The Mathematical Model of the d20 System

[Johnny Angel]

I just picked up the Grim Tales Game Mastering .pdf, which details a method for judging encounter difficulty that appears to reverse-engineer the EL system.

The idea is that the power of a creature is not its CR, but actually the square of that. Then the EL of an encounter is one plus twice the base 2 logarithm of the sum of the side's power levels. Perhaps someone can explain to me how this works, because with my weak math fu I'm getting that a CR 2 monster has an EL of 5. In any case, this does seem to characterize the model that WotC was going for, only they have it dumbed down to require only basic arithmetic operations. What I don't get is how they figure that for example the power progression of class levels is an exponential curve.

From another thread, I get that the model is something like Power = Defense x Offense. So the model seems to assume that as you level you get an amount of extra ability that amounts to an additional point of defensive ability and an additional point of offensive ability. So, defensive is hit points and saves and offensive is BAB and iterative attacks? This is all a wildly approximate model, like in the joke about the mathematician whose system for winning horseraces begins with "Assume the horses are identical and spherical" is it not?

 

[Achan hiArusa]

So its CR² = 1 + 2 log₂Sum(party character levels), which using the change of base formula (so it can be done on a calculator) is:

CR² = 1 + 2 log Sum(PCL)/log 2

So a CR 2 monster is

4 = 1 + 2 log Sum (PCL)/log 2
3 = 2 log Sum (PCL)/log 2 (Subtract one from both sides)
1.5 = log Sum (PCL)/log 2 (Divide both sides by two)
0.45 =; log Sum (PCL) (multiply both sides by log 2 =; 0.303)
3 =; PCL (rounded up) (raise both sides to 10^x power. Its actually closer to 2.8, so two real PCs and a bard )

 

[Wulf Ratbane]

I just picked up the Grim Tales Game Mastering .pdf, which details a method for judging encounter difficulty that appears to reverse-engineer the EL system.

The idea is that the power of a creature is not its CR, but actually the square of that. Then the EL of an encounter is one plus twice the base 2 logarithm of the sum of the side's power levels. Perhaps someone can explain to me how this works, because with my weak math fu I'm getting that a CR 2 monster has an EL of 5.

Hey Johnny--

I've done a whole lot of work since that PDF. I'll just say for now that I think it's incomplete and I'm now trying to wrangle Lanchester's Square Law into the whole encounter design model. It's not easy...

The PDF will serve you just fine in the meantime, but you do need to understand that the meaning of EL5 in the PDF is not the same as EL5 as WotC uses the term. You need to be able to throw out your old knowledge before applying the new. "EL5" might as well mean "Orange" for all its applicability to the term as WotC used it.

In any case, this does seem to characterize the model that WotC was going for, only they have it dumbed down to require only basic arithmetic operations. What I don't get is how they figure that for example the power progression of class levels is an exponential curve.

They = WotC, or They = Me (Bad Axe)?

For the record, the curve is quadratic (to the power of 2), not (strictly speaking) exponential.

From another thread, I get that the model is something like Power = Defense x Offense. So the model seems to assume that as you level you get an amount of extra ability that amounts to an additional point of defensive ability and an additional point of offensive ability.

That is correct.

What you're looking for is the amount of Offense that the creature can put out per action, multiplied by the Defense (which approximates the number of actions you expect it to stick around for).

The simplest and roughest model here (or as you aptly put it, the "identical and spherical horse theory"-- I like that, I'll keep using it!) is Hit Points x Average Attack Damage.

Here's a fun little process you might want to try. Pick a monster out of the SRD at random-- but one that doesn't have a lot of unusual abilities. Something straightforward. Multiply its hit points times its main attack average damage. Now divide that result by 25. Now take the square root of that. You'll be in the ballpark of its CR.

If you used a typical brute monster and your result ended up higher than expected, there's a good chance that monster is a lot rougher than expected. The CR3 Ogre weighs in at a little over CR4, which I think is probably more accurate.

