year (figures from Robert Barrow in Dragon #58; citation is Longman and Walrond, Archery, p. 240 (1894); as per Grand National Archery Series (GNAS) rules). Shooting at a motionless 2-foot radius target, the rounds in the top-level competition are held at 60, 80, and 100 yards (link) -- for which hit percentages by these champion archers are reported as 92%, 81%, and 54%, respectively. Obviously, shots at closer ranges (like under 40 yards) must be near-certain to hit. But if these are Grand Master Bowmen (GMB), then we should compare them to the lowest-level 3rd Class Bowmen -- who seem to have modern tournament scores of about 1/10 the Grand Masters (link), from which we might broadly infer that they are about 1/10th as accurate. (This is not a perfect inference by any means, because the area of the concentric scoring circles is not proportional, but it's the best I could come up with for a first-degree estimate). Say that if Grand Masters hit a target at 80 yards 80% of the time, that a minimally-trained archer will hit it about 8% of the time.

Both this data from Archery (by way of Dragon) and my theoretical model (based on bivariate normal-curve shot accuracy; see Java source code file here) assert a very sharp drop-off in hit rates, that is, about 40% loss for each doubling of range or halving of target radius (i.e., each subjective quartering of target area; a much steeper penalty than in official D&D). This is equivalent to –8 on d20, or more accurately, 8 steps on our normal-curve success chart (link), with the modifiers getting scrunched up on either end (as modeled in D&D by giving increased or auto-hits on rolls of "natural 20"). The English champion archers provide one example of this: 92% at 60 yards, but only 54% at 100 yards, for a drop-off of almost 40% with a not-quite-doubling of range (at around the 50/50 central value where modifiers are most sensitive).

 In summary: At close ranges, shooting is more accurate than represented in D&D -- shots can be aimed at small critical locations on an individual engaged in melee (not random mass targets as in the AD&D DMG rule). However, factors of target awareness and possessing a shield appear to be much greater factors in the overall success of a shot (making the difference between certain shots and impossible ones) -- granted that D&D only gives +1 for a shield (optionally +2 vs. missiles in DMG variant p. 28), or up to +4 for a stunned target (DMG p. 67). Success with shots at longer ranges drops off much more rapidly than modeled in D&D, with perhaps a –8 on d20 for each doubling of range.

The Model

Let's try to recreate the GNAS archery results with some rudimentary modifications to D&D. Assume that the tournament target is naturally AC9, with another +4 to-hit for size (about double-area compared to a man; target is 2' radius for area πr² ≈ 3(2)² = 12 sq. ft.; man abreast is about 1' wide × 6' tall = 6 sq. ft). Give an increased bonus of about +6 for an immobile, unaware, helpless target (including around +3 for target null Dexterity, and +3 for the archer being unmoving, unthreatened, and with extra aim time). Give a bonus of between +1 and +5 for shooting with modern equipment. And apply the modifier of –8 for each range doubling, at yardage 10/20/40/80/160 (and so on if needed).

Consider the novice 3rd-Class Bowman. In the bow-fire simulator (again, link to Java code here), a precision factor of 1.7 gives about expected 8% hits at 80 yards (matching GNAS data above). We'll assume this is a 1st-level fighter, with basic proficiency in the bow, and with a +2 bonus for modern equipment. Base modifier to the d20 attack roll is: 9 (AC) + 4 (size) + 6 (immobile) + 2 (equipment) + 1 (level) = +22. The results are shown in the following table:

In the table above, the first column is distance (doubling at each step). The next four columns are assessments of our D&D d20-based mechanic, with the modifiers indicated above, culminating in the percent chance for success at each range. The last column is the output of our physical bivariate-normal-curve simulator, which as you can see is matched very closely to our proposed D&D mechanic (always within 5% of each other).

Now consider a world's-best Grand Master Bowman. In the simulator, a precision of 7.3 produces the reported 80% hits at 80 yards (as per GNAS data previously). We'll model this as a 12th-level fighter, with +5 bonus for the best modern equipment, and +2 Dexterity modifier. Base modifier to the d20 roll is: 9 (AC) + 4 (size) + 6 (immobile) + 5 (equipment) + 12 (level) + 2 (Dexterity) = +38. See the table below:

This table is set up the same way. (Simulator results listed as 100% are really 99.9...% rounded to the nearest percent.) Again, the hypothetical D&D mechanic matches our physical simulator very closely.

In summary: The model reproduces the action of target shooting, across various ranges for both high- and low-level fighters, very well. And these results are also aligned with available data from GNAS long-range archery competitions.  If we back out the "immobile" bonus as used above, then we should have a moderately sensible mechanic for shooting at live characters.

Conclusions for the Game

The first and simplest point of divergence between this model and classic D&D is the +6 ranged bonus versus an immobile (helpless) target; in other words, standard D&D insufficiently values the enormous difference between an aware and an unaware target (i.e., ability to dodge or deflect missile fire). I've noted this combat bonus in my house rules (link). Compare this to DMG p. 67: I think it's a reasonable step beyond +4 for stunned/slowed/partially bound targets, but not a full-on automatic hit (which wouldn't make sense at arbitrarily long ranges).

A second point of divergence, perhaps, is the +4 bonus that I gave for increased target size. Original D&D doesn't deal with that; 3rd Edition gives a +1 to-hit for the first doubling in size, then +2, +4, and +8 (whereas I would argue that a constant bonus should be given for each doubling of length or area; see the d20 System SRD (link)). I wouldn't require that assessment, but perhaps a DM can give a bonus in this general range to hit a giant with missile fire, if desired (compare to DMG p. 63, last sentence).

But the largest single factor here from which D&D diverges is the fairly hefty –8 to hit per doubling of range category. I think it would be jarring to include such an abnormally large series of modifiers to the game, so for playability purposes in my house rules I've indicated just half of that: –4 at medium range (10-20 yards), and –8 at long range (20-40 yards) for man-to-man combat. Note that this is shooting aimed at an individual target (not randomly fired at a group); possibly modified for cover if someone is in the way, and possibly subject to a follow-up roll if a wide miss occurs. Side note: It does seem like the best-fitting model, as above, is to have the base 0 be at short range, with increasing penalties as range increases (as opposed to a base of long range and bonuses as you get closer; which was used in Original D&D Vol-1 and myself in the past).

As a fairly arbitrary game-based cutoff, I would say that man-to-man targeting is only assessed within the indoor range of 40 yards (120 feet). Beyond that, an entirely different mechanic is used -- a mass target must be selected, and the attack is made against a randomly-selected individual; no range penalty would apply as long as there are something like at least 20 or 40 creatures in the group (up to the limit of 210 or 240 yards or whatever). In reality, top-class bowmen might have a chance to hit an unaware (surprised) target at up to 100 yards, but I think for simplicity that we can ignore that.

Here's a possible rule for missed shots, solely in the man-to-man case (where the declared target is an individual, not a randomized group) -- If the total attack roll is less than 10 (as declared by the player: including any shooter bonuses, but not target AC; i.e., missed the target entirely), then the DM may opt to roll again at another target in-line with the shot. This second roll is simply d20+AC, no bonuses for level, magic, Dexterity, etc.; and no further rolls are made. (Compare to 3E DMG, p. 65: Variant: Firing Into a Crowd.)

In theory we might  consider larger bonuses for a shield vs. arrows, like on the order of +4 or so. I won't do that, but instead presume that the standard attack roll includes variation in the target sometimes being aware, and sometimes not. To conclude, the edits to my house rules are:

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• Fixed range for man-to-man fire is 30/60/120 feet for all bows.
• Shooter selects one target (not randomized, even in melee).
• Hits by missiles are at −4 for medium range, −8 for long range.
• Helpless targets are hit at +6 by missile fire.
• At DM's option, total-miss shots may be re-rolled against one random nearby target  (roll d20+AC, no other modifiers).
• If ceiling height permits, then longer bowshots can be made at mass groups (20+ individuals); ranged penalty is waived, but roll the target randomly. 

