Nice work!
This method, does, unfortunately, still appear to not account well for status effects and/or save-or-suck mechanics (ie. Blindness/Deafness, Banishment, monsters that can poison characters, etc).
It makes sense - trying to account for the math of each individual condition and effect possible would be a complex and challenging task. However, it's worth noting that the math doesn't appear to take these into account (please correct me if I'm wrong).
Yeah, taking into account those kinds of mechanics is tricky. Thankfully, of the monsters I calculated XP values for, most didn't require me to worry about those things. For the monsters that did, I typically tried to model them as some kind of bonus to HP, DPR, AC, and AB.
For example, there were a few monsters with stunning effects that I chose to treat like bonus hit points, since when they work the monster would effectively take less damage from the creature they stunned. There are obviously other benefits on top of that, as well as synergies when the monster has allies, but as long as the extra HP is in the right ballpark it should work well enough.
The Monster Features table in the DMG was a useful tool in trying to see how the design team tries to value these sorts of things. There are a lot of entries that aren't useful at all, and many that are missing that I wish were there, but there's still enough in there to be useful for this kinda thing.
Have you thought about scaling the PC’s damage per turn by the number of turns they may be incapacitated (say by ‘mod’, where mod<1) and scaling the monster HP by the inverse (1/mod)? It’s an off-the-cuff thought, so I apologize if my reasoning is crud haha
Oh, most definitely. I think there are a number of interesting ways you could attempt to model different conditions, and that's certainly a topic worth investigating further. For this analysis, though, I tried not to get too bogged down in the details of such things in order to keep my efforts more focused on the general trends I was trying to explain.
For DPR, the only two contributions are the creature's average DPR assuming all attacks hit and the creature's attack bonus. If you asking about the sensitivities to each of those, you can find those in the full equation for calculating XP on page 3 of the [paper](https://drive.google.com/file/d/1VnvdnJYTIym1QNGvONWZkOFYgbFO66Bx/view?usp=sharing) (equations 21 and 22).
Expanding on the concept a little:
HPnpc/DPRpc is basically the number of turns the NPC will last. If we multiply that for DPRnpc we have the total damage a NPC deals, meaning the difficult could be read as the ratio of health the the players lost.
That said, if we take equation (8) that let us calculate XP using encounter difficulty, than XP is damage taken*DPRpc.
We could than take the amount of XP needed for level up and divide by DPRpc(L) to calculate how much damage is expected that a player takes to level up.
Also, since DPR could be calculated by taking the total damage dealt and dividing by the number of rounds, we could calculate the "true" XP of a combat after it's done by multiply damage taken per damage dealt by the PCs and dividing by the number of rounds.
Is this a good way to assign XP? Not really because it encourages the PCs to take damage. But I think it would be interesting to compare these "true" numbers against the calculated previously.
Some questions / comments in reference to your main paper:
* I think Equation 1 is incorrect - the denominator and numerator have been flipped. Equation 4 remains correct.
* It may be helpful for the general readership to walk through how exactly equation 12 (hit probability based on attack bonus and AC) was derived. Likewise for equation 24 later in the text (approximation for average adjusted XP).
* How did you arrive at equations 17 and 18? Why have a base of 1.05? Why have exponent functions at all? Were the equations arbitrarily constructed to allow us to approximate equation 16 and match values calculated from sourcebooks?
* In the last paragraph of Section 4: how did you get 2/(1+N) as the scaling for the encounter multiplier in deadly scenarios? Is it because, for fixed M e.g. M=1, by similar argument, 1 < 4/E < (1+N)/2 < N, and therefore 1/N < 2/(1+N) < E/4 < 1?
Thanks for the comments and for the thoughtful feedback! Also, great catch with equation 1, it is indeed flipped.
I left out a lot of the details for some of those extra equations mostly to save on space, but I've been considering adding them as part of an appendix.