The CR5 Troll, meanwhile, weighs in at an expected 4.89... unless you start taking his full attack routine and Rend into account. Suddenly it jumps way up, and the troll gets a lot tougher-- as anyone who has ever stood toe to toe with a troll for a full attack can attest.

Obviously, a lot of things can make your horses less spherical. This is where the "Art vs. Science" of monster design starts to play with our mathematical expectations.

Higher than expected AC (on the order of +4 or +5 points higher than expected) is going to give the creature a lot more staying power-- akin to doubling its hit points. An area of effect attack that can catch the whole party in one action is like multiplying the creature's damage output by the size of the party. And a Save-or-Suck attack that can take one PC out of the fight in one action is as good as doubling that creature's combat power.

It is a wildly approximate model based on the average of monster abilities (statistical analysis thanks to Ryan Stoughton). So while it holds up very well across the average-- identical and spherical!-- I certainly wouldn't bet it to WIN.

Yet.

I'm working on it, though.

 

[Scurvy_Platypus]

Well, let us know when you get it straight.

I'm sure I'm not the only person that's curious to see an update to the monster system you've got for GT. Heck, I have the book and bought the pdfs just for the spreadsheets.

 

[Johnny Angel]

They = WotC, or They = Me (Bad Axe)?

Either, really, since I'm assuming that you (Bad Axe) are making explicit what WotC is making obscure. The model seems roughly likely just on the basis of some trials I've run in Excel generating values for each level's actual Defense and Offense values between 0.5 and 1.5 (just as a rough idea of how the values could vary from the ideal 1). What I found is that the results conformed closely enough to the y=x^2 curve in random generations that I concluded that reasonable deviations in the values of these hypothetical quantities evened out over time. I may be making a mistake in assumptions, but I'm getting the notion that deviation from the ideal 1 defense gain and 1 offense gain per CR kind of works itself out through the magic of the process.

It also seems to turn out that deviations from the ideal get averaged out as you go up levels, which makes me wonder why the "sweet spot" disappears after about 13th level. Seems like the averaging effect would make the risk vs. reward calculations less wonky at higher levels.

For the record, the curve is quadratic (to the power of 2), not (strictly speaking) exponential.

Somehow, I got diverted from my physics degree into a degree in philosophy, but I am dusting off a lot of mental furniture.

"identical and spherical horse theory"-- I like that, I'll keep using it!)

The version of the original joke I heard went something like this:
A gambler approached a mathematics professor looking for a way to beat the horse races. The mathematician worked on it for several months, with the gambler constantly calling for updates and preliminary results, but the mathematician wouldn't give any information until his work was done. Finally, he called the gambler and said, "I have worked out the system you wanted. The outcomes are predictable in the aggregate, and we will both be rich."

The gambeler said, "Fantastic! Tell me, how does it work?"

The mathematician said, "First of all, assume the horses are spherical..."

Later I heard the version that ends with the horses being "identical and spherical" but I'm not sure which I prefer. But it illustrates in a way that comes across intuitively the problem of mathematical models. And it also arises in less rigorous models. In a graduate course in the conflict between science and religion, I found that attempts to reconcile the two tended to assume that everybody's idea of God was, as it were, identical and spherical.

Here's a fun little process you might want to try. Pick a monster out of the SRD at random-- but one that doesn't have a lot of unusual abilities. Something straightforward. Multiply its hit points times its main attack average damage. Now divide that result by 25. Now take the square root of that. You'll be in the ballpark of its CR.

Yet, it baffles me that this is from people who, when this was TSR, approved the psionics system that came with the Dark Sun Boxed Set version 2, in which your power points were also your psionic hit points, but attacking damaged you more quickly on average then it did your enemy, and defending did more damaqge to you than not defending. After that saw print, can it really be that these people understand math better than I do?

 

[Wulf Ratbane]

It also seems to turn out that deviations from the ideal get averaged out as you go up levels, which makes me wonder why the "sweet spot" disappears after about 13th level. Seems like the averaging effect would make the risk vs. reward calculations less wonky at higher levels.

I have two theories, and they're actually interrelated.

