D&D 5E Understanding the maths behind 5e

The Crimson Binome · Jan 26, 2020

dave2008 said:
They only thing relevant that I remember off the top of my head was that: monsters are relatively weaker as they go up in CR. As PCs go up in level, the damage monsters do (at an equivalent CR) becomes a lower percentage of the PC's hit points.

Doesn't that balance against accuracy, though? At low levels, your AC is almost a reliable defense, but you can only take a couple of hits. At high levels, AC is unreliable, but you can take more hits before falling.

NotAYakk · Jan 26, 2020

I approached doing math from the encounter building perspective. I assumed the DMG knew what it was doing, and wanted an easier way to model it, build monsters, and use existing monsters.

I didn't model PCs. I wasn't intending on building ad-hoc PCs like I was monsters, and the PC building minigame wasn't somethinf I wanted to rake away from players.

But it is interesting.

What we could do is model the featless magic itemless pure fighter DPR over a model encounter day. 2 short rests, 5x hard enemies total. Champion and Battlemaster.

Some at-will some per-rest damage.

And the same with a Rogue.

miggyG777 · Jan 26, 2020

NotAYakk said:
Anyhow, you can take the XP values of various CR monsters, raise it to the (3/4) power, then futz around with it and round it. You get something like:

CR 1/8: 1
CR 1/4: 2
CR 1/2: 4
CR 1: 6
CR 2: 9
CR 3: 12
CR 4-15ish: +6 per CR
CR 16-19: +9 per CR
CR 20: 140ish
CR 21+: +15 or 20, I forget, per CR

I tried to recreate this step, however I did not get these values when I raised the XP values by (3/4). What do you mean with "futz around with it"?

dave2008 · Jan 26, 2020

Saelorn said:
Doesn't that balance against accuracy, though? At low levels, your AC is almost a reliable defense, but you can only take a couple of hits. At high levels, AC is unreliable, but you can take more hits before falling.

Possibly. That makes sense at first glance, but it was not part of the analysis I did (that I can remember). I wouldn't want to wholeheartedly agree without looking into more. And of course AC is a tricky one as it can vary quite a bit at higher levels (from less or more reliable than lower levels), so it really depends on a lot of factors (like many things at high levels)

NotAYakk · Jan 26, 2020

miggyG777 said:
I tried to recreate this step, however I did not get these values when I raised the XP values by (3/4). What do you mean with "futz around with it"?

Sorry, it is 2/3 not 3/4. And it is a calculated XP, not the rounded XP on the table, I used to fit the curve.

If take DMG XP and I raise XP to the power 0.65 then divide so that CR 1 monsters are worth 12 EBP (divide by ... 2.6?), I get

1
3
5
8
12
20
28
36
48
60
72
84
96
108
123
138
153
168
183
198
218
238
258
278
308
373
438
503
568
633
698
763
828
893

after some rounding and curve smoothing.

This is:
CR 0: 1
CR 1/8: 3
CR 1/4: 5
CR 1/2: 8
CR 1: 12
then +8 per CR up to 4:
CR 4: 36
then +12 CR up to 10:
CR 10: 108
CR 11: 123
Then +15 to CR up to 16:
CR 16: 198
Then +20 CR up to 20:
CR 17: 218
CR 20: 278
CR 21: 308
CR 22+: +65 per CR

The biggest changes here are that CR 1/8 and 1/4 are worth more, as are CR 2/3. It is also "double units" so I don't have to use fractions.

I think I over-rounded CR 1/8 and 1/4 before.

Bold parts above are where the derivative (slope) changes. They are at 1, 4, 10 and 16, and a bunch around 20 -- and 5, 11 and 17 may be familiar to people making 5e characters (points where cantrip damage scales up).

miggyG777 · Jan 26, 2020

NotAYakk said:
Sorry, it is 2/3 not 3/4. And it is a calculated XP, not the rounded XP on the table, I used to fit the curve.

If take DMG XP and I raise XP to the power 0.65 then divide so that CR 1 monsters are worth 12 EBP (divide by ... 2.6?), I get

1
3
5
8
12
20
28
36
48
60
72
84
96
108
123
138
153
168
183
198
218
238
258
278
308
373
438
503
568
633
698
763
828
893

after some rounding and curve smoothing.

This is:
CR 0: 1
CR 1/8: 3
CR 1/4: 5
CR 1/2: 8
CR 1: 12
then +8 per CR up to 4:
CR 4: 36
then +12 CR up to 10:
CR 10: 108
CR 11: 123
Then +15 to CR up to 16:
CR 16: 198
Then +20 CR up to 20:
CR 17: 218
CR 20: 278
CR 21: 308
CR 22+: +65 per CR

The biggest changes here are that CR 1/8 and 1/4 are worth more, as are CR 2/3. It is also "double units" so I don't have to use fractions.

I think I over-rounded CR 1/8 and 1/4 before.

Bold parts above are where the derivative (slope) changes. They are at 1, 4, 10 and 16, and a bunch around 20 -- and 5, 11 and 17 may be familiar to people making 5e characters (points where cantrip damage scales up).

Thanks a lot, this might be one of the most useful things I have learned about 5e so far. I still have some more minor questions though, if you don't mind.