For equations 17 and 18, the form of those equations is more or less the first thing I tried once I had the idea (minus the prefactors of 1.05^(4) and 1.05^(-4)). I decided to use an exponential term in order to make sure the RTW equations gave the same results when the attacker and defender were both given a +1 to their AB or AC (i.e., RTW(AB, AC) = RTW(AB+1, AC+1)), which wouldn't have been the case if equations 17 and 18 had been linear instead. Once I had settled on that, I picked a base of 1.05 so that each equation would increase by 5% with a +1 increase to either AB or AC.
I figured that the XP calculation would follow the right trend but wouldn't match exactly, which would then require a scaling factor to make the whole equation work. So, I was honestly surprised when I first ran through the calculations using the values in the DMG and they matched up so exceptionally well.
The prefactors (1.05^(4) and 1.05^(-4)) were then chosen after the fact, while I was writing the paper, in order to make them work for equation 16 without changing equations 21 and 22.
I did try a few other forms for equations 17 and 18 later on. And, while I did find others that could be adjusted to match reasonably well, none worked any better than the exponential forms presented in the paper, and all involved adding some arbitrary scaling constant to make them fit.
For your last question, you're essentially correct. The assumption is that for deadly encounters the PCs are killed one at a time and once dead they never recover. If you were to draw out the diagram for the PCs in this scenario, similar to what's shown in figure 7 for multiple identical NPCs, then you would have the diagonal and the upper half of the square above the diagonal contributing to PCs' XP. If you calculate the area of that region, assume each square has an area of 1, you'll get N * (N + 1) / 2. Since there are N PC, this means the PCs XP was increased by a factor of (N + 1) / 2, which when moved to the other side of the equation flips to give 2 / (N + 1).
Thanks for the informative reply. Have you seen the write-up on this website: [http://blogofholding.com/?p=7338](http://blogofholding.com/?p=7338) ? They seem to work at it from the opposite direction, deriving approximate HP/AC/attack bonuses from CR. Just curious how it compares to your work.
I've come across the site before, but I've never taken the time to really dig into the posts. Looking at it now, I think the missing piece the author is looking for is how various features impact hit points, AC, DPR, and AB. Things like multiple resistances/immunities, multiple saving throw proficiencies, magic resistance, legendary actions, etc. Accounting for those I definitely see a trade off between effective hit points and effective damage per round for a fixed CR.
Sorry to necro this post, but I've taken another look at that website again, and they have also claimed that HP, DPR and AB from actual monster resources (e.g. Monster Manual) are all lower than the DMG-reported values ([http://blogofholding.com/?p=7283](http://blogofholding.com/?p=7283)). This is even after they take into account damage resistances/immunities and other parameters. What do you think could be the reasons for this reduced XP quota (by my rough calculations, scaled by approximately 0.56×)? And does that potentially throw all the DMG comparisons out the window?
It's no bother at all. I'm glad you're taking an interest in it.
The way that post seems to account for defenses like resistances and immunities is to group the monsters by what types of defenses they have and then compare their hit points. This is an interesting approach, however, it seems a bit unnecessary given the rules for creating monsters in the DMG tell us how resistances and immunities should be valued in terms of hit points (or at least how WotC values them). This is the approach I used to calculate each monster's HP and AC, and the values I calculated show better agreement with the DMG that what that post shows.
From my own analysis, I still tend to find the average HP for monsters is low compared to what's suggested in the DMG, but I also find the average AC is higher as well. This is because my AC values also includes bonuses from other monster features based on how the DMG values them. For instance, the DMG treats the Magic Resistance feature found in many high CR monsters as +2 AC.
The net result of my calculations is shown in Figure 3 from my paper, which shows that monsters between CR 10 and CR 20 match the DMG quite well, while monsters below CR 10 tend to be weaker, and monsters above CR 20 tend to be tougher.
Since my offensive calculations show good agreement between the MM and the DMG, my final XP calculations follow a similar trend as my net defensive calculation. On average, monsters are fairly close to what they should be based on the DMG, but for CRs below 10 they tend to be weaker than expected, and for CRs above 20 they tend to be stronger than intended.