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Open Questions

How does that strike you, as regards to the theoretical model, and the derived house rules? Do you know of better data for novice-level archer hit rates? Or estimates for hit rates while in actual combat against opponents who are aware at fairly close range?

Basic D&D: On Archery

In D&D, has it ever bothered you that an archer's chance to hit a target at 200 yards is only minimally reduced from that at 10 yards? Or that lumbering giants can almost unerringly strike their tiny victims at enormous range? I'd like to make a single pointed argument here: the range modifiers for man-to-man combat in D&D are far too lenient.

Rules from Chainmail

Interestingly, in the Chainmail mass-combat rules, there are no range modifiers to hit whatsoever (CM, p. 11-12; although the traditional max ranges are immediately present on p. 10). In the man-to-man combat section, it is asserted that ranges should be split into increments of one-third (p. 25), and in the combat tables, these increments each make a difference of 1 pip to hit on 2d6 (p. 41). Simple, indeed -- however, I'll be forced to argue that the simplicity of 1-point per third-of-range was simply not thought through at all; in fact, it wildly misses the true difficulty of man-to-man archery fire at great range.

On D&D Range Modifiers

The 1-point per range increment rule is carried forward into the basic D&D rulesets. OD&D actually sets the "base" to-hit at long range, with increasing bonuses for closer distance ("Missile hits will be scored by using the above tables at long range and decreasing Armor Class by 1 at medium and 2 at short range"; Vol-1, p. 20). Gygax's Swords & Spells mass combat rules does the same (S&S, p. 24). The Moldvay/Mentzer line does appreciably the same (regular chances at medium; +1 at short, -1 at long; Rules Cyclopedia p. 108).

In the AD&D line we see an admission that these modifiers are too lenient; here, the rule is regular chances at short range, -2 at medium, and -5 at long (PHB p. 38; DMG p. 74-75). But still, I don't think these are severe enough.

On Issues of Range

First of all, let's consider an issue that is frequently overlooked: The difference between shooting at at a massed army, and shooting at a single man. The difference in the size of the target is obviously enormous; and so it's entirely possible that the former may be practically impossible to miss, while the latter may be nigh-impossible to hit, even at the same range. It's reasonable that longbows might be an effective instrument of war at 200 yards or so (against an army); while it's almost unimaginable to think that anyone could hit a given man-sized target (even stationary) at that range. So, it seems like quite an oversight in OD&D (i.e., Chainmail man-to-man rules) to switch blithely from one to the other, using the same range categories and the same chances to hit without major modification.

(Note: In Swords & Spells, Gygax did address this, with full damage only against large formations, reducing as the target unit's ranks decrease. When the "Target is single creature, about man-sized", then 90% of the normal damage is lost [p. 23]. However, no rule like this transfers into any form of D&D.)

Secondly, keep in mind that by the inverse-square law, if you double distance, the visible area of the target is reduced to just one-quarter what it was originally. For example: Say you're shooting at a man-sized target at 50 yards. Moving the target to 100 yards reduces the visible area to one-quarter. Again moving the target to 200 yards reduces the visible area to just one-sixteenth what it was originally.

On a Statistical Model of Shooting

Now, this doesn't necessarily mean that the chance to hit is reduced to exactly one-quarter and one-sixteenth in the circumstances above -- that would presume chance to hit is linear with distance -- but it should intuitively imply that hitting targets at very great ranges should be very, very difficult. What should we use as a model of shooting accuracy?

The standard statistical model would be to use a normal curve (previously developed here). For example, the article "Analysis of Small-Bore Shooting Scores" says, "... a calculation model based on the central circular bivariate normal distribution has been used to calculate the expected distribution of the the displacement of shots from the point of aim... ", and that this model was at least "partially successful" in predicting shooting competitor's scores. [A.H. Conway-Jones, Journal of the Royal Statistical Society, Series C, Vol. 21, No 3, pp. 282-296] The term "bivariate normal distribution" basically means a normal-curve model in two dimensions (mostly simply, independent normal distributions for both the x- and y-axis of the target; see here for description and simulator applets).

Let's look for some real-world data. Way back in the day, Dragon magazine published on article on broadly the same topic, "Aiming for realism in archery: Longer ranges, truer targets" (Robert Barrow, Dragon #58, February 1982, p. 47-48). Barrow begins by compiling some interesting information about modern English archery tournaments:

English archers use a 48-inch-diameter target in tournament competition. Since a 48-inch target is about the same target area as a man’s body, these archers’ scores can be examined and compared for use in game terms. A compilation of the twelve highest tournament results during a one-year period shows that the "hit" percentages of England’s finest archers at three ranges were: 92% hits at 60 yards, 81% at 80 yards, and 54% hits at 100 yards distance. The best archers for an entire year of tournament competition still scored complete misses 46% of the time when firing at a target the size of a man at 100 yards range (Archery, p. 240). And these scores were achieved using slow, deliberate fire at a stationary target. [p. 47]

Thereafter, Barrow attempts to extrapolate these numbers into a table for all different ranges. This fundamentally fails, because Barrow is trying to force the numbers into a linear progression, when our normal-curve sense (see above) tells us this certainly won't be the case. For example, Barrow's increments veer up & down irregularly: over ranges 40-140 yards, taken at 10 yard increments, the percentage chances to hit in his table decrease in these steps: 10, 8, 6, 5, 14, 13, 6, 5, 5, 4.

Let's try to do better with our normal-curve model (bivariate, in two dimensions). Write a computer program which simulates this, starting at 10 yards, and stepping back such that distance doubles over the course of 10 steps (or equivalently: shrink size of the target by the same amount in both dimensions). Pick a starting "precision" value that gives results similar to "England's finest archers" above; and fire 100,000 shots or so at each step and see how often they strike the target. A starting precision of P = 6.8 seems to do the job (see sidebar).

First, see how the key targets noted by Barrow basically match his percentages. In the table to the right, range 60 yards correlates with 92% chance to hit; 80 yards is 76%; and 100 yards is something like 58%. (Not a perfect match, but within 5% in each case.)

Let's see what this says about standard D&D longbow increments; we'll look at the middle-point of each range category, i.e., 35/105/175 yards. At short range around 35 yards, the chance to hit is nearly 100%; 105 yards, 56%; and 175 yards, about 26%. We can immediately see that the chance to hit drops off much faster than any of the modifiers in D&D or AD&D. (More specifics below.)

On Our Results in D&D Terms

So, look back and see if we can model "England's finest archers" in D&D terms. At the short range of around 35 yards, who has a 100% chance to hit (0/1 on d20)? In 1E AD&D, that's like a 12th-level fighter against AC 10 [DMG p. 74] -- and hey, that's the same as in our proposed "Normalizing Resolutions" system (level 12 + AC 10 = success level 22 = to-hit 1 on d20; see here).

Now let's derive what the range penalties "should" be for these experts. Taking the chances to-hit above (100%/56%/26%), and taking short range as the base, then the "medium" modifier could be -9 (-44% reduction), and the "long" modifier could be -15 (-74% reduction). Or alternatively we could put this in terms of our "normalized" system (probably more legitimate, granted we've used a bivariate normal curve model for our shooting) and level modifiers therein: as noted, 100% is at level 22; 56% is like level 12, i.e., -10 steps; and 26% is like level 4, i.e., -18 steps from the start.

So to make things simple again, by rounding off to convenient numbers, we see that's it's legitimate to set ranged penalties on the order of -10 at medium range, and -20 at long range. Hitting a man-sized target at 100 or 200 yards out is really, really tough! As we saw from the Barrow article above, even "England's finest archers" should be missing in our medium range about half the time -- and that's against a totally unarmored, and motionless, target. (Any additional penalties for movement are left as an exercise for the reader; or see Len Lakofka's Leomund's Tiny Hut column in Dragon #45.) Clearly range modifiers on the order of -1, -2, or -5 are fundamentally very broken.