1) How did you calculate the XP progression?
2) Why did you divide the numbers so CR1 = 12 EBP?
3) How did you figure out that the power of (2/3) gets rid of the XP multipliers?

NotAYakk · Jan 26, 2020

I copied XP progression from the DMG chart. I played with exponents until I got a roughly linear set of values from 1 to 20.

I was trying to go from raw stats to CR at the time, and after noticing the raw stats to XP connection, I figured converting XP to XR would help.

The values where not quite linear, so I examined how. Setting CR 1 to a multiple of 3 was because 1/2 and 1/4 was roughly 2/3 and 1/3 of the calue for 1.

Setting CR 1 to 12 gives really low percent errors for CR 1/8, 1/4, 1/2 and 1 as integers.

Then it is just some curve fitting. It was in my response that I noticed that putting inflection points at 5/11/17 and 20ish worked well.

The "just add them up" was a goal of mine. From looking at 4e CR I knew you can do lots to make that work.

I'll have to confirm that adding them up does the right thing with these numbers. But the test isn't that hard; take 12 CR 1/8 (36 EBP, which is CR 4). And use DMG to calculate XP multiplier etc. Then compare witb XP of CR 4 monster.

Repeat a few times.

miggyG777 · Jan 26, 2020

NotAYakk said:
I copied XP progression from the DMG chart. I played with exponents until I got a roughly linear set of values from 1 to 20.

I was trying to go from raw stats to CR at the time, and after noticing the raw stats to XP connection, I figured converting XP to XR would help.

The values where not quite linear, so I examined how. Setting CR 1 to a multiple of 3 was because 1/2 and 1/4 was roughly 2/3 and 1/3 of the calue for 1.

Setting CR 1 to 12 gives really low percent errors for CR 1/8, 1/4, 1/2 and 1 as integers.

Then it is just some curve fitting. It was in my response that I noticed that putting inflection points at 5/11/17 and 20ish worked well.

The "just add them up" was a goal of mine. From looking at 4e CR I knew you can do lots to make that work.

I'll have to confirm that adding them up does the right thing with these numbers. But the test isn't that hard; take 12 CR 1/8 (36 EBP, which is CR 4). And use DMG to calculate XP multiplier etc. Then compare witb XP of CR 4 monster.

Repeat a few times.

Thanks, I am testing it right now, so far it seems to hold up quite well. There is a bit of a deviation though, sometimes more sometimes less (I would say <10%, just from eyeballing it), which still is absolutely amazing given the simplification of the process.

Charlaquin · Jan 26, 2020

miggyG777 said:
I am currently trying to dive deeper into the mechanics behind 5e. I was fascinated with PF2s structured approach in this regard and therefore wanted to see how my favorite system (5e) compares to it.

5e overall is a little bit harder to understand, mainly because the material is not always thoroughly informative on how the underlying mechanics are implemented. Therefore you have to resort to reverse engineering to understand the actually quite beautiful mechanics that 5e employs.

Example: The monster building tutorial as outlined in the DMG is superficial at best, when it comes to explaining the reasoning behind why certain things are the way they are. The AngryGM wrote a good article on this, where he goes past the DMG and uncovers what lies beyond Monster Building 201: The D&D Monster Dissection Lab

Obviously not all concepts in 5e need this sort of reverse engineering to understand and are quite self explanatory, such as Bounded Accuracy.

But I still have some things that I want to learn.

For instance: How does HP and damage scale? For players and for monsters. This should tell me how long it takes the party to kill an appropriate level monster on average. That will allow me to understand the resource draining mechanics better and improve my own encounter building.

The monster scaling is outlined in the DMG, unfortunately the infos for the player character side are lacking, I found some spreadsheets but they are not as conclusive as I wish them to be. Therefore I do not know how much damage a player will do on average, at a certain level, and I cant infer the average TTK that the creators of 5e envisioned. That TTK is needed to calculate how much damage a certain monster will deal to a character on average, hence the resource draining.

The obvious approach here would be to reverse engineer as mentioned beforehand, but I'd be surprised if nobody had ever thought about this before. However, I haven't really found anything in the net yet. That's why I am asking you guys. Do you have good materials on the underlying mechanics of 5e? Maybe even something that deals with resource draining and encounter building? Perhaps I have missed something.

You should check out Angry’s more recent articles on monster building where he basically redesigns encounter balancing based on the principles of monsters’ and PCs’ expected TTKs and DPRs. He glosses over most of the math, so they probably won’t be super helpful in revealing those underlying mathematical assumptions, but he produces a system that works well based on those principles, and it might be another useful datapoint in reverse-engineering those values.

miggyG777 · Jan 26, 2020

Charlaquin said:
You should check out Angry’s more recent articles on monster building where he basically redesigns encounter balancing based on the principles of monsters’ and PCs’ expected TTKs and DPRs. He glosses over most of the math, so they probably won’t be super helpful in revealing those underlying mathematical assumptions, but he produces a system that works well based on those principles, and it might be another useful datapoint in reverse-engineering those values.

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I have read these. Unfortunately his monster building table, that arguably is pretty simple, is not very well suited for varying party sizes. As far as I can tell.