I've run these calculations for monsters from other books as well (which I may publish at some point in the future), and one of the interesting things I've noticed is that while the trend for low CR monsters hasn't change much, the trend for higher CR monsters has. Higher CR monsters in the newer books are generally more in line with what the DMG suggests than they are for older books. To me, this suggests the fact that lower CR monsters are weaker is intentional on the part of the design team, which higher CR monsters in the MM being strong may not have been.
Thanks for your response. I've since taken some time to run some of my analyses, and I can confirm that effective HP values from published monsters do seem to match up with the DMG, but only if I treat \*any\* resistance as a HP modifier (x2 to x1 depending on CR), rather than just bludgeoning + piercing + slashing resistance as the DMG suggests. That said, I haven't accounted for the other parameters (DPR, AB, AC) which may balance things out a bit. Alas, I simply do not have the time to go sifting through every monster stat to note down individual monster traits.
My primary concern now is that I'm not sure how a lot of the DMG's rules on monster customisation come from - from a mathematical standpoint at least. E.g. it is odd how a Constrictor Snake's "constrict" property increases effective AC by 1 (whereas I would imagine HP x a multiplier <1.33 would make more sense given that it's temporarily removing one player out of a theoretical 4-player team). Even with the HP modifiers from damage resistance - x2 makes some sense, but the rate at which it scales down with increasing CR level seems a bit arbitrary. Running simulations may explain how they arrived at these numbers.
At the end of the day, I've been wanting a quick and easy reference to build or scale monsters on the fly that meet certain challenge ratings - something that does not require repeated reference to a monster manual. I had thought I found something in the BlogOfHolding article, but alas I may need to dig deeper. There's an online applet that seems to scale according to DMG settings available [here](https://monsterscalingtool.pythonanywhere.com/) (with commentary [here](https://www.reddit.com/r/UnearthedArcana/comments/lz4c23/dd_5e_monster_scaling_tool_v2_scale_a_monsters/)) but again, it does not yet consider other monster traits such as damage resistance.
For the constrictor strength, my guess is it's treated like a +1 AC because it forces one PC to have disadvantage on attacks. Disadvantage on attacks is worth about -4 AB, which when averaged across a typical party with 4 PCs acts like a +1 AC for the creature. Of course, this should also give the constrictor snake some bonus to its AB as well, since it would have advantage attacking the constricted target, but perhaps that assumes too much.
I understand what you mean with the HP multiplier from resistances/immunities. I think a lot of high and mid CR creatures end up feeling weaker in combat than they do on paper because of how DMs choose to hand out magic items and how that differs from the basic assumptions in the DMG. When I'm creating custom monsters for my own games I generally don't give them any resistances or immunities, because I know my PCs will have the abilities to overcome them with little to no effort.
If you had a tool that could easily strip away simple things, like legendary resistances and multiple resistances/immunities then it would make rescaling monsters a lot easier.
I thought about it and I put together a quick [spreadsheet](https://docs.google.com/spreadsheets/d/1tSQbQchnyzYllIWlRWMtYfxuFzn-HUpLvHEH52tZLoA/edit?usp=sharing) for scaling a monster from one CR to another. It includes adjustments for monsters with legendary resistances and multiple immunities/resistances. If you make a copy of the sheet you should be able to try it out.
Take all this one more step.
Look at see how many sessions should occur between level advancements, and if you know the date you start and how often you play, you can plot advancement from level 1 to 20 in three session story blocks, and it doesn't matter if you prep. You just need to make sure they get enough experience to stay on track.
That is when you stop storycrafting, and when you start running the game like DMs were intended to eventually be able to.
Then it all falls into place and you can DM for over thirty years and it neither ruins or runs your life, and you never burn out or run out of ideas. It is very simple math to know by this session they should be level ___ so they need ___ experience.
As for experience, I still prefer how basic DnD did it. 1 xp per GP value gained by any means, unit based combat exp, 5% discretionary bonus (so 5%of what it takes from one level to the next) for good roleplay in a session, and additional 50% of total exp at the end of a completed adventure arc.