On Firing at Armies

Barrow is typical in including a passage like this:

Many claims are made about the greatest distance an archer can accurately fire an arrow. A modern hunting bow (for use in bagging wild game) can fire an arrow almost 300 yards; however, it has an effective range of only 60 yards. The 300-yard shots require special arrows and near-ideal weather conditions. This evidence is in sharp contrast with other sources claiming that an English longbow archer could hit a man at 400 yards. [p. 48]

Perhaps, but again this collapses the issue of firing at a man, versus firing at army (which is what would be of real interest to the English longbowman). Granted that the chance of hitting an individual man at say, 200 yards is almost negligible (20% for our top expert above). But let's consider a larger target; Barrow indicates a competition by the Royal Company at 200 yards, where any arrows within 24 feet of the target count for points.

Going to our program and increasing the target size from 2-foot radius to 24-foot radius, then the "expert" shooter cannot miss at any range (100% in every category). Even switching to "novice" capacity (Fighter level 1; i.e., starting precision P = 1.9 in our model), our shooter has 100% accuracy up to 100 yards and more, and 90% accuracy even at a distance of 210 yards. So I would conclude that the original Chainmail rule (which is to say; ignore range entirely) is a perfectly good one for the purpose of shooting at armies in mass combat -- even if anything close to that would be wildly atrocious for man-to-man combat.

Some Suggested Fixes

So in short, I would actually go so far as to recommend using this derived modifier of -10 at medium range, and -20 at long range (either as usual to the D&D to-hit numbers, or in relation to "success level" in our normalized system; it's about the same either way in the meat of the progression). Yes, this makes hitting man-to-man targets almost impossible at the longer ranges -- probably for the better, as it's (a) more realistic, (b) keeps the action within playable distance on our tabletop (say, 7" or 14" or so), and (c) leaves some room for progression by the highest-level fighters.

Of course, the preceding was all in terms of outside shots, measured in "yards", etc. What about indoors in the dungeon (where ranges are in feet, thereby closer and easier to hit in our model)? Well, you could think about giving as much as a +10 bonus to hit in that situation (to make a long story short -- literally). But, we never did take into account possible cover, low ceiling, darkness, and frantic combat movement -- so if we start adding those things up, more than likely they add up to about -10 or thereabouts, so we could probably call the whole thing a wash. (Going back to our new -10/-20 modifier.)

Probably thrown weapons -- such as spears, handaxes, and daggers -- with their much smaller range, should not be subject to these increased modifiers for great ranges. I would consider skipping any consideration of range penalties for these weapons entirely. (Or perhaps further research in axe-throwing at targets is in order...)

Open Questions

1. Would you consider using modifiers of -10/-20 or the like for man-to-man archery?
2. Can we use the same modifiers indoors as outdoors (assuming that melee movement counteracts reduced range)?
3. Should handheld missiles be without penalty?
4. Should we totally forgo ranged modifiers in mass combat rules?
   
5. How important is it to give creatures like giants separate melee and ranged attack scores?
   

C++ source code for the bivariate normal archery simulator (ZIP archive; GPL/LGPL license) can be downloaded here.

Book of War: Archery

Here's another callback to the "Basic D&D" discussions last spring, this time to the now-infamous "On Archery" blog, which generated a lot of really great comments (and is currently #4 on the list of all-time top read posts for this site). In some sense this issue was a little bit easier to deal with (somewhat fewer moving parts/coupling issues) than either cavalry or pikes. Questions posed at the end of that presentation (commentator consensus in bold):

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1. Would you consider using modifiers of -10/-20 or the like for man-to-man archery? Yes.
   
2. Can we use the same modifiers indoors as outdoors (assuming that melee movement counteracts reduced range)? Yes.
   
3. Should handheld missiles be without penalty? No.
   
4. Should we totally forgo ranged modifiers in mass combat rules? Yes.
   
5. Should creatures like giants get separate melee and ranged attack scores? No.
   

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In general, I agreed with these opinions, and I was happy to build them into the way that Book of War works. For example: I suggested Question #5 as an option (giants should have crappier to-hits for missiles than melee), but that was decisively rejected, and it meant one less detail for me to include in BOW.

One other thing: At mass scale, Questions #1 (increase range penalties) and #4 (decrease range penalties) sort of cancel each other out. I actually tested Book of War with no ranged penalties at all, which is something that previously I argued for from a realism perspective, but it didn't make great gameplay in BOW. In particular, it made archers much too strong, or in other words, they had to be crazy-expensive to be balanced with other types (like: priced similar to cavalry), which itself is not terribly realistic. Therefore I decided to go with a middle ground like AD&D penalties of 0/-2/-5, which (divide by 3 and round to closest in this case), turns into 0/-1/-2 in the BOW d6 core mechanic. This made for some very nice play.


Other Archery Issues

To the right you'll see the Book of War missile-weapons table, including rate-of-fire (ROF), and maximum standard range in inches. Rate-of-fire is the number of dice one archer figure rolls in an attack -- when motionless; or, you can make a half-dice attack with up to a half-move. These are both similar to what you'll see for classic D&D (in particular for ROF, see text of Chainmail p. 11).

But here's an interesting observation about rate-of-fire: This does not imply or require that man-to-man rates of fire be the same. In fact, in my actual D&D games, I still run archery wherein bows get 1 shot per round, and crossbows 1/2. So why do they get effectively doubled at mass scale? Well, that's a function of the presumed internal formation of each 1:10 figure. Since the 10 men are usually arrayed in 2 ranks, a missile unit with everyone firing effectively gets twice as many attacks over the same time as a melee unit (with only a front of 5 men fighting at once). Or in other words: We've established that a single die-roll in BOW represents 15 attacks across 3 rounds of D&D (link). For a full figure of bowmen shooting over 3 rounds, that's 3×10=30 attacks, or 30/15 = 2 die-rolls in BOW. (Half that for crossbows.)

Conclusion: We might say that AD&D giving multiplied shots per turn for missiles (duplicating Chainmail) was actually in error, and the Basic D&D (Holmes/Moldvay/Mentzer) rule of just 1 shot per round for bows was, in this case, both more elegant and a more clear-headed model of what should be happening. (And as usual: Scaling issues are key.)

Another thing that I was compelled to consider: Mass archery at a distance is incapable of picking out individual opponents, and should have targets picked randomly (as per DMG p. 63, and plain common sense). So: Perhaps over the course of a round, so many arrows hit duplicate (already-killed) targets, that the overall effect is reduced to some degree? This prospect was investigated by simulation in the RPGBattle program. Fortunately, the "redundant shot" effect turns out to be significant in only one case -- if there are a great many archers, and very few potential targets, i.e., the victims are about to be all wiped out anyway. So it's an effect that is relievedly negligible for our purposes. See here for the spreadsheet analysis:

ArchersV109.xls

Something else that was quite popular in the earlier post comments (re: Question #5) was the idea of treating giants throwing stones as a kind of shattering/shrapnel area-of-effect weapon. Now, this was indeed what happened in Chainmail -- but it's ambiguous in OD&D, one of two options in Holmes, and then never again treated that way in any other edition of D&D (discussion here). So in light of this, and for balance and simplicity sake, I share the latter interpretation of Holmes -- giant stones are treated as a single-target weapon for 2d6 damage, and are treated by the system the same as any other missile-fire. (Rate-of-fire is 1/2 at man-to-man scale, and thus ROF 1 in BOW.)