1) The d20 is only so large. When you reach the point that the bonuses overwhelm any possible random result, the game doesn't feel right.

2) The higher level spells are populated with far more "Absolutes." (Save or Die spells, Immovable Objects, and Irresistible Forces.) If you ran a game without spellcasters of any kind, your sweet spot would last longer.

For the record, I think the Sweet Spot ends right around 10th. Fifth level spells are a sea change. If your campaign was 13th level before you started to feel it, well... I'd say you were just cruising on post-orgasmic vibes for about 3 levels.

You just needed to come down off the climactic high before you could see that coyote for what she was.

I had quite a bit more to say about the Sweet Spot in the thread of the same name. If you can do a search for threads with title Sweet Spot and/or Quadratic Problem, with Wulf Ratbane, you should dig up some interesting... philosophy.

I've been told by folks who have played 4e that much of it feels like it came out of the Sweet Spot thread...

Yet, it baffles me that this is from people who, when this was TSR, approved the psionics system that came with the Dark Sun Boxed Set version 2... After that saw print, can it really be that these people understand math better than I do?

I am as guilty as anyone of letting my creative juices run wild with my better judgment. It happens.

I have been intermittently chiseling away at this problem for a while now and Gary's passing gave me the opportunity to dig out my old 1e books, as well as the Rules Compendium. And in addition to some other interesting things it revealed (for example, if you take a look at the things they considered "extraordinary abilities," almost all of them would qualify as asymmetric game-breakers) it mainly occurred to me: I think it must be almost literally impossible that Gary Gygax was unfamiliar with Lanchester's Square Law. We are talking about a game born from military simulations. It's unthinkable that it didn't occur to Gary-- I just need to get a better understanding of how he dealt with it. (I would guess, mainly, by throwing level appropriate, equitable-power creatures at the party in appropriate numbers.)

As for the SQRT (HP*damage/25) = CR, I think it is just an observation of emergence. I don't think there's a deliberate design behind that formula. It's definitely not going to work for every creature, and even when it hews "closely" to that model, the results are still not exact. The formula is going to take large numbers and spit out a number across such a small range that you're really aiming at some pretty big targets.

I am still playing with the numbers there but in the meantime I am not sure it accounts for much more than a parlor trick.

 

[Wulf Ratbane]

Well, let us know when you get it straight.

Don't worry, you'll know.

 

[BryonD]

1) The d20 is only so large. When you reach the point that the bonuses overwhelm any possible random result, the game doesn't feel right.

Just a minor comment: I don't think that is a problem with D20. As long as the targets and bonuses are roughly ok, it works.

But what you are talking about most certainly happens with other rolls. For example, when the high level barbarian hits for 1d12+38, the d12 roll isn't all that exciting.

 

[Wulf Ratbane]

Just a minor comment: I don't think that is a problem with D20. As long as the targets and bonuses are roughly ok, it works.

Clarification:

When the fighter hits on anything but a 1, the d20 is irrelevant.

When the wizard rolls a 20 and his result is the same as the fighter who rolls a 1, the d20 is irrelevant.

 

[BryonD]

Clarification:

When the fighter hits on anything but a 1, the d20 is irrelevant.

When the wizard rolls a 20 and his result is the same as the fighter who rolls a 1, the d20 is irrelevant.

I disagree that these are problematic.
I've never had a problem with the fighters having things easy in fights.
Easily hitting the hulking beast that is literally the broad side of a barn, only with giant teeth and a boatload of HP is a feature, not a bug.
Swatting down mooks by the dozen is a feature, not a bug.
Making the wizard's eyes get as big as saucers and go "Damn! That was impressive!! But can you do THIS!!??!!??" is a feature, not a bug.
Throwing in foes with an AC that requires a decent roll from the fighter and forces the wizard to come up with an idea better than hit it with a stick is not a problem whenever desired.
I like all of the above and would not look well on a rule that infringed upon them.

Edit: to bring it back to the point: The D20 is only irrelevant when the DM lets it be irrelevant, and in those times simply being irrelevant is neither a problem nor a source of the game not feeling right.