Monday, April 20, 2015

Dungeons & Demographics, Pt. 1

If we look at the Original D&D source material, and take up the many clues there that normal men are 0-level (even the standard men-at-arms, soldiers, brigands, pirates, etc.), then we also see several points of data as to the distribution of higher-level leaders and fighters.

In Vol-1, we see that when Clerics achieve the name level of Patriarch (8th) and build a stronghold, then they attract faithful soldiers numbering from 50-300 (p. 7):


In Vol-2, at the start of the lengthy entry for different types of Men, we read that groups of 100-300 such men will be led by an 8th or 9th level fighter, as well as numerous lower-level leaders (p. 5):


In Vol-3, in the section on the Wilderness and the various types of Castles to be encountered there, we are told that all of the castle rulers are either Name level or one less (so in the range of levels 8-11), and have from 30-180 normal men-at-arms, in addition to a small number of other heroes or monsters to assist them (p.16):


So the general commonality to all of these descriptions is this: any body of normal Men numbering at least a few hundred seem to have a Name level character leading them. The consistency of this proportion across passages in all the different booklets gives some confidence as to the sensibility meant to be taken in the stock OD&D campaign.

Now let's look more closely at the distribution given for Bandits (shared by all other types of Men), with its additional detail regarding the mid-level heroes and leaders (4th-6th level and so forth). If we wish to fit a regression curve to this data, then clearly the model to use is an exponential function of the form f(x) = a∙e^(kx), where our k will be negative and referred to as the "decay rate" (link). This will produce a proportional reduction at each level from whatever constant we start with for 0-level normal men. In theory, we could achieve a closer fit (less error) to our data via a power curve of form f(x) = a∙x^k (with negative k), but that has the unfortunate implication of an infinite number of normal men (that is, asymptotic at level 0), which is clearly not what we perceive for our campaign world. Note that we will only carry out this analysis to around name level, because after that point the X.P. charts switch from a geometric progression to linear, and so we could not expect to extrapolate the same function from one piece to the other.

Here we take a starting population of 10,000 normal men-at-arms and compute the expected number of 4th, 5th, 6th, 8th, and 9th level fighters as specified by the Vol-2 paragraph above, and then perform the regression to the best fit exponential function. (We could pick any other fixed number to start with, and it wouldn't change the output parameters in any way.) The result is the following:


That's a pretty close fit with the decay rate k = -0.60 (providing a coefficient of determination of R^2 = 0.89, that is, the regression accounts for 89% of the variation from the mean between the data points), but in trying to balance the large number of high-level data points, we've missed the starting point at 0-level by almost exactly half. Let's temporarily ignore the high-level leaders at level 8th and 9th and focus only on the initial dropoff at the data points up to the 6th level:


That's a much nicer fit; it comes very close to the first data point at level 0, and has an improved R^2 = 0.98, for nearly 100% agreement. This argues that taking a decay rate of around k = -0.80 will be a much better match, at least for the first few levels of our distribution.

Now, the funny thing to me is that this exponential fit comes close to simply halving the number of men at each level, from 0-level on up (a nearly trivial model that I've pooh-poohed in the past, looking at AD&D specifications). If you do divide the number of men by two at each level, that's precisely equivalent to an exponential function with a decay rate of k = -0.69, say -0.70 for convenience (more precisely: -ln(2)), or just halfway between the two analyses carried out above. Here's a table and comparative charts with which you can compare the two models:

 

Either of these models do about as good at fitting to the Vol-2 Bandits specification as we'll get. They are a bit high in the numbers specified at levels 4-6 (by about double), and a bit low in the numbers specified for levels 8-9 (by about half).

In conclusion: The sensibility expressed by Gygax for number and levels for leaders of groups of men in the OD&D campaign (in several different places) is surprisingly close to a simple divide-by-2 at each level method. We can fine-tune this with some other nearby exponential function, but the result will not be an order of magnitude difference. (Download an ODS spreadsheet for the work if you like: here.)

In Part 2, we'll compare these results to the distribution of characters (starting at level 0) who survive encounters with the Vol-3 "Monster Determination and Level of Monster Matrix".

