2018-05-31
Dragon Metrics By Age
Following up on Monday's post, I thought it might be nice to look at the Equivalent Hit Dice (EHDs) for Dragons, broken down by separate age categories. (In the OED Monster Database, for brevity, I have only color breakdowns, and assume ages are set by a random 1d6 method). This also includes all the different enumerations for Hydras. Power-curve ASCII graphs can be seen here. Enjoy,
2018-05-28
Harmonic Hit Dice
I recently made a change to how the MonsterMetrics program estimates monster Equivalent Hit Dice (EHDs), so as to make it more transparent and flexible; this slightly changed some of the monster EHD values (mostly for higher-level monsters). I think there’s an interesting mathematical lesson here – I learned a few things – so I think it worthwhile to lay that out here.
Consider the basis for how MonsterMetrics does its job (recently clarified in the code). We want to measure monster potency over a wide range of possible PC levels. So for each level 1-12, we compute the number of fighters that comes closest to an even fight against that monster (that is, closest to a 50% chance of either side winning; this itself entails a binary search across each possible number of fighters 1-64, or the inverse, with possibly hundreds or thousands of simulated combats at each possible number to assess the chance of victory on each side). This generates an array I call Equated Fighters (EF); and each element therein is multiplied by the level-index to generate a value called the Equated Fighters Hit Dice (EFHD).
Example: Consider the standard Ogre. We find that one is fairly matched against 2 Veterans/Warriors, 1 Swordsman/Hero/Swashbuckler, ½ Myrmidon/Champion/Superhero/Lord (that is, 2 ogres against one Superhero is fair), and so forth. So we have an Equivalent Fighter array that looks like this: EF [2.0, 2.0, 1.0, 1.0, 1.0, 0.5, 0.5, 0.5, 0.5, 0.3, 0.3, 0.3]. Multiplying each value by its associated level gives us the Equivalent Fighter Hit Dice; spot-checking a few simple values, we see that an Ogre is symmetrically worth 4 HD of Warriors (2 × 2HD each), or 4 HD of Heroes (1 × 4HD each), or 4 HD of Superheroes (½ × 8HD each), etc. Not to say that every point is exactly the same; due to the discrete nature of the matchups, there is a bit of a sawtooth artifact in the values. In total we have this array for the Ogre: EFHD [2.0, 4.0, 3.0, 4.0, 5.0, 3.0, 3.5, 4.0, 4.5, 3.3, 3.7, 4.0], and this is shown pictorially below.
Now, given that we want to present a single number to represent “monster power”, the question is, what metric do we use to crunch those numbers down to a unitary value? Granted that the EFHDs in the example above are approximately the same, this is good news for the overall idea of making a single, summary value in the first place; that is, in the graph above, they all fall roughly along a horizontal line. This is generally true for most monsters, but not all. Monsters with save-or-die abilities that short-circuit PC hit points have EFHDs that trend upward (positive slope); while monsters with area-attacks that can kill lots of low-level fighters have EFHDs that trend downwards (negative slope).
Previously I was computing an overall monster Equivalent Hit Dice (EHD) by taking the arithmetic mean of the EFHD values. That's an obvious choice, but one that has notable limitations; the arithmetic mean is sensitive to outliers, so if a monster is unusually dangerous versus PCs of one specific level, then the EHD can be wildly biased in that direction. As an extreme case, some monsters can actually face off against an infinite number of 1st-level attackers – e.g., Golems and Elementals are hit only by +2 or better magic weapons, which mere Veterans will not have – and in such a case, the mean of the EFHDs itself becomes infinite. As a result, I was forced to mark those monsters as having undefined EHDs (asterisks in the OED Monster Database).
So: Here’s where the change comes in. What other options do we have for computing an average (measure of center) of the EFHDs? The median could be considered – it’s less sensitive to outliers, but would suffer the same fate if, in theory, half of the PC levels had infinite EF’s (it has a 50% breakdown point). Instead, the silver bullet seems to be the harmonic mean, that is, the reciprocal of the average of the reciprocal EFHDs. The neat thing here is that the harmonic mean actually has no problem handling infinite values (with the understanding that 1/∞ = 0). It’s dominated by a multiple of the minimum value (see here), so the only way the harmonic mean can become infinite is if every single EFHD value is infinite (that is, the monster would be unbeatable by PCs of any level; in which case we can interpret the non-real EHD as communicating the fact that awarding XP is a non-consideration).
If the EFHD values are all equal, which is approximately true for most monsters, then the harmonic mean is exactly the same as the arithmetic mean – so we won’t get wholesale changes in our EHD values. For example: The Ogre above has an EFHD harmonic mean of about 3.5; so I set the EHD value to 4, the same as before (and the same as its actual hit dice, which is nice). On the other hand, the harmonic mean is always lesser than or equal to the arithmetic mean, so in some cases, for high-level monsters with skewed power curves, the suggested EHD was reduced by a few pips. For example: The Red Dragon presented the most extreme case of this, in which its EHD dropped from 32 to 27 (a 5-point difference). I think that this bias in the downward direction may be reasonable for a few reasons: (a) it emphasizes the most likely use-case of opposition PCs (that is: the most effective PC level), and (b) it suggests that accounting for magic spells and the like may reduce the overall danger level of such monsters.
Most importantly, this change in methodology allowed us to fill in the previously-missing EHD values for monsters like Golems and Elementals (the tricky cases mentioned above). The most dangerous monster by this metric is now seen to be the Iron Golem from D&D Supplement I, with a devastating EHD value of 104! (All other monsters top out at around 40 EHD: c.f., the Stone Golem at 41, Large Earth Elemental at 39, Vampire at 38, etc.). This allowed me to cut the number of monsters with "undefined" EHDs by more than half (adding 14 monster types to the list); those that are left are those with copious spell-like abilities (lich, titan, beholder, etc.), are invulnerable to any fighter weapons (oozes), or have attacks that don’t actually kill people (rust monster).
