The Mathematical Model of the d20 System

[AllisterH]

Obviously the DM is in control of what the PCs face. I assume that the DM sets the budget, and that he's able to go above or below that budget if he chooses.

The caution here is for the DM who thinks he's creating proportionally tougher encounters with a bigger budget, when in fact the award will be increasing at a pace faster than the commensurate risk.

It's not that the PCs are gaining XP faster than the DM wants to-- he's the one setting the XP budget, after all.

But they might be gaining XP at a rate that increases faster than the amount of risk they're actually facing.

As a DM, I don't care if the PCs get twice as much XP if the encounter is actually twice as hard.

I do mind them getting twice as much XP if the encounter is actually only 1.4 times as hard (for example). That's not ideal.

But if you use lower level monsters, wouldn't this not be a problem? For example, if xp doubles every 4 levels, instead of using a monster 4 levels higher, why not just use 2 monsters of the same level?

 

[Wulf Ratbane]

But if you use lower level monsters, wouldn't this not be a problem? For example, if xp doubles every 4 levels, instead of using a monster 4 levels higher, why not just use 2 monsters of the same level?

That's pretty much what I have been saying: When the DM moves the PCs up in "weight class" the fight isn't going to be as difficult as the XP award suggests it is.

The DM should be careful increasing the XP of an encounter, especially if he is moving from many creatures to fewer (more powerful) creatures. It's very tough for creatures to scale up in difficulty fast enough to keep ahead of Lanchester's Square Law.

If you ever had your solo BBEG get punked by a party of 3e PCs, that's why.

And in fact the converse is true: Adding more "weaker" creatures may end up being the more difficult fight, because numbers count for more than individual quality.

The "sweet spot" for encounter design is probably right in the middle: Increase the difficulty of the creatures involved, don't add or subtract creatures. There's no question in my mind that the encounters that feel the "sweetest" are going to be the ones closest to 1 monster per PC.

I'd be ecstatic if 4e actually solves this problem-- completely-- but my suspicion is that they handle it by keeping encounter design inside some very strict parameters. That's cool, too. It's certainly better than 3e.

 

[Mustrum_Ridcully]

That's pretty much what I have been saying: When the DM moves the PCs up in "weight class" the fight isn't going to be as difficult as the XP award suggests it is.

The DM should be careful increasing the XP of an encounter, especially if he is moving from many creatures to fewer (more powerful) creatures. It's very tough for creatures to scale up in difficulty fast enough to keep ahead of Lanchester's Square Law.

If you ever had your solo BBEG get punked by a party of 3e PCs, that's why.

And in fact the converse is true: Adding more "weaker" creatures may end up being the more difficult fight, because numbers count for more than individual quality.

The "sweet spot" for encounter design is probably right in the middle: Increase the difficulty of the creatures involved, don't add or subtract creatures. There's no question in my mind that the encounters that feel the "sweetest" are going to be the ones closest to 1 monster per PC.

I'd be ecstatic if 4e actually solves this problem-- completely-- but my suspicion is that they handle it by keeping encounter design inside some very strict parameters. That's cool, too. It's certainly better than 3e.

What I wonder is about Manchesters Square Law and similar approaches - how well do they really work on the smaller, tactical scale of D&D? Would they work in a dogfight between two jets? Or do they they require at least 30+ units on each side to get some reliance?
The laws might give us a statistic middle, but we're also interested in the deviation - if the deviation is to large, can we actually use them effectively?

(If I understand the Manchesters Square Law correctly, its basic look appropriate, since it assumes that every unit can bring its full strength to bear, which is usually the case for smaller units. But there is still the variance from the random factors)

Assuming they work reliably well on a small scale, how would the encounter design system actually look like? And maybe more importantly, would it look "neat" for the user, or would it require us to use calculators?

I think the Elite/Solo monster categories try to deal with the square increases of powers. But the level scale itself does not try the same. Which means the system is not a perfect fit...

 

[BryonD]

Lanchester's Square Law is a "perfect" mathematical model. As long as the conditions apply. Which is almost never.
But it is almost always close enough to be a very useful tool.

The size of the engagement does not matter. Though the larger the engagement, the better your chances that the conditions do not apply. What that probably means is that not everyone is engaged and your battle is actually a collection of smaller skirmishes, and each skirmish is better modeled by the Square Law than the battle itself.