13 comments:

  1. Very interesting. This article and "The Case for Zero Level" have caused an itch that I can't quite scratch with respect to level demographics.

    I'm looking forward to the Monster Matrix, but I would also be very interested to see what would come out of the Man vs. Man Arena using your preferred 0-level man (1-6 HP and 1,000 xp to level 1).

    I would be very surprised if 40-50% of the population advanced to level 1 (at which point they would most assuredly be classified as Veterans, having participated in enough battles to gain 1,000xp).

    I would suspect that we would actually see demographics closer to those published in Swords & Spells (1 in 10 being Elite Guards/Veterans) or even the DMG (1 in 100 being "suitable for level advancement").

    Thanks for all the thought-provoking work.

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    1. The short story is that you're correct about that.

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  2. I suppose the bulk of the population that advance in level do so via treasure acquisition as opposed to man-to-man combat.

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    1. You are correct regarding the importance of treasure acquisition, but the acquisition of treasure is accounted for in the Arena - Man vs. Man simulation.

      http://deltasdnd.blogspot.com/2014/06/arena-man-vs-man.html

      Paragraph 2 under Method: "Award experience pro-rated for standard D&D treasure awards (for example: at 1st level, multiply base experience by 20, because about 95% of XP at the level generally comes from treasure...."

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    2. I suppose my statement above, "...having participated in enough battles to gain 1,000xp...." was poorly written. I did not mean to suggest that this 1,000xp came solely from the combat.

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  3. So, at least as an approximation, we could derive the numbers for each level by multiplying the number at the previous level by the ratio of experience points of the previous and new levels, perhaps with an N% reduction to account for casualty rates. Looking at your custom differential list there, N ~= 10% seems appropriate.

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    1. That's an interesting hypothesis. Probably that's a little more complex than what you'll see here next week; I'm just going to fit what happens in the arena directly and compare to the results this week.

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    2. The advantage being, if the hypothesis remains reasonably close to your results, that people could generalize them to cover cases in similar games other than Delta OD&D (ones that include clerics, say, or Starships & Spacemen).

      Also, I'd add that obviously the 1st-2nd level jump should be halved rather than a normal ratio of 0 to 2000 or whatever. ;) The ratio of 0th-1st would be based on the specific game (e.g. AD&D specifies that only 1% of the human population, or 2% of most nonhumans, can gain levels at all, as a feature of the setting; that says something about what levels mean in AD&D, of course, but here you're only looking at the statistical map that derives from the experience charts).

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  4. Following up on Jason's observation, the idea that half the population advances to 1st level seems surprising, not to mention at odds with what is said elsewhere about how much of the population is suitable for level advancement (whatever that means). What would happen, I wonder, if we took it as an article of faith that no more than 10% of the population will be 1st level or higher, and the constructed a decay model from that starting point?

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    1. Well, keep in mind that it's half of the normal men-at-arms (fighting bandits, pirates, soldiers, etc.), not the population in general. So I'd separate out the question: what percent of the population is generally under arms in the first place? 5% or 10% or so? Granted that the "suitable for level advancement" line comes from AD&D, and is really at odds with the other details, I find it helpful to free myself from that restriction.

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    2. Ah! Thank you! Good point. In my games, I typically assume that about 20% of the population is capable of bearing arms (by local standards, so male and of "fighting age") and that, among humans, about 10% of them are actually in a profession of arms of some sort. That figure is based on my own research, but I'll admit that it is just this side of guesswork, and it does not apply evenly to all settlements. A village might only have a bailiff and a few yeomen who would count as "under arms" (accounting for maybe 5% of the village population), but the manor house associated with it might have enough men-at-arms that they make up 20% or more of the manor house's population, pushing the proportion of people "under arms" for the combined population up to 10% or higher.

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    3. Right, those are pretty much the same numbers I'd pick -- although I think I just got those from some prior D&D supplement, so if you've done research for that, then you're ahead of me on the subject. :-)

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  5. Well, we ended up about the same place, so that's good enough for me!

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