A bunch of stuff got updated to follow along with this method of using the harmonic mean on EFHD values. See the code on GitHub. In particular, the MonsterMetrics.java module received new command-line switches: -e to see the Equated Fighters array, -d to see the Equated Fighters Hit Dice Array, and -g to see the latter graphed as ASCII text. I uploaded the output for all monster evaluations and graphs (at 5 units per step on the y-axis). The OED Monster Database has been updated, and so has the Monster EHD Listing (including sorting by both name and EHD value). And more to come next week.
Thanks to CUNY/Kingsborough Professor of Mathematics Nataniel Greene for suggestions that clarified my problem and led me in this direction.
Consider the basis for how MonsterMetrics does its job (recently clarified in the code). We want to measure monster potency over a wide range of possible PC levels. So for each level 1-12, we compute the number of fighters that comes closest to an even fight against that monster (that is, closest to a 50% chance of either side winning; this itself entails a binary search across each possible number of fighters 1-64, or the inverse, with possibly hundreds or thousands of simulated combats at each possible number to assess the chance of victory on each side). This generates an array I call Equated Fighters (EF); and each element therein is multiplied by the level-index to generate a value called the Equated Fighters Hit Dice (EFHD).
Example: Consider the standard Ogre. We find that one is fairly matched against 2 Veterans/Warriors, 1 Swordsman/Hero/Swashbuckler, ½ Myrmidon/Champion/Superhero/Lord (that is, 2 ogres against one Superhero is fair), and so forth. So we have an Equivalent Fighter array that looks like this: EF [2.0, 2.0, 1.0, 1.0, 1.0, 0.5, 0.5, 0.5, 0.5, 0.3, 0.3, 0.3]. Multiplying each value by its associated level gives us the Equivalent Fighter Hit Dice; spot-checking a few simple values, we see that an Ogre is symmetrically worth 4 HD of Warriors (2 × 2HD each), or 4 HD of Heroes (1 × 4HD each), or 4 HD of Superheroes (½ × 8HD each), etc. Not to say that every point is exactly the same; due to the discrete nature of the matchups, there is a bit of a sawtooth artifact in the values. In total we have this array for the Ogre: EFHD [2.0, 4.0, 3.0, 4.0, 5.0, 3.0, 3.5, 4.0, 4.5, 3.3, 3.7, 4.0], and this is shown pictorially below.
Now, given that we want to present a single number to represent “monster power”, the question is, what metric do we use to crunch those numbers down to a unitary value? Granted that the EFHDs in the example above are approximately the same, this is good news for the overall idea of making a single, summary value in the first place; that is, in the graph above, they all fall roughly along a horizontal line. This is generally true for most monsters, but not all. Monsters with save-or-die abilities that short-circuit PC hit points have EFHDs that trend upward (positive slope); while monsters with area-attacks that can kill lots of low-level fighters have EFHDs that trend downwards (negative slope).
Previously I was computing an overall monster Equivalent Hit Dice (EHD) by taking the arithmetic mean of the EFHD values. That's an obvious choice, but one that has notable limitations; the arithmetic mean is sensitive to outliers, so if a monster is unusually dangerous versus PCs of one specific level, then the EHD can be wildly biased in that direction. As an extreme case, some monsters can actually face off against an infinite number of 1st-level attackers – e.g., Golems and Elementals are hit only by +2 or better magic weapons, which mere Veterans will not have – and in such a case, the mean of the EFHDs itself becomes infinite. As a result, I was forced to mark those monsters as having undefined EHDs (asterisks in the OED Monster Database).
So: Here’s where the change comes in. What other options do we have for computing an average (measure of center) of the EFHDs? The median could be considered – it’s less sensitive to outliers, but would suffer the same fate if, in theory, half of the PC levels had infinite EF’s (it has a 50% breakdown point). Instead, the silver bullet seems to be the harmonic mean, that is, the reciprocal of the average of the reciprocal EFHDs. The neat thing here is that the harmonic mean actually has no problem handling infinite values (with the understanding that 1/∞ = 0). It’s dominated by a multiple of the minimum value (see here), so the only way the harmonic mean can become infinite is if every single EFHD value is infinite (that is, the monster would be unbeatable by PCs of any level; in which case we can interpret the non-real EHD as communicating the fact that awarding XP is a non-consideration).
If the EFHD values are all equal, which is approximately true for most monsters, then the harmonic mean is exactly the same as the arithmetic mean – so we won’t get wholesale changes in our EHD values. For example: The Ogre above has an EFHD harmonic mean of about 3.5; so I set the EHD value to 4, the same as before (and the same as its actual hit dice, which is nice). On the other hand, the harmonic mean is always lesser than or equal to the arithmetic mean, so in some cases, for high-level monsters with skewed power curves, the suggested EHD was reduced by a few pips. For example: The Red Dragon presented the most extreme case of this, in which its EHD dropped from 32 to 27 (a 5-point difference). I think that this bias in the downward direction may be reasonable for a few reasons: (a) it emphasizes the most likely use-case of opposition PCs (that is: the most effective PC level), and (b) it suggests that accounting for magic spells and the like may reduce the overall danger level of such monsters.
Most importantly, this change in methodology allowed us to fill in the previously-missing EHD values for monsters like Golems and Elementals (the tricky cases mentioned above). The most dangerous monster by this metric is now seen to be the Iron Golem from D&D Supplement I, with a devastating EHD value of 104! (All other monsters top out at around 40 EHD: c.f., the Stone Golem at 41, Large Earth Elemental at 39, Vampire at 38, etc.). This allowed me to cut the number of monsters with "undefined" EHDs by more than half (adding 14 monster types to the list); those that are left are those with copious spell-like abilities (lich, titan, beholder, etc.), are invulnerable to any fighter weapons (oozes), or have attacks that don’t actually kill people (rust monster).
A bunch of stuff got updated to follow along with this method of using the harmonic mean on EFHD values. See the code on GitHub. In particular, the MonsterMetrics.java module received new command-line switches: -e to see the Equated Fighters array, -d to see the Equated Fighters Hit Dice Array, and -g to see the latter graphed as ASCII text. I uploaded the output for all monster evaluations and graphs (at 5 units per step on the y-axis). The OED Monster Database has been updated, and so has the Monster EHD Listing (including sorting by both name and EHD value). And more to come next week.