Finally, one detail to keep in mind about the decision to use range modifiers that look like 0/-1/-2 in Book of War (analogous to AD&D mods): Note that the scale-switch from classic AD&D 1"=10 yards outdoors to BOW 1"=20' means that using the same range-in-inches (like 18" for a light crossbow) actually discounts the "real" distance for missile attacks by 2/3. As a result, the basic rule in BOW is that we really only deal with only two possibilities: under half range (no to-hit modifier) and over half range (at -1). For simplicity, in both gameplay effect and conversions using identical numbers, this was considered very desirable. The Optional Rules section contains the possible variant of extending out to long range (add another half-range, like a further 9" for crossbows), although at -2 to hit that makes for impossible shots in most cases anyway, so there's yet another reason to not worry about the missing extreme range.

And as a final final detail note this important item: Targets in heavy armor (plate, AH6 in BOW) are effectively immune to normal D&D missile attacks beyond short range (with -1 to hit, it's like AH7 versus d6). So keep that in mind for your tactical play.

Lack of Scale Considered Harmful

The first and biggest mistake in the foundation of D&D was not specifying any units of scale in the Chainmail Man-to-Man Combat rules.

In general, specifying units of scale (for figures, time, and distance) was usually among the very first things stated in traditional wargames, often before the actual start of the rules themselves. For a specific example, these are given in the third paragraph of the Chainmail mass-combat rules, preceded only by a statement on what the Middle Ages were, and what size and brand of miniatures are recommended (p. 8). These Chainmail mass-combat rules came with a respectable pedigree of playtests, refinement, and editorial corrections; in the second paragraph, they reference the prior "LGTSA Medieval Miniatures Rules... the rules have been thoroughly play-tested over a period of many months..." (p. 8). Jon Peterson in Playing at the World tells us, "The LGTSA medieval miniatures rules resulted from Gygax's expansion of Jeff Perren's original four-page ruleset..." (p. 30), published in Domesday Book #5 (July 1970). Also, "The core system of Chainmail adheres closely to the earlier LGTSA rules; for example, the movement and missile combat system charts are copied verbatim..." (p. 40). Furthermore, these rules themselves were influenced by older systems such as those of Tony Bath, etc. (p. 31). As a result of this legacy of a wargame refined by diverse hands, we find that the scales of time and distance, movement and missiles, found in LGTSA/Chainmail mass-combat are fundamentally reasonable and match well with real-world data. 

But with the publication of Chainmail, we also get a new 3-page section on Man-to-Man Combat which, relatively speaking, appears to come out of nowhere. It does not claim to come with a history of playtesting, appears relatively slapdash, and is likely the conceptual work of a single author (Gygax?). Notably, while one figure now represents one man, there are no specifications given for time or distance units on the table; it seems to have not even been considered at all. Broadly speaking, these rules try to "cheat" the issue by silently assuming that the mechanics for mass-combat can be used without alteration in man-to-man combat ("Generally speaking, the rules for 1:20 scale apply to man-to-man missile fire...", p. 25) and so forth.

Now, there are some things in the world for which, when we "zoom in" on them, the characteristics appear the same as when we "zoom out"; the technical term for this is "scale invariance" (for example: fractals). You can almost get away with the assume-everything-is-the-same approach for movement (if distance and time are changed in proportion), and melee combat. But it's precisely with missile combat where the problems and contradictions spring into plain sight -- ranged combat is distinctly not "scale invariant".

Here are two of the top absurd positions that this oversight forced Gygax to defend constantly in later a years as a result of this initial oversight: (1) That man-to-man combat took place at the same 1-minute action cycle as Chainmail, and that therefore only one sling or crossbow-shot could be made per minute, etc.; and (2) That effective missile ranges were at the same 10-yards-per-tabletop-inch scale, such that it was feasible to shoot an individual man at 210 yards outdoors with a longbow, which is patently ridiculous. (See OD&D Vol-3 p. 8 and 17; AD&D PHB p. 39).

Of course, a claim is made that the distance scale shortens to 1" = 10 feet in the indoor/underworld environment (OD&D Vol-3, p. 8). As a result of this, magic spells likewise grow and shrink depending on whether they are used indoors or outdoors. In Dragon #15 Gygax writes what seems to be a correction and apology on the issue after Len Lakofka points out the problem here. Gygax calls the existing result "ridiculous" and that "the blame for the possible ignorance of player and Dungeon Master alike rests squarely on my shoulders" (read the article and my past analysis on it here). This altered rule, that magic ranges change indoors-to-outdoors, but areas-of-effect do not,  then gets incorporated into the AD&D PHB (p. 39), in a rather screechy all-caps passage, below:

For purposes of the game distances are basically one-third with respect to spell and missile range from outdoors to indoors/underground situations. Thus most ranges are shown as inches by means of the symbol ", i.e. 1", etc. Outdoors, 1" equals 10 yards. Indoors 1" equals 10 feet. Such a ratio is justifiable, to some extent, regardless of game considerations.

Actual effective range of an arrow shot from a longbow is around 210 yards maximum, in clear light and open terrain. Underground, with little light and low ceilings overhead, a bowshot of 210 feet is about maximum. Archery implies arching arrows. Slings are in this category as are hurled darts and javelins, all arching in flight to achieve distance. Crossbows are a notable exception, but under the visibility conditions of a dungeon setting, a yards to feet conversion is not unreasonable.

Magic and spells are, most certainly, devices of the game. In order to make them fit the constrictions of the underground labyrinth, a one for three reduction is necessary. It would be folly, after all, to try to have such as effective attack modes if feet were not converted to yards outdoors, where visibility, movement, and conventional weapons attack ranges are based on actual fact. (See MOVEMENT.)

Distance scale and areas of effect for spells (and missiles) are designed to fit the game. The tripling of range outdoors is reasonable, as it allows for recreation of actual ranges for hurled javelins, arrows fired from longbows, or whatever. In order to keep magic spells on a par, their range is also tripled. IT IS IMPERATIVE THAT OUTDOOR SCALE BE USED FOR RANGE ONLY, NEVER FOR SPELL AREA OF EFFECT (which is kept at 1" = 10') UNLESS A FIGURE RATIO OF 1:10 OR 1:20 (1 casting equals 10 or 20 actual creatures or things in most cases) IS USED, AND CONSTRUCTIONS SUCH AS BUILDINGS, CASTLES, WALLS, ETC. ARE SCALED TO FIGURES RATHER THAN TO GROUND SCALE. Note that the foregoing assumes that a ground scale of 1" to 10 yards is used.

Now, a couple things to note about this passage. One: a cursory justification for the feet-to-yards conversion is made for missiles ("little light and low ceilings overhead"). Two: absolutely no justification is attempted for the expansion of magic spell ranges; it is purely a matter of raw game balance ("devices of the game... designed to fit the game"). In fact, to my knowledge, Gygax never attempted any in-world explanation or rationalization for this phenomenon. (You can of course make up your own: Do magic energies follow ballistic trajectories and get limited by ceiling height? Is every underworld locale uniformly imbued by dark counter-energies that reduce magic effects? Not to say that any such claim is in any rules text.) Ultimately Gygax hangs his hat on, "It would be folly, after all, to try to have such [magic] as effective attack modes if feet were not converted to yards outdoors, where visibility, movement, and conventional weapons attack ranges are based on actual fact." But this ignores the actual actual fact that shooting an individual man at 210 yards with a longbow is sheer lunacy in the first place.

Here's the thing that occurred to me a few days ago, and that I'm embarrassed at how many years it took me to observe: The whole notion of indoors-versus-outdoors is a false path and a distraction. The real issue is whether the action is at mass-scale or man-to-man-scale. Which again, is the original error, the essential oversight in the new section of Chainmail.

Let's look at some data. There's a notable real-world circumstance in our favor; modern archery competitions in the United Kingdom have the exact distinction that we're looking for here. There's standard target archery, at a fairly close range, with a target passingly close to the size of man; and separately, clout archery at a distance near the limit of a classic longbow, with a relatively huge target area (fundamentally simulating shooting at an army). Specifically: standard target sizes are 122 cm in diameter (approximately 4 feet, or 2 foot radius). Clout archery for adult men is held at 180 yards range, with a target area 12 feet in radius, and a central "clout" (bullseye) of 18 inches radius (that is, about the size of the entire short-range target, or roughly a single man's area).