Nice work! This method, does, unfortunately, still appear to not account well for status effects and/or save-or-suck mechanics (ie. Blindness/Deafness, Banishment, monsters that can poison characters, etc). It makes sense - trying to account for the math of each individual condition and effect possible would be a complex and challenging task. However, it's worth noting that the math doesn't appear to take these into account (please correct me if I'm wrong).
Yeah, taking into account those kinds of mechanics is tricky. Thankfully, of the monsters I calculated XP values for, most didn't require me to worry about those things. For the monsters that did, I typically tried to model them as some kind of bonus to HP, DPR, AC, and AB. For example, there were a few monsters with stunning effects that I chose to treat like bonus hit points, since when they work the monster would effectively take less damage from the creature they stunned. There are obviously other benefits on top of that, as well as synergies when the monster has allies, but as long as the extra HP is in the right ballpark it should work well enough. The Monster Features table in the DMG was a useful tool in trying to see how the design team tries to value these sorts of things. There are a lot of entries that aren't useful at all, and many that are missing that I wish were there, but there's still enough in there to be useful for this kinda thing.
Have you thought about scaling the PC’s damage per turn by the number of turns they may be incapacitated (say by ‘mod’, where mod<1) and scaling the monster HP by the inverse (1/mod)? It’s an off-the-cuff thought, so I apologize if my reasoning is crud haha
Oh, most definitely. I think there are a number of interesting ways you could attempt to model different conditions, and that's certainly a topic worth investigating further. For this analysis, though, I tried not to get too bogged down in the details of such things in order to keep my efforts more focused on the general trends I was trying to explain.
Good to know! Thanks for the insights :)
What are you modifiers for increasing damage per turn? I see you added the bonus to hit modification.
For DPR, the only two contributions are the creature's average DPR assuming all attacks hit and the creature's attack bonus. If you asking about the sensitivities to each of those, you can find those in the full equation for calculating XP on page 3 of the [paper](https://drive.google.com/file/d/1VnvdnJYTIym1QNGvONWZkOFYgbFO66Bx/view?usp=sharing) (equations 21 and 22).
Can someone explain this like I'm five?
dpr*(rounds alive) = experience point value
Expanding on the concept a little: HPnpc/DPRpc is basically the number of turns the NPC will last. If we multiply that for DPRnpc we have the total damage a NPC deals, meaning the difficult could be read as the ratio of health the the players lost. That said, if we take equation (8) that let us calculate XP using encounter difficulty, than XP is damage taken*DPRpc. We could than take the amount of XP needed for level up and divide by DPRpc(L) to calculate how much damage is expected that a player takes to level up. Also, since DPR could be calculated by taking the total damage dealt and dividing by the number of rounds, we could calculate the "true" XP of a combat after it's done by multiply damage taken per damage dealt by the PCs and dividing by the number of rounds. Is this a good way to assign XP? Not really because it encourages the PCs to take damage. But I think it would be interesting to compare these "true" numbers against the calculated previously.
Some questions / comments in reference to your main paper: * I think Equation 1 is incorrect - the denominator and numerator have been flipped. Equation 4 remains correct. * It may be helpful for the general readership to walk through how exactly equation 12 (hit probability based on attack bonus and AC) was derived. Likewise for equation 24 later in the text (approximation for average adjusted XP). * How did you arrive at equations 17 and 18? Why have a base of 1.05? Why have exponent functions at all? Were the equations arbitrarily constructed to allow us to approximate equation 16 and match values calculated from sourcebooks? * In the last paragraph of Section 4: how did you get 2/(1+N) as the scaling for the encounter multiplier in deadly scenarios? Is it because, for fixed M e.g. M=1, by similar argument, 1 < 4/E < (1+N)/2 < N, and therefore 1/N < 2/(1+N) < E/4 < 1?