Thanks to CUNY/Kingsborough Professor of Mathematics Nataniel Greene for suggestions that clarified my problem and led me in this direction.
2018-05-21
Underworld Overhaul, Pt. 5: Dungeon vs. Monster Treasure
Throughout our investigation of the OD&D underworld stocking system, we've taken for granted that the treasure table in Vol-3, p. 7, with the first column titled "Level Beneath Surface" -- what I call the "dungeon treasure" table -- is indeed the standard in use for treasures in the dungeon. We have many points of evidence that the Vol-2, p. 22, "Treasure Types" table is for use only in wilderness adventures. However, here we entertain the thought: what if we used monster-keyed treasure types in the dungeon anyway?
There is an undeniable attraction to that idea. In particular, the treasure types system seems to give a certain "flavor" or preference to the treasures commonly found with different types of monsters. For example: Men (Type A) have large amounts of treasure and prisoners. Dwarves (Type G) have large amounts of gold, and reject any baser metals. Dragons (Type H) have overwhelmingly huge piles of treasure. Rocs (Type I) may have gems but can't collect coins. (Note the extreme specificity of those latter types; the monsters I just named are the only ones keyed to those treasure types.) On the other hand, a great many monsters are entirely lacking any keyed treasure type; e.g., skeletons/zombies, most fliers, extradimensional creatures, oozes, insects, and animals.
And here the same for monster treasure types:
Note, however, that all rolled values are highly variable; in particular, the vast majority of expected value comes from the rare and high-value gems and jewelry (esp. the latter) in each case. For example: At the 1st dungeon level, 95% of treasures lack any jewelry, and such treasures have an expected value of only 90 gp. On the other hand, the 5% of 1st-level treasures that do include jewelry have a class expected value of 12,000 gp. In this perspective, we might say that the core D&D game is one of PC adventurers searching strictly for jewelry -- after about 10 treasures (at 1st level) they have a 50% chance of having secured jewelry and so leveling up; whereas any other treasure is effectively negligible. (Later levels have somewhat less variance than this, as the chance of gems/jewelry rises; but note that chance maxes out at 50%.)
We might compare the average treasure-type values seen here to the similar chart presented in Moldvay's Basic rules, p. B45 (repeated in Cook Expert, p. X43). He has overall lower expectations (about half what we see here), and yet the main treasure type table looks to have copied all the entries from OD&D, even adding new columns for electrum and platinum. This would seem to imply equal or higher value, so how can that be? Again, this is explained by the majority of value residing in the gems/jewelry. Moldvay has modified the gems table -- emphasizing chances for the lower-valued types; and even more radically restricted the jewelry -- allowing only the lowest-value category from OD&D. And hence overall lower expected values for the same treasure types.
Note the following: When dungeon treasure is used, it is generated for every encounter (waived the 50% for treasure, but also not including any DM-placed high-value "important treasures"). Likewise, if monster treasure types are switched on, then we consult the type table for every encounter (waiving the "% In Lair" chance designed for wilderness adventuring). On the other hand, we honor the Monster Manual rule that monster treasure types be scaled in proportion to the average monster number appearing in the wilderness; whereas there is no such suggestion for dungeon treasures, whose distribution is thereby fixed (regardless of monster numbers). Note that since we scale monster numbers by party size, the average shares of monster treasure stay fixed for larger parties, but shares of dungeon treasure decrease for larger parties.
The following repeats (from the last few posts) the demographics from parties of size 4 adventuring in our dungeon, with monster numbers fixed in proportion to party size (no variation), using the default dungeon-treasure setting (Arena switches -n=10000 -v -z=4 -rs):
And here are the demographic results if we instead use the monster treasure types (adding the -t switch):
Conclusions: First of all, the monster treasure types generate significantly less treasure than the dungeon treasure system. This is reflected in the fact that dungeon treasure supplies about two-thirds of XP when in use; but monster treasure supplies slightly less than half. That might come as a bit of a surprise -- I think there are some misconceptions, based on impressions of the very large Dragon treasure (for example), forgetting that the vast majority of monsters in the system have little or no treasure whatsoever.
Secondly, and as a result of that, the use of the monster-treasure system would make it harder to advance in levels past the 1st. Comparing the two charts above, the monster-treasure population has roughly half the numbers at levels 3-5 (as compared to the top dungeon-treasure chart); and only about one-fifth the numbers at levels 6-7 (and in fact, the solitary 7th level character may be something of an outlier, with their excellent Dexterity and Constitution scores).
In summary: While there is some charm in the flavor of bending the rules to use monster treasure types for dungeon adventures, this results in overall less treasure, and makes things even harder on the PCs in terms of advancement opportunities (and is also more calculator-intensive, in that it requires scaling down from the table in proportion to number of monsters encountered). We would recommend sticking with the OD&D Vol-3 dungeon-treasure tables for underworld adventures.
There is an undeniable attraction to that idea. In particular, the treasure types system seems to give a certain "flavor" or preference to the treasures commonly found with different types of monsters. For example: Men (Type A) have large amounts of treasure and prisoners. Dwarves (Type G) have large amounts of gold, and reject any baser metals. Dragons (Type H) have overwhelmingly huge piles of treasure. Rocs (Type I) may have gems but can't collect coins. (Note the extreme specificity of those latter types; the monsters I just named are the only ones keyed to those treasure types.) On the other hand, a great many monsters are entirely lacking any keyed treasure type; e.g., skeletons/zombies, most fliers, extradimensional creatures, oozes, insects, and animals.