Here are results for the Yorkshire Archery 2018 Clout tournament. For more, here are results from the National Clout Championships of 2016. Here's a data analysis by myself (ODS spreadsheet) of the latter tournament for the Gentleman's Longbow event . Some results of that analysis (N = 30, discounting last two outliers with only one point between them): The average hit rate on the 12-foot radius target at 180 yards was just 42% (ranging from 11% for the bottom-performers, to 83% for the winner). The average hit rate on the central bullseye/clout -- about the size of a man (assuming a totally immobile, defenseless one) -- was only 1%!. (Even the winner himself only scored a 1% hit rate on the centermost target; the two runners-up scored 6% and 8% bullseye rates, but these may be considered pure luck since their overall accuracy was not as good as the winner's, and in any event represent the equivalent of natural-20's for these almost-England's-best-archers).

The central lesson here is that it can be effectively impossible to hit a target in man-to-man combat at long range (1% vs. the central clout), while being completely feasible against a larger area/group of men (42% vs. the larger target, roughly the same chance D&D gives a 1st-level man to hit an unarmored opponent). If we take the small 18 inch = 1.5 foot radius as roughly the area of a single man, then the larger 12-foot target is equivalent to some 64 men in formation ((π(12)^2)/(π(1.5)^2) = 64). If we were to double the target radius again, to 24-feet and some 256 men, then this would be a 90%-something shot, nearly unmissable (using ArcherySim on GitHub). For emphasis: With a longbow at around 200 yards, hitting an individual man is a 1% shot or less, while hitting an army is a 99% shot or more. The cases are exact binary opposites. Note that Chainmail mass-combat had no rules or penalties for missile range, and we find this to be completely reasonable; but keeping the same or a minimal range penalty for man-to-man combat is, as Gygax would say, "ridiculous".

Some conclusions: One, the maximum effective range for man-to-man missile combat should be set at around 40 yards; this is especially true for a target that is mobile and defending itself (note that the real-world data above assumes a completely immobile, defenseless, unarmored target; hit rates should obviously be lower if that is not the case). This is true whether indoors or outdoors. Note that the legacy of this glitch has led to ranged attacks never being close to right in any edition of D&D. Over 40 years later, and in 5E D&D (from what I can tell), a 1st level fighter shooting a longbow at a mobile, active single man at maximum range of 200 yards still has a 42% chance to hit (AC 10 with +2 attack bonus, i.e., target 8 on 20, with disadvantage); coincidentally exactly the same rate that the UK's champion clout competitors actually have against an immobile, barn-sized target. That's twisted.

Two, in classic D&D, there should have been greater care taken in specifying scales, and distinguishing between the man-to-man and mass-combat situations; the two are not equatable. A random example: In D&D Vol-3, the Aerial Combat and Naval Combat sections seem to be closely related; they refer to each other in places, stipulate the same playing area, turn sequence, and written orders (compare p. 25 and p. 30), etc. But in truth, Aerial Combat is intended for man-to-man scale (each figure a single creature), whereas Naval Combat is intended for mass-scale (each ship model carrying tens or hundreds of men; missile fire as per Chainmail mass rules on p. 30, etc.; at least until a boarding action occurs and then we are directed to switch maps and rulesets to the man-to-man basis on p. 31). These are very different situations, requiring different distance and time units, tabletop missile ranges, turning radii (another characteristic that is definitely not scale invariant), etc., and this qualification should have been highlighted in the original rules.

Three, by Gygax's logic in the PHB, we should also calibrate the range and effect of magic spells on the reduced basis for game-balance purposes ("It would be folly... [for magic to contradict] where visibility, movement, and conventional weapons attack ranges are based on actual fact"). Again, irrespective of being indoors or outdoors; that is not a relevant distinction. Note that once this physical reality of the magic spell range is set by man-to-man scale, it implies that apparent range and usefulness in the mass-scale context is much reduced. Example: In Gygax's later Swords & Spells mass combat rules, spell ranges in inches are copied verbatim from the D&D rules, and hence have extraordinarily long-range effects on the battlefield (e.g., 24" for a fireball, i.e., 240 yards). Reversing the reasoning, we now consider that if the spell range was fixed at 40 yards, as per man-to-man missile fire, then on the mass-scale battlefield a fireball would only be usable at 4" or something like that. That's a fairly major change (perhaps in Book of War?), but upon reflection, it may be a better simulation of magic effects as seen in pulp literature and similar traditions. Some of the higher-level spells meant to influence large areas outdoors might prove troublesome, however.

To wrap this up, we look at a quote from Gygax in The Strategic Review, Vol. 2, No. 2 (April 1976), in his article "The Dungeons & Dragons Magic System" (p. 3), that some of us have been considering recently:

Magic in CHAINMAIL was fairly brief, and because it was limited to the concept of table top miniatures battles, there was no problem in devising and handling this new and very potent factor in the game. The same cannot be said of D & D. While miniatures battles on the table top were conceived as a part of the overall game system, the major factor was always envisioned as the underworld adventure, while the wilderness trek assumed a secondary role, various other aspects took a third place, and only then were miniatures battles considered.

This is somewhere between a strange thing to say and a ghastly oversight (that underworld adventures came first in the calibration of D&D magic, and miniatures battles a distant fourth), because in terms of time and distance, exactly the opposite is the case. The mass-combat miniature scales were carefully figured, and the underworld scales simply taken by theft of the same and without any real consideration. Even in the SR 2.2 article quoted above the issue continues to entirely escape Gygax's attention (the topic being only a defense of the Vancian memorization and daily-limit conceits). If only some assistant had been able to point that out at the earliest date.

Edit 9/1/18:  Around the time I was writing this, Jon Peterson had a new post about some of the prior systems that fed into Chainmail man-to-man combat. It doesn't exactly address my main criticism here, but it's quite interesting to know about for its own sake. Thanks to Jon for pointing this out to me.

Edit 9/20/18: Geez, I may be stuck in a fugue on this issue.

Archery – Field Experiment

My current house-rules for archery are based on a combined mathematical model and game simulation program (link). This is the fruit of quite a bit of analysis on the archery game (search the blog for "archery", you'll find lots of posts). The most important observation is this: shooting man-to-man and shooting at an army are totally different tasks (the former may be impossible to hit, while the latter impossible to miss, at the same range).

This is based on some pretty good data in Dragon #58, originally from the book Archery, on hit rates for Grand Masters at different ranges (think: top-level fighters with several bonuses). Not having specific hit data, my model for beginning bowmen (1st level fighters?) was very roughly estimated by recent GNAS scoring, and guessing they're maybe one-tenth as accurate as Grand Masters. The table below recaps the expected hit rates from both the game rule model and the bivariate normal physics simulation. But were those reasonable assumptions?

Obviously, the only way to approach this scientifically is to run a test in the field (like, an actual field). At the end of August I visited my folks' place in Maine and got out my very old bow and arrow kit and set up a target to see whether my own accuracy broadly matched this math model. The equipment is a 30-year-old Bear compound bow with a 28" draw at 60 pounds (with no maintenance ever having been done in those 30 years; in fact, it spent several winters in an outdoor shed), shooting 32" target arrows. I made a 2-foot radius target (to match the old GNAS competition), and took a series of 10 shots each from 10, 20, and 40 yards distance (to match the increments in the prior D&D model). Due to time constraints, I didn't take any practice or warmup, and I haven't done any shooting in at least several years (and I've never done it with this target size or range).

