[Math] WotC Challenge Ratings

[Cheiromancer]

The contretemps with Upper_Krust about the "new" formula relating CR to EL has drawn me back to the basics; the official rules for challenge ratings and encounter levels. Specifically I'm been trying to figure out the mathematical basis behind Table 2-6: Experience Point Awards (Single Monster) (DMG p. 38) and Table 3-1: Encounter Numbers (DMG p. 49).

It seems that there is a variable that can be calculated by figuring out how many creatures of various CRs there are in an encounter, and that this number can be used to determine the EL, and also average experience. Call this number M (for monster). It is defined as 2^(CR/2). Or you can use the following table (the M values for odd numbered CRs are about 6% too high):

CR     M
---------
1       1.5
2       2
3       3
4       4
5       6
6       8
7      12
8      16
9      24
10     32
11     48
12     64
13     96
14    128
15    192
16    256
17    384
18    512
19    768
20   1024

Given an encounter with monsters of various CRs, add up the M values, and find a CR whose M value is close; that is the EL of the encounter.

For example, if your encounter has 6 CR4 creatures in it, look up the M value of CR4 (which is 4) and multiply by 6 to get 24. 24 corresponds to a CR of 9, so this is an EL 9 encounter.

To work out experience (like in table 2-6) divide the encounter's M value with the character's M value, and multiply by 300 times the character level. For example, if an 11th level character (M=48) in a four-member party helps defeat an encounter whose CR is 13 (M=96), divide 96 by 48 to get 2, then multiply by 300 to get 600, and multiply by 11 to get 6,600. Which is the number given by table 2-6.

I don't claim any understanding of what M means, but if you manipulate it in the indicated ways, all the tables come out right. (Well, except for fractional CR- but in an epic game I'm not going to worry about that. Just group the fractions until you get a complete CR.)

---

Alright. Now here's a major puzzle: if four identical characters (10th level fighter/rogues, say, conveniently named One, Two, Three and Four) are attacked by Five, they'll defeat their attacker without difficulty. So what CR (and M value) should each of these characters have? It doesn't seem like a moderate encounter (25% resources used), does it?

It's easy to see that, terrain permitting, the party will get 4 attacks in on Five for every attack that Five makes. In other words, if One and Five would take eight rounds to reduce each other to zero hit points, the party of One to Four will defeat Five in two rounds; in this time Five will have reduced One to 75% of his starting hit points. Five will have lost all his hit points and the party will have lost 1/16 of their total hit points. The 4:1 advantage has become a 16:1 advantage. (I'm assuming fine granularity in attacks and average damage, etc. There would be much more variance in a real combat, and initiatives etc. would play a role.)

A similar argument will establish that a five person party will have a 25:1 advantage over an identical attacker, and, generally, a n person party will have an n²:1 advantage.

So what PC level attacker would a four person party have a 4:1 advantage against? One that it will lose 25% of its hit points against? I think it would have to be one that is 2 levels higher than the party- in other words, the CR of a PC is level-2.

Which means that the M value for a 10th level character is 16. Four such characters would have an M value of 64; so a 10th level party is equivalent to a CR 12 monster: EL +2 (very difficult). By reasons of symmetry, such a challenge should have a 50/50 chance of being a TPK.

Incidentally, the fraction of resources used can be worked out by dividing the M scores and squaring. If your 10th level party has an M score of 64 and you fight a creature whose M score is 48, you should use up (48/64)² of your resources; 56% of them. A tough fight, but winnable.

All this assumes, of course, that the tables in the DMG are correct. Which is doubtful. If M is not an exponential curve, but something else, something that an exponential curve approximates, then these various numbers would have to be adjusted.

More later.

 

[Cheiromancer]

Incidentally, the fraction of resources used can be worked out by dividing the M scores and squaring. If your 10th level party has an M score of 64 and you fight a creature whose M score is 48, you should use up (48/64)² of your resources; 56% of them. A tough fight, but winnable.

This rule has very dramatic implications when the CRs are a long ways apart. According to the chart, if a CR8 creature squares off with a CR18 creature, the CR8 creature will end up dead and the CR18 creature will have used up (16/512)² = 1/1024 of its resources.

E.g. for CR 18 take a Nightcrawler with 212 hp, AC 35, DR 15/silver and magic, and two attacks: +29 bite (4d6+21/19-20, avg 35) and +24 sting (2d8+11/19-20 plus poison, avg 20). Plus haste, which would give it a second bite. If it hits with its bite it can swallow a Huge or smaller creature whole. Grapple +45

For CR 8 take an Athach with 133 hp, AC 20, and four attacks: Morningstar +12/+7 melee (3d6+8), and 2 morningstars +12 melee (3d6+4), and bite +12 melee (2d8+4 plus poison). Grapple +26.