Expected Values
As a lead-in, it may be helpful to inspect the expected values of each of the different treasures possible. In each case this is done via the Arena code package, with the unit tests built into the appropriate code modules, which perform simple Monte Carlo simulation methods. Expected value for dungeon treasures, by level (rounded to two figures of precision):And here the same for monster treasure types:
Note, however, that all rolled values are highly variable; in particular, the vast majority of expected value comes from the rare and high-value gems and jewelry (esp. the latter) in each case. For example: At the 1st dungeon level, 95% of treasures lack any jewelry, and such treasures have an expected value of only 90 gp. On the other hand, the 5% of 1st-level treasures that do include jewelry have a class expected value of 12,000 gp. In this perspective, we might say that the core D&D game is one of PC adventurers searching strictly for jewelry -- after about 10 treasures (at 1st level) they have a 50% chance of having secured jewelry and so leveling up; whereas any other treasure is effectively negligible. (Later levels have somewhat less variance than this, as the chance of gems/jewelry rises; but note that chance maxes out at 50%.)
We might compare the average treasure-type values seen here to the similar chart presented in Moldvay's Basic rules, p. B45 (repeated in Cook Expert, p. X43). He has overall lower expectations (about half what we see here), and yet the main treasure type table looks to have copied all the entries from OD&D, even adding new columns for electrum and platinum. This would seem to imply equal or higher value, so how can that be? Again, this is explained by the majority of value residing in the gems/jewelry. Moldvay has modified the gems table -- emphasizing chances for the lower-valued types; and even more radically restricted the jewelry -- allowing only the lowest-value category from OD&D. And hence overall lower expected values for the same treasure types.
Adventuring Demographics
Using the current Arena program, it's simple to explore using the two different treasure tables; by default the dungeon treasure system is used, while the -t switch forces use of the monster treasure types instead.Note the following: When dungeon treasure is used, it is generated for every encounter (waived the 50% for treasure, but also not including any DM-placed high-value "important treasures"). Likewise, if monster treasure types are switched on, then we consult the type table for every encounter (waiving the "% In Lair" chance designed for wilderness adventuring). On the other hand, we honor the Monster Manual rule that monster treasure types be scaled in proportion to the average monster number appearing in the wilderness; whereas there is no such suggestion for dungeon treasures, whose distribution is thereby fixed (regardless of monster numbers). Note that since we scale monster numbers by party size, the average shares of monster treasure stay fixed for larger parties, but shares of dungeon treasure decrease for larger parties.
The following repeats (from the last few posts) the demographics from parties of size 4 adventuring in our dungeon, with monster numbers fixed in proportion to party size (no variation), using the default dungeon-treasure setting (Arena switches -n=10000 -v -z=4 -rs):
And here are the demographic results if we instead use the monster treasure types (adding the -t switch):
Conclusions: First of all, the monster treasure types generate significantly less treasure than the dungeon treasure system. This is reflected in the fact that dungeon treasure supplies about two-thirds of XP when in use; but monster treasure supplies slightly less than half. That might come as a bit of a surprise -- I think there are some misconceptions, based on impressions of the very large Dragon treasure (for example), forgetting that the vast majority of monsters in the system have little or no treasure whatsoever.
Secondly, and as a result of that, the use of the monster-treasure system would make it harder to advance in levels past the 1st. Comparing the two charts above, the monster-treasure population has roughly half the numbers at levels 3-5 (as compared to the top dungeon-treasure chart); and only about one-fifth the numbers at levels 6-7 (and in fact, the solitary 7th level character may be something of an outlier, with their excellent Dexterity and Constitution scores).
In summary: While there is some charm in the flavor of bending the rules to use monster treasure types for dungeon adventures, this results in overall less treasure, and makes things even harder on the PCs in terms of advancement opportunities (and is also more calculator-intensive, in that it requires scaling down from the table in proportion to number of monsters encountered). We would recommend sticking with the OD&D Vol-3 dungeon-treasure tables for underworld adventures.
Important Treasures
A final consideration; granted that we plan to follow the dungeon-treasure placement rules in Vol-3, it is important to recall (again) that the foundation for that system starts with the DM placing "important treasures" by fiat, prior to any random methods being used (p. 6):It is a good idea to thoughtfully place several of the most important treasures, with or without monsterous guardians, and then switch to a random determination for the balance of the level. Naturally, the more important treasures will consist of various magical items and large amounts of wealth in the form of gems and jewelry.So: How big should these important treasures be? (We ask, of course, because this can have a larger impact on PC survival and advancement than anything else we've analyzed in the system to date.) If we take literally the recommendation that they include "large amounts of wealth in the form of gems and jewelry", then we note the following. A dungeon-treasure including gems has an expected value of double the basic treasure; and a treasure including jewelry has an expected value of twenty times the base. So it would be legitimate to glance at the dungeon-treasure expected values above, and place special treasures (including gems/jewelry) at something like a 10-fold or 20-fold multiplier (or more) on each level. Be sure to stoutly guard such treasures with copious monsters and fiendish traps, of course!
2018-05-14
Underworld Overhaul, Pt. 4: Monster Numbers Appearing
Here we get to an issue that's a little bit more tricky, because the advice given in the OD&D books are somewhere between ambiguous and self-contradictory: how many monsters appear in an encounter? Again I'll quote the foremost passage from Vol-3, p. 11:
For the following, we make the following baseline assumptions: (1) As specified in the last post, we use a formula for numbering appearing of NA = scaleFactor × level / EHD, where level = dungeon/character level (presumed equal). That is: we interpret "level of monster" in the quote above to be EHD, not the 1st to 6th level tables on the same page (contrast usage of "level [of] monster" on Vol-3 p. 11 with Vol-1 p. 18, say). (2) Standard party size is 4 characters/fighters (this seems like a common expectation; is about median for the sample party sizes in the book quote above; and seems like a reasonable balance between safety and fast advancement seen in last week's article). In the Arena code, scaleFactor is the same as party size, so 4 in this case. (3) We are simulating dungeon lair encounters, each with a roll on the dungeon treasure table (waiving the 50% chance for treasure, but also no DM-fiat high-value "important treasures"), not wandering encounters.
