D&D 5E (2014) Tidbit for monster design

[Sword of Spirit]

I came up with that factor by noting the game's math seemed to be centered around one monster against a party of four PCs, but it's possible the actual calculation uses something else that just happens to be close to 0.25. For example, if the actual calculation uses 0.65/2.5 = 0.26 then a base AC of 12 and a base AB of +4 fits the published XP values quite well. Why these numbers? Well, 0.65 is the base chance to hit, so our effective DPR would now include that factor. And 2.5 is the average number of rounds a monster tends to live for against a baseline party of four level appropriate PCs, so our effective HP is treated more like an effective HP per round.

I had a thought and wanted to share immediately in case anyone wants to make use of it before I get to it.

What if it's just x1/3, since combat is expected to last 3 rounds?

That might sound overly simple, but we know they oversimplify from the input form (they don't include initiative bonus at all, for example).

EDIT: That's not actually looking very promising.
@tomedunn do you have any other possibilities than 14/13 that might work for an exponential version? Playing around with some other things and realizing having more flexibility there might be useful.

 

[Steady Vane]

A couple other thoughts have just occurred to me after messing with the formulas as I was writing this. One is that stegosaurus stands out as a problem. The only formulas that can deal with it are your exponential one (which has a steeper slope, and is too high on the cyclops), and your linear one with a -1 rather than -2 (which somehow magically works for everything, even though, if I understand correctly, it relies on two approximations in its derivation). It fails on everything else. I'm starting to believe that maybe a mistake was made in the data entry for the stegosaurus when they were making Volo's Guide. In this kind of analysis, I'm loathe to dismiss something as an anomaly, as it could just as easily be evidence that the formulas are all wrong. But I'm at least considering the possibility. If the developer had entered a 36 instead of a 26, or a 9 instead of a 7, or a 15 instead of a 13 or a 96 instead of a 76, it would give us the listed CR 4. Going forward, I may flag the stegosaurus as a potential error in the published math and test out formulas that work on everything except the stegosaurus to see how they hold up with more complicated monsters.

One of the funny things about 5e is that it has come out how certain "iconic" spells do too much damage on purpose. They had a spell formula, and purposely ignored it for certain spells to make them more "fun" - fun meaning overpowered for their level.

If I recall correctly, this also applies to most Dragons in the CR calculations. They are flat out stronger than they should be for their CR level. Now this has not been stated, but presumably that choice would follow the same logic. They are iconic and should be more powerful to be "fun".

When it comes to spells, Fireball is the most notorious, but I think I recall people who did spell cleanups for 5e stating that there were one or two per spell level that were chosen as well, if simply not as well known.

All that to say, could the Stegosaurus be another such monster case, where it's simply more powerful than it should be, because it's an "iconic" monster?


Addendum: This is why I rely on averages when making 5e monsters, it tends to drown out the outliers. Tomedunn's blog has been a great source of such math crunching

 

[Ondath]

I've got a few questions for you two if that's alright: @Steady Vane and @tomedunn

I've been trying to understand the game's key assumptions about how combats are supposed to go, and I'm wondering if you also found the answers on some of these assumptions.

So from what I understand, we know that monsters are meant to survive for 2-3 rounds, that there is a hidden variable of average PC AC and average PC save bonus... But does this also give us a clearer picture of what the fabled "adventuring day" is supposed to look like? It's widely known that the ideal-type of an adventuring day contains 6-8 encounters of an appropriate challenge. But if we take the calculations you made into account, can we paint a clearer picture of what that adventuring day looks like?

How much HP are the players supposed to end the adventuring day? Someone had said in an earlier thread that the maximum amount of total damage balanced encounters can be expected deal at the end of an adventuring day are 1.5x the party's total HP, since you only regain half of your Hit Dice (and thus half of your HP recovery potential short rests) at each Long Rest, and if you use those to recover half HP during short rests, then get to a point where you're nearly at 0 HP at the end of the adventuring day, you'll make a full recovery with a Long Rest. Do your calculations confirm that? If so, this would be very interesting since it'd mean that WotC assumes that a "balanced" adventuring day beats the party nearly to a pulp, and I think many players would think that they are facing unfair challenges if every adventuring day ended with them at ropes.

