Friday, April 15, 2016

A Frightful Simulator Symmetry

Let's say you take our Arena simulator (see previous posts here), and run the most generic model possible. Generate 10,000 men of 0-level, ability scores 3d6-across-the-board, dress them in plate, shield, and sword, and then pair them off to fight. Award XP according to the simple Vol-1 system (i.e., 100 XP for a 1 HD enemy; with a ×4 multiplier for presumed treasure awards), and if they level up, apply a 5% chance that any of their equipment gets a magic boost by 1 pip. Refill the list with new recruits and repeat, say, 500 times. Here's the level distribution that results from that simulation:


But consider the leaders specified for groups of Men in OD&D Vol-2; recall that it asserts for Bandits and likewise all other types of Men:
BANDITS: Although Bandits are normal men, they will have leaders who are supernormal fighters, magical types or clerical types. For every 30 bandits there will be one 4th level Fighting-Man; for every 50 bandits there will be in addition one 5th or 6th level fighter (die 1-3 = 5th level, die 4-6 = 6th level); for every 100 bandits there will be in addition one 8th or 9th level fighter (die 1-3 = 8th, die 4-6 = 9th).

So, 4th level fighters appear in a ratio of 1/30 = 0.03; 5th-6th level fighters in the ratio 1/50 = 0.02; and 8th-9th level fighters in the ratio 1/100 = 0.01. Let's compare that to the numbers that fall out of our Arena simulator above:


At the first step and the last step, the results match surprisingly, almost supernaturally, well. At the middle step the simulator asserts there should be about twice as many 5th-6th level fighters combined, as compared to the book rule. But nonetheless the book ratios constitute a basically reasonable distribution, and our simulator generates roundly the same kind of population. (Remember that this is not the same as the population of adventurers exploring underworld dungeons, which is exponentially more dangerous.)

This, then, motivates the development of another software tool that you'll see here on Monday.


18 comments:

  1. Funny, that.

    I am wondering what the cause is of the steep drop after level 8. Could it be that 500 runs is barely sufficient to gather enough xp to reach lvl 9 at all? In other words, has the distribution converged?

    What is the chance that a lvl 0 fighter wins from a lvl x fighter, with x>1 (well x>0, but I have this feeling that lvl 0 vs lvl 0 ends in a 50/50 result ;). Does that give a similar table?

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    1. You're right that the dropoff at level 9 is largely an artifact of the chosen number of fights run. If we increase the number of fights, then characters can continue earning XP to any level we wish (barring some extra rule to cap it off). The 500 number was selected as representing approximately 1 fight every 2 weeks over a 20 year career.

      A table of chances of level 0 beating another character would look very different, I assume.

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  2. Very interesting results! And I always thought the number of leveled leader-types for men were overstated.

    This series has been fascinating as hell, and it's really causing me to re-think how to set up a campaign. Thanks.

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  3. The 5-6 levels could be better simulated by a roll of d8, with 1-5 giving 5th level, 6-8 6th level, if someone wanted to do that.

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    1. I agree; I had something very similar to that annotated in my rulebook.

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  4. So if we wanted to correct this to match the probability curve from your combat simulator, we could do something like the following:

    Say that there are two mid-level leaders for every 50 men. One will be 5th level, and roll a d6 for the other one. On a roll of 1 or 2, the second one is also 5th level; on a 3 or higher, he is 6th level instead.

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  5. This is interesting, I also thought there were too many high levels in the men types.
    But if I understand correctly there were 5,000 casualties in each round of fighting, so 2.5 million men died. You just wiped out something like the adult male part of the medieval population of the British Isles to make 9 Lords! From that perspective, maybe they should be rare indeed.
    I know this is meant to be something like a steady state, but even so no one in real life would ever get through enough fights to make it that far before old age also took its toll, that's one fight a week for ten years assuming no time off for healing.
    Not meant as critical, I really appreciate the effort and am interested, just thinking through the implications.

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    1. I do agree with that; it takes a crazy number of fights, even with the generous Vol-1 XP system (and a ×4 multiplier for presumed treasure).

      When I initially made the simulator I looked at historical gladiators, who as I understand it might fight seasonally (4 times a year, maybe for 20 years for the oldest-ever gladiator; ~80 fights max). But it turns out that number is nowhere near adequate to produce high-level D&D characters.

      We might alternately say that with real-world numbers of fights (a few dozen max in a lifetime?), the realistic limit for "level" would be around 5th.

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  6. Of course people might object that they could also be fighting more exotic monsters, but how many are there? I don't like the idea of a world crawling with enemies and bodies piling up everywhere. To be frightening, such encounters should be unusual. D&D implies a low-population environment to me.

    I think your treasure multiplier is good, possibly even too generous since most ordinary people won't have 400 XP worth of treasure with them to loot.

    Real world fights probably aren't a good guide. I suspect few people survived more than a handful of battles and skirmishes before they ran out of luck. Even Lords in the Wars of the Roses with good armour and bodyguards tended to end up on the wrong end of a pole-arm within a few years if they took an active part. And wounds and disease were worse than the game, but who'd want to simulate that?

    So, the answer in real life must be that we get nearly all our XP from treasure and it gets awarded regardless of risk in obtaining it ;-)

    More seriously, it shows the crazy number of games that must have been played in the early days (assuming rules weren't being bent), and possibly why I never got to high level playing fair. And Greyhawk turned low HD monster experience down!

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    1. I agree. I tend to have a mental metric of the Red Baron (80 kills) or Flamma (34 gladiatorial fights); but of course these would be outliers, and the majority of fighters would not really survive past their second fight.

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    2. Just FYI, WWII pilot Erich Hartman is the worlds leading ace with 352 kills. :)

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    3. It's true, he's just such a crazy outlier he tends to not be the first person I think of. :-)

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  7. Sure, very many sigmas away from the mean in those cases! And they were both in somewhat ritualised, 'clean' combat where you could come away unscathed, unlike the medieval face-to-face situation. The more I learn about that, the more glad I am I wasn't there (though I'd obviously have been a peasant legging it at the first chance)!

    High-level D&D characters are beyond mass murderers really, unless they are pure thieves, but it's the thieves who can't be lawful...

    Anyway, thanks for these insights, I'm enjoying your blog and just starting to catch up on some of the older posts.

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  8. I recalled something yesterday; something that could significantly improve the likelihood of fighters surviving those low levels. Specifically, I recalled what Michael Mornard (Gronan of Simmerya on the ODD74 boards) had said about EGG's games circa 1972 (Source: http://odd74.proboards.com/thread/7313)

    "Gary used to give us the option of rolling an additional die, or rerolling all your hit dice. However, if you rerolled them all, you took the new number, period.

    You could also reroll at the beginning of an adventure, rerolling them all."

    So those lucky fighters who win despite their low HP rolls would have a shot at replacing their wimpy rolls with better ones. This would increase the odds of surviving to attain a "safe" number of hit points where the probability of losing to a given 0-level challenger becomes negligible.

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    1. That would certainly change things a lot. I guess you'd need a decision process about when one decided to take that option or not.

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    2. With only a brief look at the odds, a convenient decision rule that occurs to me would be to take the between-adventures reroll if you have less than 4 hit points per (current, prior to leveling up) character level. That gives a 75% or greater chance of improving your total.

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