Granted that, Let's investigate the effect of a few different options for monster numbers appearing in the dungeon. First we repeat the demographic results from the last post; a party size of 4, and equivalent scaleFactor fixed at 4 (so, if dungeon level = EHD, monster numbers are equal to party size, and otherwise proportionally adjusted):
Now we consider if we change the "basic number" in this case, i.e., the scaleFactor, to a variable 1d6. This has an average of 3.5 (slightly less than our prior 4); in 3 cases less, in 2 cases more than our prior fixed value. (This is done by a one-line code change to Arena, so can't be directly executed via the current release application). This results in the following population:
Despite what we might have expected (with a lower average encounter size), we see here that the extra variation in the 1d6 actually makes for a significantly more dangerous campaign. Compared to the prior chart, there are only one-half or fewer fighters at the 5th, 6th, or 7th levels. Even if the PCs may be happy to sometimes fight only 2 or 3 same-level monsters, they will easily be overwhelmed sometime when they are outnumbered by 5 or 6 same-level monsters. (On the flip side, it's a bit weird to occasionally have a treasured lair with only 1 single monster.) Also, the Strength scores for the high-level fighters start to hint that the game may have turned into something of a random meat grinder, regardless of character ability, which we do not want. So let's dial down the variation a bit and look at instead rolling 1d4+1 for the scaleFactor:
Even with the same average as on a 1d6 (i.e., 3.5), this is clearly better for the PCs. Compared to the fixed scaleFactor = 4 table, there are roughly the same number of fighters at 1st to 5th level. The numbers at 6th and 7th level are reduced, but, e.g., there are over three times as many characters at 6th level as in the 1d6 experiment. And the ability score averages at high level look reasonable. And we avoid having single-monster lairs. So: This looks pretty good for dungeon lair numbers. (Side note: In each case, XP from treasure is roughly two-thirds of the total, with monsters accounting for the other one-third.)
Based on this, now consider wandering monster encounters. Seems like this could be around half the size of a "lair" encounter, say: scaleFactor × 1d3. Obviously this is somewhat subjective, because these encounters don't generate treasure, are not critical to advancement, and hopefully avoided entirely by discriminating PCs (and not simulated in our program in any meaningful way). Note that in this case, with our default party size of 4, and an average wandering encounter with 2 monsters, we exactly match the guideline text in Vol-3, p. 11, repeated here:
But here's a complication: We assume that encounters (esp. wandering ones) will be scaled in proportion to both party size and monster EHD on the fly, and in practice this would require a calculator for the number-crunching (and also the complete list of monster EHDs). Here's a shortcut rule-of-thumb to make that more practical on the fly, based on the average EHDs at each level. (Bunch of spreadsheet number-crunching occurred here, not shown.) Look at the revised Monster Level Matrix we're using. There are six "tiers", and for each, a die-roll of 3-4 lands on a level-column which provides a "median monster", where EHD approximately equals the dungeon level (and so, presumably character level). The number appearing can then be adjusted by where the result is to the left or right of that median 3-4 result column:
You may note that a result of "Left 2" can only ever result from a die roll of "1" on the matrix; a "Left 1" result only from a "2"; and so forth. The multiplier shown might also be used for lair encounters (recommended base 1d4+1) and so forth. For clarity, the exact number appears for each pip of the 1d3 wandering encounter roll. I've got this jotted into my copy of OD&D Vol-3, p. 11.
Final thought: In contrast to this system, broadly in synch with what's related in that key page of OD&D Vol-3, Gygax's module creations tend to be a lot more dangerous, with larger numbers of monsters than we see here (even in proportion to party size). For example, the suggested monster numbers in Mike Carr's module B1 are much less than Gary Gygax placed in module B2. Another example: The wandering monster groups in the DMG Sample Dungeon have an average EHD of about 6 total, any one of which is an existential threat to a group of only 5 1st-level PCs, as depicted in the example of play, to say nothing of the variation which allows them to regularly be up to twice that size. (If the lair groups are any larger than this, then it's hard to see how they'd even fit in the rooms indicated.)
Number of Wandering Monsters Appearing: If the level beneath the surface roughly corresponds with the level of the monster then the number of monsters will be based on a single creature, modified by type (that is Orcs and the like will be in groups) and the number of adventurers in the party. A party of from 1-3 would draw the basic number of monsters, 4-6 would bring about twice as many, and so on. The referee is advised to exercise his discretion in regard to exact determinations, for the number of variables is too great to make a hard and fast rule. There can be places where 300 Hobgoblins dwell...In particular, in the first sentence, the parenthetical comment ("Orcs and the like will be in groups") makes no sense as a reasonable balancing factor; if a "single creature" at Level 1 is a reasonable challenge for some (small) party, then any multiple number of Orcs will be deadly (to say nothing of 300!).
For the following, we make the following baseline assumptions: (1) As specified in the last post, we use a formula for numbering appearing of NA = scaleFactor × level / EHD, where level = dungeon/character level (presumed equal). That is: we interpret "level of monster" in the quote above to be EHD, not the 1st to 6th level tables on the same page (contrast usage of "level [of] monster" on Vol-3 p. 11 with Vol-1 p. 18, say). (2) Standard party size is 4 characters/fighters (this seems like a common expectation; is about median for the sample party sizes in the book quote above; and seems like a reasonable balance between safety and fast advancement seen in last week's article). In the Arena code, scaleFactor is the same as party size, so 4 in this case. (3) We are simulating dungeon lair encounters, each with a roll on the dungeon treasure table (waiving the 50% chance for treasure, but also no DM-fiat high-value "important treasures"), not wandering encounters.