Secondly (and this is probably very hard to analyse), do we know how many of your limited-use resources (things like spell slots and X/Short or Long Rest class abilities, not consumables or magic items) you're expected to expend at the end of the adventuring day? I'm guessing we could maybe guesstimate this number by looking at monsters' expected HP, get a total of the amount of HP 6-8 encounters' worth of monsters make, then see what amount of "feature use" would get an ideal-type party to get that amount of total damage. I know that @mearls mentioned in a Bluesky post that they assumed an adventuring day would last 20 rounds, and that they "spread" the additional damage from limited-use features to a class's "at will damage per round", so I'm wondering if this kind of "expected ability use in an ideal adventuring day" is also something we can reverse engineer. Again, the point would be to see how many of your spell slots/class features WotC assumed you'd have used up at the end of the day. And again, I'm guessing that many current 5e players would think that an adventuring where every single one of their spell slots/limited-use abilities are used up would be an unfair amount of challenge.

And a final thing, have you guys ever looked into including Lanchester's Laws in your calculations? There is a guy who argues that 5e CR could be used in a way that'd give us a score per monster that accounts both for that monster's overall quality, and how overbearing that monster would be when that monster is used in larger numbers. If such a calculation seems reliable to you as well, I think that could be one way to make calculating the balance of an encounter much more precise.

Anyway, I've been thinking about the deep fundamental math of 5e a lot lately, so I was very excited when I saw that this thread is ongoing and still active, so apologies if this was too many questions in one go!

 

[Steady Vane]

Combat is assumed to last 3 rounds, I can answer that.

As for adventuring day, it depends on the strength of combats, but I believe the number sits at around 3 to 4 with appropriate encounter strength. Casters at least are assumed to use one of their highest-level spell slots per encounter, and they generally have about three to four of those slots, depending on character level.

Going for easier combats, and the number the party can chew through before resting increases. That's where the 6-8 figure comes from.

 

[Ondath]

Combat is assumed to last 3 rounds, I can answer that.

As for adventuring day, it depends on the strength of combats, but I believe the number sits at around 3 to 4 with appropriate encounter strength. Casters at least are assumed to use one of their highest-level spell slots per encounter, and they generally have about three to four of those slots, depending on character level.

Going for easier combats, and the number the party can chew through before resting increases. That's where the 6-8 figure comes from.

So is the idea that there should actually be 3-4 Medium difficulty encounters in an adventuring day, and the Oft-cited 6-8 encounters actually has the assumption that some of that 6-8 will be easier stuff?

I think an interesting point of comparison might be Shadow of the Weird Wizard by Rob Schwalb (who was also working on 5e when it first came out) who mentions in the GM book that the game assumes 12 rounds of combat in a regular adventuring day. If the expected number of encounters in 5e is 3-4 Medium ones like you said, then the two systems are actually quite alike! But that also contradicts with Mearls' bluesky post about how they expected an adventuring day to have 20 rounds of combat...

 

[Steady Vane]

3 to 4 combats is at "deadly", which in the playtest versions of the game was instead named "hard".

If the combats are easier, you can go through more of them. 6-8 would be during a dungeon crawl, which is what the assumed playstyle was when 5e launched. Presumably the difficulty would be lower, probably around "medium" or "hard". Dungeon crawling implies wandering monsters, and those are often not that deadly, but more of a resource tax when exploring the dungeon for loot.

If you look at the DMG exp budgets, they match about 3 1/2 "deadly" encounters per adventuring day. That's the assumed.

 

[Sword of Spirit]

In my XP calculations I've generally used (AB + AC - 15) which comes from a base AC of 12 and a base AB of +3. These seem to fit the XP values in the DMG the best, but, as you point out, they don't result in a 65% chance to hit. A possible way of explaining this inconsistency is that the factor of 0.25 at the front of the XP equation is wrong.