Now technically the first thing I did was set up the target and several bales of hay in back of a metal trailer and shoot one flight of arrows that I had to start with. Each of these shots went entirely through the target, hay, into the side of the trailer, and entirely shattered from head to tail. So I didn't count those, and had to go down to the Kittery Trading Post to get another batch of arrows. Later that afternoon, I was back with this setup:

Here are the two flights of 5 arrows each, shot from 10 yards. I easily hit all ten times, although several of the arrows flew entirely through the target. Notice that even on the second set of 5 my accuracy was noticeably improved, grouping the arrows closer to the center of the target.

Here are the  shots from 20 yards. Again I hit the target all 10 times -- although more of the arrows are disappearing through the target. I think I started getting a fuller draw at this point, because on the second set of 5 every one of them flew entirely through the target.

At this point I started shooting from 40 yards away (from way down in the field, actually). Here I only got 2 arrows out of 10 to actually go in the target. Generally I was aligned correctly, but my shots were mostly falling low/short, although I'm sure that would improve if I got more practice and got the ascension right.

So, an admittedly small sample size, with very old equipment and an unpracticed shooter, but that's all I could accomplish on the particular day. Let's compare the results to our prior model:

That's not a perfect match, but the numbers do seem to moving in generally the same direction. Based on my experience that one day, I really couldn't miss an immobile target of that size from 10 or 20 yards distance. From 40 yards I was hitting a bit less than predicted for a "3rd class bowman", but I'm pretty sure with a little more practice at that range I could start doing much better than that, likely above 30%. (As a comparison, the next day I spent most of the afternoon shooting from 30 yards, and I felt like I nearly couldn't miss once I had the range down; over about 100 shots I missed only 2 or 3 times, for a 97-98% hit rate. The house rule game model would predict a 55% success rate, and the physics-model simulator about 48%.)

Generally it seems like with a little practice, my hit rates are better than predicted, which could be due to a number of factors: (a) I underestimated the skill of GNAS 3rd-class bowmen, (b) I have better equipment than that used for the base data (the Archery book was published in 1894), (c) I'm a better archer than expected -- seemingly the least-likely hypothesis.

Coincidentally, a neighbor's son who is entering high school as freshman came over the next day with his own archery equipment, and it was kind of stunning to see how the equipment and style has all changed completely in the 30 years since I got my own bow. His bow is much lighter and smaller (half the draw weight) with a bunch of built-in sights and range-finders and whatnot. Shooting is done with a bent left arm (whereas I need a big leather guard to prevent injury) and a loose grip, letting the bow fall out of the hand loosely on a cord after the shot (whereas my heavier bow would likely break my wrist if I tried that). Plus a trigger-button release is used, whereas I mostly chewed a hole through a leather glove and my fingers over the afternoon. He took a few shots and obviously had much greater accuracy than I was getting, which speaks to the rapidity of how much the discipline can change due to technology in a fairly short time.

Anyway: What we clearly see in the experiment is that hit rates may be 100% at close ranges, and then very rapidly drop off to near-zero with just a few doublings of range. Even if I could practice more at the 40-yard range, I'm sure that shooting from 80 yards out at a single man-size target would be practically impossible for me. But the flip side is that it would only take a few doublings in target size (4 or 8 men deep or wide) and I'd be back to automatically hitting on almost every shot. So in broad strokes the math does seem to be winning, and generally predict the overall dropoff in accuracy from shooting at a small target at distance.

Got any more data?

Interpolating the Man-to-Man Scale from Chainmail Hit Rates

As a follow-up to last week's post where we considered the yawning gap in that Chainmail Man-to-Man Combat had no units of scale specified (and the years of headaches that followed), today let's try to compute what the distance scale should have been. Of course, more than one argument has been advanced in this direction. The most obvious one is to just take the figure scale in use and use the same or similar scale for ground distance; done. But here I'll take an approach I've never seen considered -- granted that Chainmail Man-to-Man Combat has no scale, it does have specified hit probabilities, and we can use those to back-calculate what distance scale is implied by those.

First, here's our standard simulation of archery hits at range, using a bivariate normal model (both error in the x- and y-axes simulated by a normal distribution), and now uploaded as ArcherySim on GitHub. An earlier version of this simulator was used numerous times on this blog, such as here and here. The simulator has been calibrated to match results seen last time from the UK National Clout Championships: at 180 yards range, a hit rate of 1% against a 1.5-foot radius target (equivalent to one man), and a 42% hit rate against a 12'-foot radius target (equivalent to a group of 64 men). Here's a table compiling the results from that program run on several different target sizes:

A couple comments about this table: Internally, all the simulator really does is for each doubling of distance, shrink the apparent radius of the target by half -- and likewise to interpolate any other distance. Therefore, each doubling of distance is perfectly offset by a like doubling of the real target radius (and hence by a quadrupling of the number of men in formation), which can be seen by matching numbers running diagonally in the table. Of course, this presumes an immobile, defenseless, unarmored target (an aware and mobile man on the battlefield should shift these probabilities downwards by some amount). All of this is reasonable.

Next, let's look at the "Individual Fires With Missiles" chart from Chainmail. It looks like this:

For simplicity, we'll take the median missile range of 18 (as for a horsebow, light crossbow, or arquebus) as exemplary; therefore our supposed range categories fall at distance 6/12/18 inches, per the footnote on the table. Also note in the first column of results (armor class 1, which on this page indicates "no armor"), most of the hit targets are 5-6-7 (indicating the minimum score on 2d6 for a hit at each range category), with the outliers either up or down by one pip.

Now we'll take a few theoretical different scales for these "inches", see what the hit rates against a 1.5-foot radius target (one man) would look like according to ArcherySim, and convert those back to a 2d6 basis (for example, using a table like the one that appears at the bottom here). We will consider the possibilities that 1" = 10 yards, 1" = 10 feet, 1" = 5 feet, or 1" = 3 feet.

• 1" = 10 yards: Therefore the median missile range categories fall at 60, 120, and 180 yards. According to ArcherySim (using the detailed output option with the -L switch), the hit rates against the 1.5-foot target should be 7%-2%-1%. On the 2d6 basis, these percentages convert to target scores of 11-12-/ (i.e., effectively impossible at the longest distance).
• 1" = 10 feet: In this case, the range categories are 60-120-180 feet, equivalent to 20-40-60 yards. The simulator says the hit probabilities should be 47%-15%-7% (you can see two of these values in the topmost table, in the leftmost column). On 2d6 these convert to 8-10-11. 
• 1" = 5 feet: Here the range classes are 30-60-90 feet, equal to 10-20-30 yards. ArcherySim estimates the hit rates at 92-47-24%. On 2d6 these become 4-7-9. 
• 1" = 3 feet: Range categories become equivalent to 6-12-18 yards.  Our simulator computes the chances to hit at 100%-83%-54%. On 2d6 that looks like 2-5-7.

These results are illustrated below:

The first thing that is visibly obvious here is that the Chainmail modifiers for range adjustments, at just a single pip per category, are far too small. In the chart this appears as the Chainmail hit targets (the green line segment) being a tiny little span compared to the real-world models at around the same values. A better simulation would be to alter the hit targets by around 2 or 3 pips per range category (equivalent to something like −6 to hit in D&D with the d20 system). Also, this could be better if the range classes weren't assumed to be linear (that is: 6-12-24 range units would be a better physical model than 6-12-18 units; then the long-range chance segment wouldn't be shrunk compared to the medium-range chance segment in each case).

The second only slightly less obvious fact is that the 1" = 10 yard and 1" = 10 feet scales (the ones actually used in D&D) are terribly poor matches; they don't even overlap the Chainmail 5-6-7 targets at all. The 1" = 3 feet proposal at least overlaps it, but is skewed off almost wholly to the left side (e.g., should be effectively unmissable at such a short range). The one that is best centered over then Chainmail hit interval is the 1" = 5 feet scale, and we can therefore use this as the true "implied" distance scale for Chainmail Man-to-Man Combat.