Granted that, Let's investigate the effect of a few different options for monster numbers appearing in the dungeon. First we repeat the demographic results from the last post; a party size of 4, and equivalent scaleFactor fixed at 4 (so, if dungeon level = EHD, monster numbers are equal to party size, and otherwise proportionally adjusted):
Based on this, now consider wandering monster encounters. Seems like this could be around half the size of a "lair" encounter, say: scaleFactor × 1d3. Obviously this is somewhat subjective, because these encounters don't generate treasure, are not critical to advancement, and hopefully avoided entirely by discriminating PCs (and not simulated in our program in any meaningful way). Note that in this case, with our default party size of 4, and an average wandering encounter with 2 monsters, we exactly match the guideline text in Vol-3, p. 11, repeated here:
If the level beneath the surface roughly corresponds with the level of the monster then the number of monsters will be based on a single creature... 4-6 [party size] would bring about twice as many...In order to make this synch up, we've had to: (1) interpret "level of the monster" as meaning Equivalent Hit Dice, (2) strictly read the "based on a single creature" phrase, and (3) entirely ignore the parenthetical note about groups of Orcs, and the follow-up example of hundreds of Hobgoblins. Our construction gives "lair" encounters again twice this size, which seems to be the upper bound for what PCs can confront and survive more than a few times.
But here's a complication: We assume that encounters (esp. wandering ones) will be scaled in proportion to both party size and monster EHD on the fly, and in practice this would require a calculator for the number-crunching (and also the complete list of monster EHDs). Here's a shortcut rule-of-thumb to make that more practical on the fly, based on the average EHDs at each level. (Bunch of spreadsheet number-crunching occurred here, not shown.) Look at the revised Monster Level Matrix we're using. There are six "tiers", and for each, a die-roll of 3-4 lands on a level-column which provides a "median monster", where EHD approximately equals the dungeon level (and so, presumably character level). The number appearing can then be adjusted by where the result is to the left or right of that median 3-4 result column:
You may note that a result of "Left 2" can only ever result from a die roll of "1" on the matrix; a "Left 1" result only from a "2"; and so forth. The multiplier shown might also be used for lair encounters (recommended base 1d4+1) and so forth. For clarity, the exact number appears for each pip of the 1d3 wandering encounter roll. I've got this jotted into my copy of OD&D Vol-3, p. 11.
Final thought: In contrast to this system, broadly in synch with what's related in that key page of OD&D Vol-3, Gygax's module creations tend to be a lot more dangerous, with larger numbers of monsters than we see here (even in proportion to party size). For example, the suggested monster numbers in Mike Carr's module B1 are much less than Gary Gygax placed in module B2. Another example: The wandering monster groups in the DMG Sample Dungeon have an average EHD of about 6 total, any one of which is an existential threat to a group of only 5 1st-level PCs, as depicted in the example of play, to say nothing of the variation which allows them to regularly be up to twice that size. (If the lair groups are any larger than this, then it's hard to see how they'd even fit in the rooms indicated.)
2018-05-07
Underworld Overhaul, Pt. 3: Character Party Size
Here we investigate the effect of party size on success in the dungeon environment, and overall adventurer demographics, assuming our core derived dungeoneering system (parts one, two). Recall that OD&D suggests in multiple places that encounters be scaled to the size of the PC party (Vol-2, p. 4; Vol-3, p. 11). Therefore, the code in Arena sets encountered monsters at a number appearing of NA = scaleFactor × level / EHD, where scaleFactor = party size, and level = the dungeon or character level, which are presumed to be identical (more on this later). Also, we are using the core Vol-3 "dungeon treasure" table (link) for rewards; while we are not placing any high value DM-fiat "important treasures", we are waiving the 50% chance for treasure (i.e., every encounter generates treasure), so maybe that roughly balances out on average. Note well: Although monster numbers, and also monster treasure types, are recommended to scale with party size, no such suggestion appears for the dungeon treasure table; so in the dungeon, treasure distribution is apparently fixed per encounter, regardless of how many PCs or how many monsters are fighting over it.
It bears saying, regarding the basic number-appearing formula, that a lower-bound of one monster is set per encounter (or, obviously, a null encounter could occur). This actually has a major side-effect if a small party runs into a high-level monster, as is permitted by the monster determination matrix. For example: Say 1 PC fighter of 8th level encounters a Purple Worm (calculated EHD 32); even against just a single such monster, the PC is overmatched by a factor of ×4, and will pretty much automatically perish if they engage in combat. A party of 2 such 8th-level fighters (total 16 levels), again versus one Purple Worm, is overmatched by a factor of ×2, and similarly is probably dead. It takes at least 4 such 8th-level fighters to have an even match against a Purple Worm in a normal fight, by our estimate. The same is true for many of the 5th- and 6th-level monsters; even if encounters are nominally scaled to party size and strength, the top-level monsters have a fundamental irreducible danger in this way that usually wipes out small parties when they meet.
Granted that, here are some experiments to look at the effect of different party sizes on resulting adventurer demographics. This is accomplished in the current Arena simulator with the switches -n=10000 -v -z=1 -rs (adjusting the party-size z value as shown below; and reducing the overall population n value if you want faster results).
Observations: The overall survivability increases monotonically with party size, as we might expect: the total number of living fighters in the tables above are, respectively: 6141, 7000, 7626, and 7881 (this out of 10,000 fighters alive prior to the last encounter). But the peak level achievement is not monotonic: at party size 1, there is a single Lord; at party size 2, an increase to 3 Lords; at party size 4, a decrease such that there are no 9th or even 8th-level fighters, with only a half-dozen at 7th-level; and at party size 8, another decrease to just a single 7th-level fighter. This is fairly easy to interpret: compared to a solo adventurer, a party of size 2 is better equipped to survive the irreducible high-level monster danger described above; but past that, the more the fixed treasure awards are divided up, the harder it is to gain levels. Likewise, we see that the ratio of XP from treasure declines with higher party size (treasure stays fixed but monsters multiply), respectively: 86%, 77%, 64%, and 47%.
Another interesting effect: At small party sizes, we see that the abilities of Strength and Dexterity are more critical for survival and advancement in level (these scores noticeably increase with level at party size 1 and 2; more need to kill fast and avoid any hits at all?). But with larger party size, this effect fades away and Constitution becomes more important (as at party size 8; more need to tank and shield the rest of the party from attacks?). Although at the highest levels the sample size is small, so this might be illusory.