I came up with that factor by noting the game's math seemed to be centered around one monster against a party of four PCs, but it's possible the actual calculation uses something else that just happens to be close to 0.25. For example, if the actual calculation uses 0.65/2.5 = 0.26 then a base AC of 12 and a base AB of +4 fits the published XP values quite well. Why these numbers? Well, 0.65 is the base chance to hit, so our effective DPR would now include that factor. And 2.5 is the average number of rounds a monster tends to live for against a baseline party of four level appropriate PCs, so our effective HP is treated more like an effective HP per round.

What does the 14/13 in your exponential equation represent?

I just spent quite some time writing up a post explaining my current work for anyone reading along, and then accidentally closed my brower. Sigh Sorry internet.

So here's a tiny bit that might help someone.

If you use the conceptually accurate math (XP = 1/4 eHP x eDPR where HP and DPR are both adjusted from a baseline assumed chance to hit by referring to the actual values of the monster) on the DMG Table, the best fit based on a baseline 65% chance for a PC to hit a monster is obtained by entering in a baseline 50% chance for a monster to hit a PC. I think that might be meaningful.

Then testing which approximated equation (since the conceptually accurate one doesn't accord 100% accurate to what they used, for reasons I mentioned before) from a bunch of variations of equations both linear and exponential that I tried all over the place is a best fit for the DMG Table assumed monsters, the winner is this:

That isn't, however, the best fit equation for the 61 basic monsters (slightly expanded number from before). The honor for that one (selected for reasons you'll see from several that will all produce only 3 failures) is this:

The only difference is the lack of the -2. But what's further is that when applying the conceptually accurate math to the actual 61 basic monsters, the best fit by far is to assume that the monsters have a 65% hit chance.

I don't know why that all works out the way it does, but it certainly seems like it means something.

 

[Charlaquin]

What does the 14/13 in your exponential equation represent?

I just spent quite some time writing up a post explaining my current work for anyone reading along, and then accidentally closed my brower. Sigh Sorry internet.

So here's a tiny bit that might help someone.

If you use the conceptually accurate math (XP = 1/4 eHP x eDPR where HP and DPR are both adjusted from a baseline assumed chance to hit by referring to the actual values of the monster) on the DMG Table, the best fit based on a baseline 65% chance for a PC to hit a monster is obtained by entering in a baseline 50% chance for a monster to hit a PC. I think that might be meaningful.

Then testing which approximated equation (since the conceptually accurate one doesn't accord 100% accurate to what they used, for reasons I mentioned before) from a bunch of variations of equations both linear and exponential that I tried all over the place is a best fit for the DMG Table assumed monsters, the winner is this:

View attachment 428125

That isn't, however, the best fit equation for the 61 basic monsters (slightly expanded number from before). The honor for that one (selected for reasons you'll see from several that will all produce only 3 failures) is this:

View attachment 428126

The only difference is the lack of the -2. But what's further is that when applying the conceptually accurate math to the actual 61 basic monsters, the best fit by far is to assume that the monsters have a 65% hit chance.

I don't know why that all works out the way it does, but it certainly seems like it means something.

My best guess would be that those 61 are intended to be “solo” monsters. Legendaries, dragons, and other monsters the designers expected to most often be used as bosses. Does that jibe with the 61 you observed to have the higher hit chance?

 

[Steady Vane]

If you use the conceptually accurate math (XP = 1/4 eHP x eDPR where HP and DPR are both adjusted from a baseline assumed chance to hit by referring to the actual values of the monster) on the DMG Table, the best fit based on a baseline 65% chance for a PC to hit a monster is obtained by entering in a baseline 50% chance for a monster to hit a PC. I think that might be meaningful.

I couldn't tell you the relation between xp values and anything else, but I can tell you those assumptions are wrong.