Given that an Athach can only hit a Nightcrawler on a 20, it would make sense for it to Power Attack for 10 (its BAB). However, I'm not sure how it could best use its 3 hands to maximize its damage. In any event, the Nightcrawler could also power attack for 10 and still hit except on a 1. The damage reduction of the Nightcrawler is a formidable obstacle to the Athach's attacks, and the swallow whole ability is difficult to overcome; when it swallows a creature its energy drain ability gives it 5 hp per round. The Nightcrawler can only hold 2 Huge creatures (like an Athach) in its gullet, though. I wonder if, once the swallowed opponent is dead, it can regurgitate it and swallow another one?

Anyway, with just the standard tactics, an Athach will hit a Nightcrawler once every five rounds (because of DR the average damage is a little over 2 hp), while the Nightcrawler will kill an Athach once a round or so (with the extra attack from haste, and using cleave when an Athach is dropped). It will take almost 500 rounds for a line of Athachs to kill a Nightcrawler, in which time the Nightcrawler will have disposed of more than 400 Athachs. That's without the summoning of incorporeal undead, which the Athachs have no defense against.

A different pair of monsters might yield different results; there are some scissor-paper-stone combos where, for example, one creature can't harm another (a flier vs non-flier, an incorporeal creature vs one that doesn't use magic, etc.). But the Athach/Nightcrawler example illustrates how dramatic the combat disparities can be between monsters in D&D.

On the other hand, when the extremes of the d20 play less of a role, the WotC system is less accurate. A playtest between 3 young adult brass dragons (CR 10 each) and an old white (CR 15) found them to be very nearly equal, while the M-scale (and table 3-1) suggests that you'd need 6. The cubic formula was more accurate in that case.

 

[Cheiromancer]

I'll say it here, so it isn't buried in the post somewhere: if a PC's CR is two less than the character's level in the WotC exponential system, this translates to being 63% of its level in a cubic system. And in place of M in the first post, use CR³ divided by 30. (Or 31.25, if you want it to match exactly at CR 10. But it doesn't matter; it's only the ratio of M values that are important).

So anyway, if AC and attack bonuses both scale at +1/CR (or faster!) then it is very easy to see how a low CR creature can be utterly demolished by a higher CR creature, who is demolished in turn by one higher yet. Having a 20 to 1 advantage (missing only on a 1 vs hitting only on a 20) multiplies any other advantages the more powerful creature might have; hit points, damage dealt, number of attacks, etc..

It doesn't only have to be combat stats; spell resistance and caster level exhibit the same mechanic, and (to a lesser extent) save DC and saving throw bonus. Abilities tied directly to hit dice (the blasphemy mechanic is the most obvious example) also reinforce this mechanic.

So what? Well, if you want to design monsters so that the WotC paradigm continues to function, you'll have to use these d20 mechanical factors; SR, AC, saves and attacks. If you don't, then you'll have to let them slip aside. Have it so that AC falls behind attacks, say, or saves increase faster than save DCs. Or something- anything that makes that 20:1 advantage disappear.

If damage and hit points are roughly proportional to CR (and hitting isn't a problem) then the combat effectiveness of a creature will scale according to the square of its CR. Still, a higher CR creature will probably have a bigger attack bonus than it knows what to do with; some power attack will come into play. Or other kinds of special abilities. These will probably work out to be a third linear factor, and that would yield a cubic progression.

I'll have to think about what rate these things will have to change with. 2/3 would probably work (or 0.63, the cube root of 1/2). So AC would increase more slowly than attacks, save DCs would increase more slowly than saving throw bonuses. Caster level more slowly than SR.

Spells are for buffs, healing, transportation, dealing with mooks and lackeys - but there are too many save or die effects for the system to favor offense. Offensive combat options, though- that's a different story. Fighters should be able to hit an equal CR challenge on a 2, at least with their first attack. And, at high enough levels, with the second and subsequent attacks too. If the critter's CR is more than 50% of their own, however, this should not be so straightforward.

That's probably doable.

Or maybe one could dump the cubic system and try to stay within the WotC framework. Make it so that creatures do double in power every 2 CR. Make it so that a 40th level character outstrips a 20th level character in the same way that a 20th level character outstrips a 1st level character. 1000 times as powerful- not merely 8 times as powerful, as is presupposed by the mathematics underlying UK's system. (Let alone the x4 multiple of the quadratic design mode). To do this you'll have to have AC, attacks, SR and caster level all increase at a rate of at least 1/2 levels. So the 40th level character can hit the 20th level character except on a 1, but the 20th level character needs a 20. And similarly for saves, SR and so on.