Finally, a comment on age: One may note that all the fighters in our experiment are fairly youthful, almost all between 19 and 23 years. In the code, every fighter starts at age 18 (and since the year ends on the last iteration, everyone in the final list is incremented to 19), and a default of 12 fights/year is simulated (note that this synchronizes with the OED healing rule: one month to heal up fully from any fight). On the one hand, this an unrealistically large number of combats for real-world humans (compare to Roman gladiators: maybe one event per season); and on the other hand, far fewer than most PCs engage in (bolstered by magical healing and other factors). Many of us have surely observed PCs advancing levels in our games at a temporal rate that seems counter-intuitive. For the simulator, you may consider dialing down the fights/year to a more realistic level (via the f switch); for PCs, this is part of the reason I'm in favor of not accelerating natural healing, and also possibly limiting adventuring to certain seasons, say (e.g., only in the summer, or skipping over the winter, at least).
Conclusions: With the system at hand, adventurers must in some sense balance the following risk-reward calculus: bigger parties increase safety from death, but maximal rate of advancement occurs at a party size of around two. Choose wisely!
It bears saying, regarding the basic number-appearing formula, that a lower-bound of one monster is set per encounter (or, obviously, a null encounter could occur). This actually has a major side-effect if a small party runs into a high-level monster, as is permitted by the monster determination matrix. For example: Say 1 PC fighter of 8th level encounters a Purple Worm (calculated EHD 32); even against just a single such monster, the PC is overmatched by a factor of ×4, and will pretty much automatically perish if they engage in combat. A party of 2 such 8th-level fighters (total 16 levels), again versus one Purple Worm, is overmatched by a factor of ×2, and similarly is probably dead. It takes at least 4 such 8th-level fighters to have an even match against a Purple Worm in a normal fight, by our estimate. The same is true for many of the 5th- and 6th-level monsters; even if encounters are nominally scaled to party size and strength, the top-level monsters have a fundamental irreducible danger in this way that usually wipes out small parties when they meet.
Granted that, here are some experiments to look at the effect of different party sizes on resulting adventurer demographics. This is accomplished in the current Arena simulator with the switches -n=10000 -v -z=1 -rs (adjusting the party-size z value as shown below; and reducing the overall population n value if you want faster results).
Observations: The overall survivability increases monotonically with party size, as we might expect: the total number of living fighters in the tables above are, respectively: 6141, 7000, 7626, and 7881 (this out of 10,000 fighters alive prior to the last encounter). But the peak level achievement is not monotonic: at party size 1, there is a single Lord; at party size 2, an increase to 3 Lords; at party size 4, a decrease such that there are no 9th or even 8th-level fighters, with only a half-dozen at 7th-level; and at party size 8, another decrease to just a single 7th-level fighter. This is fairly easy to interpret: compared to a solo adventurer, a party of size 2 is better equipped to survive the irreducible high-level monster danger described above; but past that, the more the fixed treasure awards are divided up, the harder it is to gain levels. Likewise, we see that the ratio of XP from treasure declines with higher party size (treasure stays fixed but monsters multiply), respectively: 86%, 77%, 64%, and 47%.
Another interesting effect: At small party sizes, we see that the abilities of Strength and Dexterity are more critical for survival and advancement in level (these scores noticeably increase with level at party size 1 and 2; more need to kill fast and avoid any hits at all?). But with larger party size, this effect fades away and Constitution becomes more important (as at party size 8; more need to tank and shield the rest of the party from attacks?). Although at the highest levels the sample size is small, so this might be illusory.
Finally, a comment on age: One may note that all the fighters in our experiment are fairly youthful, almost all between 19 and 23 years. In the code, every fighter starts at age 18 (and since the year ends on the last iteration, everyone in the final list is incremented to 19), and a default of 12 fights/year is simulated (note that this synchronizes with the OED healing rule: one month to heal up fully from any fight). On the one hand, this an unrealistically large number of combats for real-world humans (compare to Roman gladiators: maybe one event per season); and on the other hand, far fewer than most PCs engage in (bolstered by magical healing and other factors). Many of us have surely observed PCs advancing levels in our games at a temporal rate that seems counter-intuitive. For the simulator, you may consider dialing down the fights/year to a more realistic level (via the f switch); for PCs, this is part of the reason I'm in favor of not accelerating natural healing, and also possibly limiting adventuring to certain seasons, say (e.g., only in the summer, or skipping over the winter, at least).
Conclusions: With the system at hand, adventurers must in some sense balance the following risk-reward calculus: bigger parties increase safety from death, but maximal rate of advancement occurs at a party size of around two. Choose wisely!
2018-05-05
OERAD Offering: Monster EHD Digest
Here's my new offering for today: the OED Monster EHD Digest. It's an extract of the primary results of the Monte Carlo simulations we do here to balance our games (i.e., computer simulations of monsters vs. fighters of various levels, run a few million times each, to dial in the danger levels). Previously you'd have to hunt through the OED Monster Database spreadsheet for this information; now you can print it on a single sheet for use at the table. Use for encounter balancing if you wish, and probably more importantly, XP awards at the end of each game. Also please check out other stuff at the OED Games website while you're there. Happy OERAD. Fight on!
Edit 5/26/18: Updated with more monsters and sorted by both name and EHD value.
Edit 7/24/18: This particular format turned out to be a pain to maintain and has been removed. Please see the full OED Monster spreadsheet that contains all the most-up-to-date EHD estimations (as well as having them now inserted into the automatically generated stat blocks):
The Master's Monastery, Ep. 4
Maia 1, 4729.
- Personae: Long Tim (Hobbit Ftr3), Tahj Birdfoot (Elf Ftr1/Wiz2), Tia Birdfoot (Elf Wiz1/Thf3), Penrod Pulaski (Human Thf3), Banjo Saskin (Dwarf Wiz 2), Ruff Sharktrainer (Hobbit Thf 3). Brother Maccus unavailable, still recuperating from injuries 3 weeks ago. The group buys as many healing potions as they can afford and heads to the ruined monastery.
- The group decides to explore the upper ruins more. Takes a stairway to an upper-floor cloister with books and scrolls. Carefully searches the works and finds a gold ring and a partial map of the cellars, most of which they have explored, but showing several additional rooms. Attacked by several animated skeletons with swords in the dead-end room. Party manages to fend them off. Banjo takes a skeleton's arm with attached sword. (Reasons he should swing this and count the skeleton as wielding the sword.)
