## Monday, April 14, 2014

### 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:

• 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.

### 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?

1. Say 30' range, shooting into melee. You're saying it's possible, accurate, and unlikely to hit your party members or teammates in an open melee?

That said, wholesale adoption for any future classic D&D games is likely.

1. Yes, that's totally been my recent epiphany. In addition to the bow-hunting and SCA evidence, I had a lengthy discussion last weekend with one of the top national laser-tag players who also asserted that must be the case, based on his analogous experience.

2. I use a different mechanic
-3 for medium range
and
roll d12 (not d20) for long range....
d12 allows master archers to still make simple shots at long ranges, but difficult shots are still mostly unobtainable..
..
In ranged combat, the natural ‘1’ automatically misses and signifies that an attacker has EITHER run out of ammo, broke his bow string OR that a magical weapon has lost its enchantment ( DMs discretion).

3. 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).

Would the advantage/disadvantage mechanic of the D&DNext playtest do? Against an unaware opponent, roll 2d20, keep the better score.

1. Urh, I never deal with reroll or dice-pool mechanics. Partly for tradition and partly because the probability assessments (e.g., synching with real-world research) starts getting unboundedly complicated. Just not something I'd personally consider. That sounds very 4E-ey.

4. I think you have immobile, mobile – unaware, mobile – aware. On top of that I think aware shield is another important bonus, it’s a big reason to have a shield. Not sure if you want to make the distinction between split focus aware and sole focus aware. That might be a bridge too far, but in the video it’s pretty clear, if someone has a sword in your face, you may be aware of the archer, but your defense is compromised. If the shield is between the attacker and target, i.e. not a rear shot, I would add the shield bonus to ac normally, and allow the additional bonus for aware shield. Not sure what bonus level each would have, but I would consider making them all the same, even though some seem to be more important.
Immobile 0,
Mobile -2,
Mobile and aware -4,
Mobile and aware with shield -6 (again this is on top of shield bonus to ac)
If archer is sole threat additional -2
Attacker safe from retaliation +2
The other question I have, is there really a difference between not being in combat, but mobile (flat footed) and being in combat, but unaware of the threat? (So immobile, flat-footed, mobile, aware, sole focus aware.)

1. Yeah, that's more or less exactly what's in the AD&D 1E DMG. I think the evidence points to a bigger real-world difference between mobile-aware and mobile-unaware. And importantly the D&D system expects the base case to be mobile-aware, giving bonuses in other cases (i.e., scale your numbers by adding 4 to each).

2. I’m not sure why it would be like the DMG. I’m talking about keeping all your rules and range penalties. I’m just saying the +6 Helpless mod you have could be broken out into 2 or 3 cumulative mods. +2 for distracted (aware, but someone else also threatens target, +4 for total un-aware and +2 for immobile (+6 if unaware and immobile). Or for a simpler set of mods +3 for immobile and +3 for unaware. I should have made it clearer.

I also think shields would be very important, but I would not fiddle with separate ranged and melee armor class, I would just give the shooter a -2 penalty if the target is not un-aware and has a shield

3. Certainly the strongest part of that is bigger modifier for the shield; no question that core D&D vastly undervalues it. If anyone took a "next step" I'd agree in prioritizing that.

5. Another thing occurred to me, I know you did not want to do the doubling range penalty, but motion, and dodging become more effective at range too. Perhaps you have 3 or 4 modifiers (immobile etc) all at -1. You add all penalties together and double them all at range. Motion at 100 yards is more effective than at 10 yards.

1. You're right realistically, but as you guessed, that's just a little too complicated for my taste at the game table. I'd prefer to batch those numbers up into a single modifier (in fact, I have a personal general rule to try and avoid situational +/-1 modifiers at all times).

6. Yes, in my home brew all ranges are for "skirmish" combat (typical DnD encounter, scrambling dodging etc), with a lot hand waved into the abstraction of HP and AC.
I take a separate approach for "Field" combat and "Sniper" attacks. This gives some good numbers for me to tweak my approach.
I could see mixing in increased bonus for shields, might be a nice way for me to quantify bucklers vs. medium and large shields without resorting to AC inflation.
P.S. it is good to see from your above reply, we can use similar rules for laser weapons.

1. I agree! I think the biggest thing is that the system really needs to separate out long-range-mass "field" and close-up-individual "sniper" shooting, as you call them.

7. I was revisiting this for my house rules and I think I may say the first 6 squares have no range mod. All following squares are -1. The results in -6 for 10-20 yard range and -12 for the 20-40 yard range, total -18 vs -16 for the first 2 range bands, which seems like a reasonable fit and avoid the big chuck modifiers. Of course determining exact range can be a pain. 40-80 yards should be -1 per 3 squares, but I doubt anyone will shoot beyond the -18 range.

1. Yeah, that's totally reasonable. You're right that the exact distance is an issue -- I run almost all my fights narratively and don't expect to have minis in play, so I wouldn't ever expect to have that level of precision in the measurements. But the math is solid there.

8. Delta, thanks for your exhaustive address of this topic. I have two questions:

Given your conclusions and the sometimes problematic conversion, though you suggest a 2 to 1 ratio, how might you provide a redress to range in Chainmail?

As a majority of encounters in our campaign are skirmishes occurring within the wilderness/overland battery, what is your estimation of the arbalest, the heavy crossbow, within an open field as it has a powerful linear, rather than arced trajectory? Thanks.

1. Good questions. The principal thing about Chainmail is that the man-to-man rules should have stipulated a 1" = 5 feet scale. Then I'd use the given modifiers above. (The mass shooting rules don't look immediately incorrect to me, assuming shooting at mass targets.)

I'm not really sure that I'd handle the heavy crossbow any differently. I guess I'd want some data on that in the field before guessing?

9. Delta, my math-fu does not measure with your skill, but my sense of how range modifiers would translate for Chainmail would be 0/-4/-8 (s/m/l). What do you make of that? Also, as mentioned previously, the heavy crossbow/arbalest: as it does not suffer the limitation of an arced range in the manner as other bows would not its range categories outdoors be longer? Lastly, hand thrown weapon weapons, am I correct in reading in other posts on the subject that you decided their range would always be considered long?

1. Thanks for the questions! I agree with your conversions on Chainmail man-to-man adjustments (from a pure realism standpoint). I don't see much to argue for a change from the heavy crossbow, really (if anything maybe the indoor roof-limited ranged could be modified upwards). Currently in my house rules I have missile weapons following the same rules as other weapons (which winds up effectively being only at short or medium range).

10. Inspired by your work, I took a look at archery using similar monte carlo methods, but with archery scores from longbow tournaments as input.

Ended up with a similar scale of penalties as you, but going with a system of increasing range increments, rather than increasing attack bonus.

https://spellsandsteel.blogspot.com/2018/08/archery-variable-range-increments.html

1. Nice work, thank you for sharing!

11. Very interesting!