Archery Revisited

I've written about refining the D&D archery rules a few times, in regards to indoors ballistics and normalized probabilities (e.g.: here, here, and here). A while ago I again looped back to thinking about them, because a few parts of the prior stuff I've written have started to bother me. Here's the revised rule what I've been using for a while now:

---

Bows: Bows can be fired every round; slings and crossbows every other round. Indoors, all missile weapons have an effective range of 6"/12"/24" (30/60/120 feet; assume a 10' ceiling). Attacks are +4 at short range, +2 at medium range.

Throwing: A spear, dagger, or hand axe may be thrown up to 12" (60 feet) indoors. These are always treated as long range (no bonus to hit).

Long Distance: High ceilings allow longer bowshots, but these are at -10 to hit individual targets. Shots at great distance outdoors are only effective against armies or the like.

---

Commentary: The primary issue that drove the change is as follows. Even with the modifications I've suggested in the past, I was still trying to hang on to the hand-wavy system in D&D that you can arbitrarily switch from scales in tens-of-feet (indoors) to scales in tens-of-yards (outdoors) and still use the exact same range modifiers. In retrospect, that's unsustainable, and I'm going to stop using it; different scales simply must recognize different chances to hit. (This would be one of the "distortions" mentioned by Gygax in Dragon #15 [todo: link], and one that most later systems sensibly avoided.)

Let's think about why that was done in the first place. Again, the scale 1" = 10 yards was originally established for historical, mass-combat Chainmail, and in so doing, created realistic scaling for mass figures, movement rates, and bowshots on the tabletop. Later, 1" = 10 feet was used by Arneson in his man-to-man games, and included by Gygax in D&D as the "underworld" scale (Vol-3, p. 8), with 1" = 10 yards maintained as "wilderness" scale (Vol-3, p. 17). But there are two major problems with this retention. First, the 1" = 10 yards scale is less about being outdoors, and more about the mass-combat scale, and so irrelevant for the man-to-man RPG. Second, while it enables a realistic-length bowshot outdoors, it overlooks a colossal and critical fact -- no one in the world can possibly hit a single man at maximum distance with a longbow. Hitting an army in formation, yes, easily so; hitting a single man, no, not even close. And hence this consideration is also irrelevant, and even permitting it is one of the "proud nails" that will irk many about the system for years to come.

So let's agree to abandon the separate scale for outdoors action, and consider some physics for a better rule: Due to the inverse-square-law, if we were being really honest, range categories should work by doubling the distance in each category -- in fact, you see this in a lot of gun-based systems like Boot Hill, Star Frontiers, etc. The D&D system (splitting ranges into equal thirds, linearly) is an outlier in that regard, and just plain incorrect. The best I can figure is that, taking a shot of about 40 yards as a base, every halving or doubling of distance should modify to-hits by about +/-8 on a d20 roll (for example: look at the expert archery table we made before; compare the chances shown at 80 yards and 160 yards; 76%-30% = 46%, i.e., 46%/5% = -9 in that case). Technically this modifier should be applied on our normalized table, but in the meat of the progression it's the same as standard D&D.

Now, it's good to be aware of the correct real-world success chances involved; but at the same time, it's best to take those insights and massage them into easy-to-use, highly memorable mechanics that are convenient at the table. What we've calculated in the past for indoor missile ballistics is a 150' maximum shot for basically any weapon under a 10' ceiling (see here). But I figure it's nice and simple to smooth it out to 120', and so have the range in inches revert back to our familiar multiples-of-three: 6"/12"/24". Also, this happens to line up perfectly for range in inches indicated for the heavy crossbow weapon (the longest in the game). And also the 30' lower limit is the same as that identified in AD&D Unearthed Arcana as "point blank range" (UA, p. 18). And also our bonuses are like those in man-to-man Chainmail/OD&D, except doubled for the conversion factor we agreed on in the past (here). So I think there's a lot in favor of this simple setup.

The scale and the mechanic will be used identically both indoors and out (removing one of the "distortions", as Gygax put it). We observe that single men simply cannot be hit by a bowshot outdoors at hundreds of yards distance. If an indoor area has a very high ceiling (cavern, giant-hall, etc.), then allow a shot perhaps up to 48" (240 feet), but at a massive -10 penalty. For simplicity, the same range rule is used for any missile weapon in the game (presenting something easy to memorize, and reflecting again that range of the weapon has little or no bearing on accuracy against a man-to-man target).


Below are some ideas for optional rules you might also consider using in conjunction with this rule.




Optional -- Other Penalties: The scores above assume best-case conditions. Be sure to apply other penalties for darkness or low-light, cover and obstructions, and possibly high-speed movement lateral to the shooter's field of fire. (See Lakofka article in Dragon #45, for example.)

Optional -- Weapon Variation: If you want to treat various weapons differently for indoor, man-to-man combat, then split ranges in inches into thirds as customary, and apply the +4/+2/+0 bonuses as shown above. For example: a longbow will be 7"/14"/21" (35/70/105 feet). This is not totally accurate, but fairly close, simple, and playable (although not so simple as the unified ranges above).

Optional -- Longer Shots: If you want to permit very long-range shots outdoors against man-size targets, keep in mind that this will be an epic feat achievable only by very high-level warriors. Longer shots can be allowed outdoors at ranges up to 50"/100"/200" (i.e., 250/500/1000 feet, or about 80/160/320 yards), at to-hit penalties of -8/-16/-24, respectively. Of course, the standard maximum range of the missile weapon still applies.

Optional -- Shots at Groups: While I'm thinking about it, here's a possible rule for shots at groups of size N. Step 1: Identify a target for the shot by random method. Step 2: Roll d20 to hit, but as long as the natural die-roll is less than or equal to 4×log₂N, then re-roll any miss. See here for the exact upper bound: N=1:0, 2:4, 4:8, 8:12. (The basic observation here, again based on the inverse-square-law, is that each doubling of range category is balanced by each quadrupling of area/people in the group, i.e., +/-8 to hit. Therefore each doubling of people should be effectively half this, or +4. I don't actually do this, but perhaps you'd like to try it.)

OED Update (v0.8)

Getting ready for some D&D games at HelgaCon 2011 this weekend, I went and updated my Original Edition Delta house rules (now version 0.8). Some refinements, edits, and clarifications. Folded in some miscellaneous house rules, and also the fighter Feats that I use. Added some tables to make some things clearer.

See the mini-website here. Or get the PDF directly here (5 pages). Or see the updated player-aid card here.

One thing that rather surprised me as I did my editing process was the rule for archery that I decided to use (based on prior blogs discussing things like indoor missile ballistics and archery range penalties). What I decided to run with was this:

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Bows: Bows can be fired every round; slings and crossbows take one round to reload between shots. Underground, missile weapons all have an effective range of 30" (assume 10' ceiling). Attacks are −10 to hit at medium range (over 1/3; 50') and −20 to hit at long range (over 2/3; 100'). Note: Thieves can only use a sling or light crossbow.

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In the past I didn't think I'd use that option (from the indoor ballistics article), but as I did my pass-through tonight it seemed really elegant and attractive. It's short. It's accurate to the physics (at a 1"=5 ft scale). It's trivial to remember (one number for all missile weapons indoors). It gets rid of one of the tables you'd otherwise need. It's easy to divide into range increments (10/20/30 inches). It's simple to eyeball on a 10'-scale-map (50/100/150 feet). Coincidentally, the penalties are actually the same as the range increments: -10 at 10"; -20 at 20" (which then suggests the idea that you might just apply a -1 per 1" penalty; although it really shouldn't be linear like that, maybe it's close enough).

So that seems like a lot of reasons pulling for it. We'll see how the players respond to it this weekend (or if they even notice a difference). More to come...