- Looks inside a central garden area and finds a giant plant growing 60' away, with something metallic glinting in the sunlight. The group stays at distance and Ruff shoots it with one crossbow bolt; one of the large growths on the thing explodes, firing a stream of high-projectile seeds back at him for damage. The party responds with a volley of more missiles, all rolling well and scoring hit after hit. Two more volleys of seeds fire back and the the plant slumps to the ground, killed. A silver statuette half-buried in the soil is retrieved (1,000 sp).
- An exterior room is explored and two giant demonic hogs spring to attack; the party defeats them easily, but avoids the other filthy droppings in the room. The next room has a sagging ceiling and a makeshift beam propping it up; Ruff checks for whether it is load-bearing, and several flakes of ceiling plaster tumble down. The group considers whether it could fit in the wall slots in mysterious dead-end the 3rd room of the cellars, but decides it is too big a turn aside.Then, a kitchen with a mounted crossbow trapped aimed out the exit; the party disarms this and moves on.
- One of the last exterior buildings, the party goes to move through an empty room. Unfortunately, dice show that the covered trap is sticky and only gives way as the last row crosses it. Ruff makes his save and jumps aside; Banjo falls 10' and takes damage. Immediately a large squad of hobgoblins attacks from the entrance on the other side of the pit, by surprise. Tim and Tahj take sword-hits. Penrod runs from the room, trailing a rope into the pit after him. Hobgoblins get initiative again, and press the attack with swords and spears. Tia takes two spear-hits and goes down (save vs. death successful; narrowly clinging to life). Tahj casts sleep and 4 hobgoblins fall, but 6 remain. Banjo climbs out of the pit. Tahj is hit twice, one of which is a double-damage critical; damage dice rolled -- snake-eyes for a total of only 3 points, leaving her with 1 hit point; she flees from the line of melee. The two hobbits, Tim and Ruff, hold the line; 2 hobgoblins are killed. Penrod slings stones from the entryway, bloodying the larger ones at the back. Banjo successfully casts charm person on a hobgoblin. Morale for hobgoblins narrowly succeeds, and they strike more blows. Two more fall and the chief, at 1 hit point, tosses his sword and falls to his knees, begging for mercy in hobgoblinese.
- Tia is given several healing potions (as do Tahj and Ruff). The group try to interrogate the chief but lack any shared language, and so put him to the sword. The charmed hobgoblin, Ukt, is discovered to know Common and pressed for information. He takes them to the group lair and informs them of the large metal net trap over the other entrance, which they spring with a 10' pole. Party retrieves a silver medallion and chest of 1,000 sp. Ukt seems to have grandiose plans of raising a goblin army and overtaking the entire kingdom, and seems to see himself as the new leader of the party group. Also discover a row of cells with a human child, Phillip, frightened and cowering, held for the next meal. Party debates what to do with this unfortunate. The warlike Tim argues forcefully for strapping him to his own back and giving him a weapon to fight in that direction. Banko suggests simply releasing him to escape on his own. Other party members try to give him various playthings to cheer him: the gold ring, a drink of wine, the skeleton arm and sword, a potion bottle with a painted face, a.k.a. "Mr. Potion Bottle", allegedly a favorite toy for city children. Phillip is then locked back in his cell for "protective custody".
- Last few rooms of the upper ruins are searched, finding nothing but ruined furnishings. Group makes it way back to the cellars/dungeon, with Ukt leading the way.
- Passing stealthily, the group avoids goblin and beetle lairs, and proceed to a long, cool room to the southeast of the main storage chambers. Inside find shelves of old vegetable matter and large urns. Ukt is directed to check one, at which point 4 giant centipedes spring out him! In the first round, all miss him with their poison mandibles. Ukt throws one off and cuts it in half with his scimitar. Ruff runs forward and daggers a second. Tia pierces one with her spear. And Tim fires an arrow scoring 6 points of damage, cleaving through the last centipede and also Ukt's chain mail, killing him instantly.
- Party proceeds to the wine cellar, avoiding the pit with the ooze-monster that killed Aslak the Unclean. Enter back into the large L-shaped room with scores of broken casks and barrels, many of which are turned into nests for many horrid bat-mosquito monsters. 5 creatures fly into the air to attack; missiles and daggers mostly miss them; Tia's last spell, magic missile, partly injures one. One sticks in Banjo's backside, and falls and rolls to squish it; it dislodges and gets speared by Tia. Two fliers are cut down, but another flies up out of its nest. A few more are cut down, and then 6 more fly out, swamping the parties' attempts at attacks. The buzzing creatures dive and strike at all the party members, and now every party member has a bat-creature stuck in them and sucking blood rapidly.
- Becoming desperate, Penrod lights a torch and fires one of the nests. Yet more spring into the air. One is stuck deep in Tia's ear, blood spurting up and the thing lapping it greedily. One is stuck on Tim's foot and he misses it with his sword. Several are pulled out and the party runs for their lives, creatures still pierced into Penrod, Tahj, and Tia, each rapidly turning white and near death. Tim plucks the one out of Tahj and squashes it. Tia has 3 hit points and is about to take 1d3 blood-suck damage. Banjo tries and fails to grab the thing. Tia herself fails to pull it out. Damage die comes up: 3 hit points and Tia is out of points. Save vs. death: failed, and Tia sadly expires before the others can save her. Tim slashes his sword at the bloated monster and it explodes, splashing the entire party with Tia's life-blood.
- The party retrieves Phillip and returns to the village. His well-to-do aunt, Heide, weeps with joy and rewards them with 500 sp. Total haul for the excursion: 3,500 sp, and a total of 7,000 XP (so: 700 sp and 1,400 XP for each survivor). Tahj is promoted to a 2nd-level fighter. Banjo achieves 3rd level as a wizard and looks eagerly forward to wielding a magic spell of the 2nd level on his next adventure!
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