I don't think epic characters and monsters quite follow this progression, do they? For it really to work then the typical balanced combat PC would hit its own AC 50% of the time; a spellcaster PC would have a 50% chance of penetrating their own SR, and a 50% chance of making the save if they did so. And similarly for monsters. I don't recall anymore what the "self-attack" percentages are for epic characters and monsters, but I think they are more off-center than that. And if the self-attack percentages are off-center, then we're probably talking about a cubic progression. Maybe only quadratic, but likely a higher degree than that.

But if we did wrestle with treasure, feats and class abilities so that the self-attack numbers were centered, then you could extend the exponential progression out quite a way. A 40th level exponential character would be equivalent to a 200th level cubic character in Krusty's system. And a E-30th level would be like a C-63rd level character. You could certainly introduce powerful feats a lot sooner!

I probably got some of those improvement rates wrong. I think caster level is easier to increase than SR, isn't it? That'll probably cause problems. I'm just thinking out loud, really- but when Sep and I get going on this again, it will probably be a helpful reference.

Maybe to figure out baseline stats for a monster- and if one of its critical stats (AC, SR, whatever) is out of whack, then another stat can be moved to compensate. Same for characters- if they spend feats or treasure to improve one critical stat, another should be weaker than the baseline.

[edit] When it comes time to work out the baseline stats for monsters, rycanada's DM's Best Friend Table definitely deserves attention. Regularizing the table so it follows certain theoretical trendlines would be nice. The stat block of monsters could be distilled down (see thread) and any exceptions noted. Rakshasas would have unusually high SR for their CR, for example.

 

[Cheiromancer]

I've been thinking of how particular kinds of modifications affect a creature's CR. For instance, what would it involve if you doubled its hit points? Assuming it is defeated in combat and is defeated by being reduced to exactly -1 hp, then doubling its hit points increases its CR by +1. If it is defeated by non-combat means (a save or die spell, say) or by two blows with (massive overkill on the second blow) then it isn't quite a +1. A 1/day save reroll option would help vs save or die spells, but not much vs the overkill. Not quite +1 CR.

If a creature doubles its actions (either via a choker's quickness ability, or something) then this would also add +1 CR. A little more, actually, since if it wins initiative it could get some instakills off. But assuming a combat that would last long enough so that the opposition gets half the actions the attacker does, it would be +1. Stipulating that the second set of actions happens at the end of a round (but not during a surprise round) would help ensure this condition is met.

Combining both these modifications should add about +2 to a creature's CR. Which suggests a handy way of turning a PC into an appropriate challenge for a party of the same level; double the character's hit points and actions. I'm assuming the challenge will be a combat challenge, and so the modified PC should be a combat PC- not the party skill monkey and trap springer.

I imagine this would be a quick way of testing this theory- and if bears out, it would also extend the usability of test creatures. So if you have a CR 10 monster that is your gold standard, you could double its actions and hit points and use it as a CR 12 gold standard monster.

Note: I *think* that the creature might need a 1/day reroll saves feature for it to be exactly +2 CR. That's presuming that the codicil about its extra actions coming at the end of a normal (not surprise) round are included. But I'm not really sure. Maybe a kind of "slippery saves" mechanic would be better? (Like slippery mind, but applicable to all saves)

[edit]I think a 1/day reroll *and* +2 to all saves would be just about perfect for a +2 CR.

Also, I reposted an edited version of the first post in the rules forum: CR and EL Calculations. No comments as of this posting, and it will soon be off the first page. Oh well.

I ran down an observation that gestalt characters are (in the level 5 to 15 range) about a LA+2 class. And also that Bo9S characters play much like gestalt characters; they are rather more powerful than characters out of the PHB. Perhaps that means that a party could have straight Bo9S characters as well as gestalt PHB characters.

Anyway, just a few thoughts. Not worth the *bump*, but a stealthy edit is a different story.

 

[Cheiromancer]

Note that the previous post has been edited.

The discussion in the OotS 439 thread made me realize that the xp mechanic in the first post is not what is given by the DMG. I was treating a multi-creature encounter as if it were a creature whose CR was the EL of the whole encounter. And it isn't. The special rule that gives no xp for creatures more than 8 CR less than you is also not reflected in the first post.

Oh well.

Anyway, I was wondering about converting CR to ECL. I think that a CR 8 creature will be a CR 10 character, just based on how combat works. But that presumes that the monster will not gain any extra equipment.

In UK's system, the conversion from ECL to CR is accomplished by multiplying by 2/3. In the M system it is represented by a -2 CR. In UK's system the transition from PC equipment to negligible equipment (a few percent of recommended wealth) would be handled by a further multiplication by 2/3. In the M system I'd guess that it would be a further drop of -2 CR. If the character had NPC wealth it would only be a -1 CR drop.

Working backwards, it would seem that a CR 6 monster would be ECL 10 if it had PC equipment. I wonder if that is the case? A game designer might fudge the LA by 1 or so to ensure that a campaign remains humanocentric, but it would seem to be a good place to start.