PC accuracy is assumed to rise, it only starts at 65% at level 0 with a +0 weapon. Accuracy rises with every +X weapon, to 85% assumed PC accuracy, at least in the math. It is why both DMGs say explicitly you should use +4 weapons and armor.
DMG 2014 p284 and DMG 2024 p58 if you want to check

Monsters also grow more accurate, they go from hitting on ~13s (40% accuracy) to hitting on ~9s (60% accuracy). Your 50% works as an average of this. Both sides have the same increase of 20% accuracy as you level. If you are assuming XP relations based on accuracy, you need to start there.

When I ran the math, the levels where accuracy rises for PCs are assumed to be 2, and every 5 levels after. So 2, 7, 12, 17 for +1, +2, +3, and +4 weapons. That worked the best for PCs, but I couldn't say for monsters.

Hope this helps!

 

[Sword of Spirit]

My best guess would be that those 61 are intended to be “solo” monsters. Legendaries, dragons, and other monsters the designers expected to most often be used as bosses. Does that jibe with the 61 you observed to have the higher hit chance?

Oh, no, no, no! That's not what's going on at all (clever guess from my scanty info though!)

You probably already know this, but I'll briefly explain for anyone following. There are only 4 values that go into the equation that produces CR: HP, DPR, AC, and AB. That's it. There most definitely are many other factors that effect CR, but the way they do it is by altering those 4 values into the effective values that actually end up being plugged into the equation.

The DMG describes what, for purposes of the not terribly accurate DMG version of the monster building rules, those other values are that alter the basic 4 and what alterations they make. The official CR calculator spreadsheet's data entry page seems to show that at least some of those alterations are similar to the real monster math, but not all of them are.

Exactly how all of those alterations to the basic 4 values are done is unknown. In order to figure that out out, we need to first discover the correct equation, and then test--as far as is possible--each alteration independently.

Those 61 basic monsters are all of the monsters that I can be sure have no features that would affect their HP, DPR, AC, or AB. They just have the raw stats and nothing that might modify them. No saving throw bonuses. No damage resistances or immunities. No features that cause you to make a saving throw, or impose a condition. Etc. They are your "boring bags of hit points". Many of them are beasts. None of them have a CR above 7. Honestly, if there were as many monsters like that as people think there are this would be a lot easier to do! (It's actually likely that there are other monsters whose additional features have no effect on CR, but from the input form and info we do have, and lacking the full instructions of how to use that form, I can't be sure.)

So what's the importance of the basic monsters? Limiting initial trials of proposed equations to them is the only way to make sure the equation works right. The correct equation should work on 100% of basic monsters. Then I can move on to trying to discover how all the other variables actually adjust the 4 basic values, in order to fully reverse engineer the math.

However, I'm stuck at 95% of those monsters, and can't find any way to do better.

The 2 monsters that won't work with anything I can even think of are the twig blight and the needle blight. They both always come out weaker than the CR the MM gives them, regardless of what variant of what equation I use. Since they were in the original Starter Set that predated the MM by like half a year it's possible that they hadn't refined all the math (and, AFAIK, they never change the listed CR of a monster after publication), or even that they just decided to cheat, because they preferred that the CRs of the three blights ended up as the published 1/8, 1/4, and 1/2, rather than the 0, 1/8, and 1/2 they probably actually are by the official math. If it's the latter, it's quite possible they did that before they decided to stop doing that and be strict about using the results the equation gives them--which I think they have been doing since the MM.

The other anomaly depends on which equation I use. I've developed several which end up with only 3 failures, but can't get it down to just those blights. The one I posted also fails on the grimlock--which is the most common one to take that honor. I have no explanation whatsoever for it. It just doesn't come up with enough XP to push it into the listed CR 1/4. Now, there are some equations that will get it there, but they are at least a little more complicated, and instead fail on different monsters. The one I'm most comfortable failing on is the myconid sprout, because there is an easy mistake that could have been made when entering its data that could cause it. But again, it just seems like the equation(s) that give us the grimlock as an inexplicable failure are more likely to be accurate.

So that's what those 61 basic monsters are and why they are absolutely critical to everything--they are the ones that all have to work.