D&D 5E Understanding the maths behind 5e

miggyG777

Explorer
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.
 
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dave2008

Legend
I've done an analysis of Monster and PC damage across levels before. I know the PC information was in a spreadsheet it posted on this forums somewhere. If I can find it, i'll link to it here.

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. Basically, a CR 1 monster requires fewer hits to kill a lvl 1 PC than a CR20 monster requires to kill a lvl 20 PC.
 

Jacob Lewis

Ye Olde GM
One of the many things I absolutely loved about 4th Edition (DnD) was how easy it was to figure out the maths. Sure, the numbers bloated to insane amounts, but the principles were solid. It was well structured and highly organized, and yet retained immense versatility making it easy to scale and adjust as needed.

The digital tools helped, of course. Still, it didn't take much effort to do it by hand. We just got spoiled and lazy with automated software. I miss that. I miss 4e. Sigh.
 

dave2008

Legend
One of the many things I absolutely loved about 4th Edition (DnD) was how easy it was to figure out the maths. Sure, the numbers bloated to insane amounts, but the principles were solid. It was well structured and highly organized, and yet retained immense versatility making it easy to scale and adjust as needed.
PF2e is structured in a similar way (math wise) as 4e. You might give it a try.
 




NotAYakk

Legend
Warning Edit: I did the math in this from memory, not from my spreadsheet. The actual exponent is 2/3 not 3/4, and there are a few other errors (corrected in later posts).

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As for encounter building, all of that futzy "double XP" crap can be gotten rid of by just raising XP values and budgets to the ^(3/4). This changes the 25%/50%/75%/100% easy/medium/hard/deadly to 40%/60%/80%/100%.

The basic idea is, how hard is a monster with 2x the HP and Damage? Is it 2x (I mean, you doubled both), or 4x (2x * 2x = 4x)? If you just double a monster's damage, it is twice as hard, right?

So how about 2 monsters? Well, they are more vulnerable to AOE effects, and it tends to be harder to focus fire with 2 monsters, and if you deal 1 monsters worth of HP their damage output halves. How do 2 monsters compare to one with 2x HP?

That (3/4) factor is 5e D&Ds answer to that question.

Suppose we have X monsters on the left and one monster with Y times the HP and Damage on the right.

(HP*Damage)^(3/4) (adjusted for defences and accuracy) answers "how many X monsters does it take to be just as hard as a Y".

X * (HP*Damage)^(3/4) = (Y*HP*Y*Damage)^(3/4)

X * (HP*Damage)^(3/4) = (Y^2)^(3/4) * (HP*Damage)^(3/4)

X = (Y^2)^(3/4)
X = Y^1.5

So if Y is 4, X is 4^1.5 which is about 6.

So 6 "small" monsters are just as hard as one "4 times size" monster.

How this works in the 5e encounter balancing system is this.

Suppose the small monsters are wroth 400 XP each. The big monster will be worth 400*4*4 = 6400 XP or so.

400 * 6 is only 2400 XP. But the encounter size multiplier then boosts 2400 XP by 2.5-3, making it reach 6000-7200 XP. Roughly the same budget.

As far as I can tell, all of those encounter size multipliers can be ignored by first raising XP to the (3/4) power. Now XP is additive.

But if we are doing this, we don't have to use the XP values posted. I mean, many people advance by milestones instead anyhow.

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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 call those "encounter building points". They let you build encounters by just adding up numbers and no multiplication.

---

On the PC side, you can add up player EBP and divide by 4 to work out what their medium budget should be. Then an easy encounter is 2/3, hard is 4/3, deadly is 5/3 of that, and "crazy" is 2x.

If we divide by 3 instead of 4 we get even nicer math. Easy is 1/2, medium is 3/4, hard is that EBP, and deadly is 5/4 (+25%) and crazy is 3/2 (+50% over budget).

As the EBP stuff above is already divisible by 3 we get:
L1: 2 EBP
L2: 3 EBP
L3: 4 EBP
L4-15: +2 EBP per level (or Level*2-2)
L16-19: +3 EBP per level
L20: 45 EBP

then just add up the party's EBP to give an EBP budget (at "hard" difficulty).

(the cutoff at 15-16 might be a level layer, I'm doing this from memory).

There is modest differences in the encounters build this way compared to the DMG, but I find it ridiculously easier.

Turning this into scene building (budget for time-between-short-rests) and daily budgets isn't hard. Really, 5 times your EBP budget (as calculated above) isn't a bad number to at level 3 and above (at level 1-2, use 4x).

So a party of 4 level 5 PCs has an EBP budget of:
16 easy (don't use anything easier than this)
24 medium
32 hard
40 deadly
48 crazy
160 per day

If you have 2 short rests per day, that is about 56 EBP per "scene" (period between short rests).

Lets build a set of encounters. We are in a village. The baron's daughter is visiting. Mercenaries attack, pinning down the PCs.

1. 6 guards (EBP 6) plus PCs against 6 orcs (24 EBP) (18 EBP net, easy fight)
2. Guards leftover from first 1 (say, 4), plus 5 new guards (so 9 EBP), plus PCs, vs 8 orcs (32 EBP) and a veteran (CR 2, 18 EBP) 41 -- deadly-crazy fight.

Short rest. The attackers have kidnapped the baron's daughter! They flee on the water.

PCs get on boat and give chase (short rest).

3. Neried (36 EBP, hard-deadly) engages PC's boat.

PC continue chase (short rest).

4. Enemy boat is pulled up on a dock. They left 6 orcs behind to ambush PCs (24 EBP, medium encounter).

A boat with 20 guards (EBP 20) and a Guard Captain (EBP 12) arrives after the fight is finished, as they got going slower than the PCs did.

5. PCs catch up to fleeing bad guy. They are a mage (CR 6, EBP 30) with a veteran (EBP 18) and 10 orcs (EBP 40) with the daughter (Scout, CR 1/2, EBP 4, but restrained and unarmed). The guards engage the orcs while the PCs+Guard Captain have to fight the mage and veteran. (EBP 36, hard-deadly)

(each round 1d6 guards die and 1d3-1 orcs; if the PCs take too long, the orcs fall on them.)

6. Clean up leftover orcs. They flee if reduced to 3 or fewer orcs (EBP: 20ish)

Total budget: 155-175

Now just need to work out what happens if the PCs don't follow the railroad. ;)
 
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jsaving

Adventurer
PF2e is structured in a similar way (math wise) as 4e. You might give it a try.
PF2e brought in a fair number of 4e elements, from the abolition of free-form multiclassing to a simpler skill-point system to increased "guardrails" on player choice so you won't inadvertently create a gimped character. Some players who adopted PF1e because they disliked 4e are mystified by Paizo's decision to go in that direction (which they are now doubling down on with the recently announced promotion of 4e's design guru). But yes you are completely right that math structure is another area where PF2e drew from 4e.
 


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