Tournament Archery

I like archery, I like tournaments in D&D, and I like having a good concise rule for both of those things. There was an article in &-Magazine #7, this past January, by Len Lakofka on the "Archer/Archer-Ranger PC Class" -- an update to his article originally in Dragon #45. I was tickled to see at the end of that article that Len included rules for a tournament archery competition:

Cool idea, although it's not exactly how I'd do it. One: it's another of these big table-based mechanics from the old days, which generally turn out to be unnecessary, and requires another page of paper at your table just for this. Two: The game-world target that Len presents has just 4 rings, as opposed to real-world targets with 5. Third: Len doesn't include any points-scoring system (which wouldn't be bad if the target was the same as real-world, but it isn't, so it's unclear to what resource we should turn).

A Second Shot

So here's my take on it. An initial, maximally-concise option would be this: just have the archers roll d20's and add their attack bonuses, and add the totals all up as their "points" (however many rolls you like). That's in line with the core mechanic and requires nothing to look up or memorize, really. However, maybe that's a little too abstract.

A more concrete option would be this: Say in-game targets are the same as real-world ones, viewed by Imperial scoring rules -- the targets have five concentric regions on them: white, black, blue, red, and gold, for point values 1-3-5-7-9 respectively (in Metric scoring the regions are subdivided into two halves each, for point-values from 1 to 10). Roll an attack as normal against the target, and for each +5 increment, say that an improved scoring region has been hit (i.e., each smaller region has +5 better armor class).

Based on our model of archery for D&D from last week (link), the effective AC for a large, immobile tournament target is approximately 10 (base AC) + 4 (size) + 6 (immobile) = 20. Note that this exactly cancels out the normal "Target 20" requirement for a hit, so we can simplify matters by just looking at the raw roll from the player, d20 + attack bonus (including level, Dexterity, equipment, etc.) and not even bothering with the normal AC or to-hit requirement. Say the levels of success are then as follows, for convenience:

Again, this is just standard real-world archery scoring. I would have each archer in the competition shoot perhaps 12 times (real-world tournament have contestants shoot 2-3 dozen arrows per round, but you'll probably have other players at the table waiting to do something). Also: this is at very short range (10 yards; Lakofka sets his at 40 feet) -- if you run a longer-range competition at a distance of 20, 40, or 80 yards (as in real life), then subtract a penalty of -4 from the attack roll for each doubling increment of range. (Optional simplification: just say hits are one ring worse per range increment.)

So I think that mechanic is pretty attractive, and you can probably memorize it for use without any table lookups -- Just make a standard attack roll for each archer, and remember that each increment of 5 is an improved hit location (with real-world Imperial scoring of 1-3-5-7-9 points). Easy!

More Missile Modeling

I've written so many letters on the physics and statistics of missiles, archery, and ballistics that it could sink a warship (search the blog, you'll see). So much so, sometimes it's easy to lose the plot at this time. I figured I'd summarize some of our findings to date.

We have two primary sources of data. One is from Longman and Walrond, Archery (1894) -- as reported by Barrow in Dragon #58, "Aiming for realism in archery" (Feb. 1982). He writes (first noted on the blog here):

English archers use a 48-inch-diameter target in tournament competition... A compilation of the twelve highest tournament results during a one-year period shows that the “hit” percentages of England’s finest archers at three ranges were: 92% hits at 60 yards, 81% at 80 yards, and 54% hits at 100 yards distance.  

A second source of data is from more recent UK "clout" long-distance longbow competitions. Results from a competition in 2016 show that at a range of 180 yards, competitors hit a 12-foot radius target 42% of the time, and an 18-inch radius central target only 1% of the time. (Full data and spreadsheets on the blog here.)

Using that as a guideline, we've developed a simple physical simulation to model archery shots, using an idea I first saw in Conway-Jones, "Analysis of Small-Bore Shooting Scores", Journal of the Royal Statistical Society (1972). The idea is fairly simple: model shooting error in both the x- and y-axes directions as two independent normal curves, which we call the "bivariate normal distribution". (First noted on the blog here.)

The simulation of that is written as a Java program and posted to a public code repository at GitHub (here). If we run that program with settings of precision = 6.7 (extremely high skill!), target radius = 2 feet, and long output form (that is: parameters 6.7 2 -L), then we get to-hit results very close to the 1894 Archery figures (compare highlights to quote above):

Likewise, if we run the program with precision = 1.6, target radius = 12 feet (parameters 1.6 12 -L), then we get results very close to the recent UK clout tournaments:

Also, if we set the target radius of this latter experiment to 1.5 feet (that is, 18 inches), then the hit rate at 180 yards becomes 1%, exactly as seen in the real-world data. Comparing these two data sources, we might be led to think that English archery skill has dropped off precipitously between 1894 and 2016 (precision 6.7 in the former and 1.6 in the latter). But based on the short quote regarding the first data source, we might say that it was cherry-picking its data; the best dozen results across all tournaments in England in a year. Contrast that with the second data source which includes all 30 competitors in one single tournament, whether they performed well or not. So the jury is still out on that issue.

That ends the recap. Now for a new thought: What is the best statistical model for these numbers? Clearly it's capped above and below: the chance to hit (or miss) cannot possibly be more than 100%, or less than 0%. Presumably we want a smooth, continuous curve, and one that can theoretically handle any arbitrary distance. Effectively we have just given the definition for a sigmoid curve, that is, an S-shaped curve seen in many probability cumulative distribution functions. The simplest model for this is the logistic function, as applied in logistic regression analysis.

One problem with this observation is that logistic regression of this sort is not built into standard spreadsheet programs (Libre Office, Excel) like many other types are (linear, polynomial, exponential, etc.) So what I've done below is this: Used the model derived from 2016 clout shooters (second experiment above; precision = 1.6, set target radius = 2 feet); increased granularity of the output to increments of 2 yards (for added detail); converted hit chances to miss chances (because the logistic curve expects numbers to be increasing from left-to-right), and used the online Desmos graphing calculator site (here; thanks immensely, guys!) to regress it to a logistic function. We get the best possible fit as follows:

Note that our regression (orange curve) has an R² = 95.87% match with the numbers from our simulated physical model of UK long-distance clout shooters (black dots). One possible downside: the logistic formula shown in the bottom-left is probably too complicated to use in a standard D&D gaming session. However, a second observation occurs to us: in the central part of that curve, at distances from around 20 to 40 yards (that is, ignoring the parts that are close to 0% or 100%; i.e., the part with maximal rate-of-change), the curve is practically a straight line.

Let's find an approximating line for that "critical" part of the curve. Our regression formula generates the points (20, 0.28) and (40, 0.73) -- so, this is the region where hit-or-miss chances vary from about 25% to about 75%. Solving for an equation of a line through those points (using Wolfram Alpha or good ol' college algebra) gives: y = 0.0225x − 0.17. Note the slope m = 0.0225, which means the chance to hit drops by 2.25% per yard on that region. Converting to feet we get 0.0225/3 = 0.0075, so: 0.75% per foot, or 7.5% per 10 feet. Note that this is freakishly close to the 7.6% per 10 feet figure we saw in the Milks spear-throwing experiment a few weeks ago.

In conclusion: It seems like our data and multiple models are telling us that there's a consistent dropoff in hit rates of around 7.5% per 10 feet, in the part of the range where it matters (neither a near-automatic hit or miss). This is why in the last few months in my D&D game I've shaved this number off to 5% and simply said there's a −1 chance to hit per 10 feet, on a d20 attack roll. But how to account for the extended upper and lower parts of the sigmoid S-curve distribution (where the chances are almost, but not quite, 0% or 100%)? Well, the classic rule to auto-miss on natural "1" and auto-hit on "20" (or something close to that: say they count as −10 or +30) does a fair job of recreating the rest of that model.

(P.S. Keep in mind that the exact hit-or-miss numbers shown above assume a single unmoving, undefended, man-size target of radius 2 feet or so. In practice, we need all kinds of extra modifiers to account for aware, defensive men in the field; shooting at a clustered army of bodies; and so forth. But from what we can tell the specific range modifiers increments would be generally consistent regardless of other considerations.)