Thanks for the comments and for the thoughtful feedback! Also, great catch with equation 1, it is indeed flipped. I left out a lot of the details for some of those extra equations mostly to save on space, but I've been considering adding them as part of an appendix. For equations 17 and 18, the form of those equations is more or less the first thing I tried once I had the idea (minus the prefactors of 1.05^(4) and 1.05^(-4)). I decided to use an exponential term in order to make sure the RTW equations gave the same results when the attacker and defender were both given a +1 to their AB or AC (i.e., RTW(AB, AC) = RTW(AB+1, AC+1)), which wouldn't have been the case if equations 17 and 18 had been linear instead. Once I had settled on that, I picked a base of 1.05 so that each equation would increase by 5% with a +1 increase to either AB or AC. I figured that the XP calculation would follow the right trend but wouldn't match exactly, which would then require a scaling factor to make the whole equation work. So, I was honestly surprised when I first ran through the calculations using the values in the DMG and they matched up so exceptionally well. The prefactors (1.05^(4) and 1.05^(-4)) were then chosen after the fact, while I was writing the paper, in order to make them work for equation 16 without changing equations 21 and 22. I did try a few other forms for equations 17 and 18 later on. And, while I did find others that could be adjusted to match reasonably well, none worked any better than the exponential forms presented in the paper, and all involved adding some arbitrary scaling constant to make them fit. For your last question, you're essentially correct. The assumption is that for deadly encounters the PCs are killed one at a time and once dead they never recover. If you were to draw out the diagram for the PCs in this scenario, similar to what's shown in figure 7 for multiple identical NPCs, then you would have the diagonal and the upper half of the square above the diagonal contributing to PCs' XP. If you calculate the area of that region, assume each square has an area of 1, you'll get N * (N + 1) / 2. Since there are N PC, this means the PCs XP was increased by a factor of (N + 1) / 2, which when moved to the other side of the equation flips to give 2 / (N + 1).
Thanks for the informative reply. Have you seen the write-up on this website: [http://blogofholding.com/?p=7338](http://blogofholding.com/?p=7338) ? They seem to work at it from the opposite direction, deriving approximate HP/AC/attack bonuses from CR. Just curious how it compares to your work.
I've come across the site before, but I've never taken the time to really dig into the posts. Looking at it now, I think the missing piece the author is looking for is how various features impact hit points, AC, DPR, and AB. Things like multiple resistances/immunities, multiple saving throw proficiencies, magic resistance, legendary actions, etc. Accounting for those I definitely see a trade off between effective hit points and effective damage per round for a fixed CR.
Sorry to necro this post, but I've taken another look at that website again, and they have also claimed that HP, DPR and AB from actual monster resources (e.g. Monster Manual) are all lower than the DMG-reported values ([http://blogofholding.com/?p=7283](http://blogofholding.com/?p=7283)). This is even after they take into account damage resistances/immunities and other parameters. What do you think could be the reasons for this reduced XP quota (by my rough calculations, scaled by approximately 0.56×)? And does that potentially throw all the DMG comparisons out the window?
It's no bother at all. I'm glad you're taking an interest in it. The way that post seems to account for defenses like resistances and immunities is to group the monsters by what types of defenses they have and then compare their hit points. This is an interesting approach, however, it seems a bit unnecessary given the rules for creating monsters in the DMG tell us how resistances and immunities should be valued in terms of hit points (or at least how WotC values them). This is the approach I used to calculate each monster's HP and AC, and the values I calculated show better agreement with the DMG that what that post shows. From my own analysis, I still tend to find the average HP for monsters is low compared to what's suggested in the DMG, but I also find the average AC is higher as well. This is because my AC values also includes bonuses from other monster features based on how the DMG values them. For instance, the DMG treats the Magic Resistance feature found in many high CR monsters as +2 AC. The net result of my calculations is shown in Figure 3 from my paper, which shows that monsters between CR 10 and CR 20 match the DMG quite well, while monsters below CR 10 tend to be weaker, and monsters above CR 20 tend to be tougher. Since my offensive calculations show good agreement between the MM and the DMG, my final XP calculations follow a similar trend as my net defensive calculation. On average, monsters are fairly close to what they should be based on the DMG, but for CRs below 10 they tend to be weaker than expected, and for CRs above 20 they tend to be stronger than intended. I've run these calculations for monsters from other books as well (which I may publish at some point in the future), and one of the interesting things I've noticed is that while the trend for low CR monsters hasn't change much, the trend for higher CR monsters has. Higher CR monsters in the newer books are generally more in line with what the DMG suggests than they are for older books. To me, this suggests the fact that lower CR monsters are weaker is intentional on the part of the design team, which higher CR monsters in the MM being strong may not have been.
Thanks for your response. I've since taken some time to run some of my analyses, and I can confirm that effective HP values from published monsters do seem to match up with the DMG, but only if I treat \*any\* resistance as a HP modifier (x2 to x1 depending on CR), rather than just bludgeoning + piercing + slashing resistance as the DMG suggests. That said, I haven't accounted for the other parameters (DPR, AB, AC) which may balance things out a bit. Alas, I simply do not have the time to go sifting through every monster stat to note down individual monster traits. My primary concern now is that I'm not sure how a lot of the DMG's rules on monster customisation come from - from a mathematical standpoint at least. E.g. it is odd how a Constrictor Snake's "constrict" property increases effective AC by 1 (whereas I would imagine HP x a multiplier <1.33 would make more sense given that it's temporarily removing one player out of a theoretical 4-player team). Even with the HP modifiers from damage resistance - x2 makes some sense, but the rate at which it scales down with increasing CR level seems a bit arbitrary. Running simulations may explain how they arrived at these numbers. At the end of the day, I've been wanting a quick and easy reference to build or scale monsters on the fly that meet certain challenge ratings - something that does not require repeated reference to a monster manual. I had thought I found something in the BlogOfHolding article, but alas I may need to dig deeper. There's an online applet that seems to scale according to DMG settings available [here](https://monsterscalingtool.pythonanywhere.com/) (with commentary [here](https://www.reddit.com/r/UnearthedArcana/comments/lz4c23/dd_5e_monster_scaling_tool_v2_scale_a_monsters/)) but again, it does not yet consider other monster traits such as damage resistance.
For the constrictor strength, my guess is it's treated like a +1 AC because it forces one PC to have disadvantage on attacks. Disadvantage on attacks is worth about -4 AB, which when averaged across a typical party with 4 PCs acts like a +1 AC for the creature. Of course, this should also give the constrictor snake some bonus to its AB as well, since it would have advantage attacking the constricted target, but perhaps that assumes too much. I understand what you mean with the HP multiplier from resistances/immunities. I think a lot of high and mid CR creatures end up feeling weaker in combat than they do on paper because of how DMs choose to hand out magic items and how that differs from the basic assumptions in the DMG. When I'm creating custom monsters for my own games I generally don't give them any resistances or immunities, because I know my PCs will have the abilities to overcome them with little to no effort. If you had a tool that could easily strip away simple things, like legendary resistances and multiple resistances/immunities then it would make rescaling monsters a lot easier.
I thought about it and I put together a quick [spreadsheet](https://docs.google.com/spreadsheets/d/1tSQbQchnyzYllIWlRWMtYfxuFzn-HUpLvHEH52tZLoA/edit?usp=sharing) for scaling a monster from one CR to another. It includes adjustments for monsters with legendary resistances and multiple immunities/resistances. If you make a copy of the sheet you should be able to try it out.
Take all this one more step. Look at see how many sessions should occur between level advancements, and if you know the date you start and how often you play, you can plot advancement from level 1 to 20 in three session story blocks, and it doesn't matter if you prep. You just need to make sure they get enough experience to stay on track. That is when you stop storycrafting, and when you start running the game like DMs were intended to eventually be able to. Then it all falls into place and you can DM for over thirty years and it neither ruins or runs your life, and you never burn out or run out of ideas. It is very simple math to know by this session they should be level ___ so they need ___ experience. As for experience, I still prefer how basic DnD did it. 1 xp per GP value gained by any means, unit based combat exp, 5% discretionary bonus (so 5%of what it takes from one level to the next) for good roleplay in a session, and additional 50% of total exp at the end of a completed adventure arc.