Reinforcements are also not at all accounted for. To account for them you would have to run the model forward until the first reinforcement arrived and then stop and reset the battle with new initial conditions and repeat that every time a new reinforcement arrived. Not easy.

Another point is that, being a perfect mathematical model, it assumes that every combatant is perfectly engaged continuously until the battles end. Obviously that is never going to be true. However, the formula is square with respect to the number of units and only linear with respect to their fighting capacity. So if you assume that the inefficiency is fairly constant for both sides, then it doesn't matter, it just changes how long the fight will take. If one side is better than the other at bringing its units to bear, then you must account for that tactical improvement in terms of a higher fighting capacity parameter. Which can be hard to quantify.

The bottom line seems to be that it is virtually impossible to find real battles and cleanly model them with the Square Law, but it is easy to show that the influence of this mathematical truth is present, modified by numerous other variables.

Taking it back to D&D, it actually works better for D&D than for reality. That is because in D&D you can quantify fighting capacity. You can state attack bonuses, and average damage, and you can state precise ACs and HP totals. So the fighting capacity is pretty well defined in a quantitative manner. And the expectation that every unit is fully and continuously engaged is “more” true in a typical D&D battle than a real battle. When you have reinforcements it is still a problem, but that is less common in D&D.

Its weaknesses in D&D include initiative order, healing, and odd-ball magic stuff. If a wizard gets initiative and tosses out a fireball, then the presumption of everyone engaged throughout is out the window. For the goblins that get blasted their time of engagement is zero. Healing is pretty much like having reinforcements, though not as bad. That is because the healer is giving up his attacks to restore someone else. So if you have one fighter and one cleric healing him, it kinda fits as if you have two fighters. But not as cleanly. It certainly isn’t what the math assumes. But it isn’t going to send you way off into huge error either. And the magic stuff should be obvious. If a Finger of Death save is failed, then the fighting capacity of a spellcaster can suddenly skyrocket. Dimension Door is going to wreck havoc on the engagement assumptions. You can come up with a list of exceptions here. But again, it doesn’t do as much harm to the final conclusion as you may think. It just makes a blur around the edges. You can not tell exactly how things will end, but who will win will still be the same. These errors are probably going to be less significant than the time the fighter suddenly can’t roll over a 6.

 

[Wulf Ratbane]

What I wonder is about Manchesters Square Law

Lanchester.

and similar approaches - how well do they really work on the smaller, tactical scale of D&D?

The problem with D&D is the number of absolutes and asymmetrics in the system may very well decide the issue: Lanchester's Square Law isn't a sufficient model.

Unless you can quantify the absolutes and asymmetrics so that you have a clear understanding of their contribution to the overall effectiveness of a unit.

Would they work in a dogfight between two jets?

I assume you mean 2-on-2, since a dogfight between two jets (1-on-1) is both Linear and Squared. "1" works out neatly like that on both sides of the equation.

Anyhow IIRC Lanchester developed his ideas at the birth of aerial combat specifically to make those predictions.

Assuming they work reliably well on a small scale, how would the encounter design system actually look like? And maybe more importantly, would it look "neat" for the user, or would it require us to use calculators?

As far as I am concerned, if you can't find a model that works reasonably well without a calculator (although tables are OK from this designer's standpoint) then the gains are negligible. It's far better at that point to keep the encounter constrained within certain boundaries that ensure that the differences the Square Law would inject into the system are kept to a minimum.

However,

• We have been told that the power curve in 4e has been stretched and flattened, and
• 4e removes many absolutes and asymmetric abilities, which should actually make them hew closer to the Lanchester model, and
• the number of creatures in a combat are determined by the XP budget, and
• the XP awards are definitely not smooth or flat,

... I have concerns. A DM faithfully following the system may find himself overpowering the PCs with swarms of lesser creatures, and his "tough fights" with Elites and Solos will feel just the opposite.

Yet the PCs are going to get comparatively less XP for the toughest (swarm) fights and comparatively more XP for the easier (boss) fights, with respect to the actual difficulty they have with the encounter.

I think the Elite/Solo monster categories try to deal with the square increases of powers.

I think so too, and I really like the looks of it so far. Giving Elite and Solo creatures extra actions and "interrupts" means that you can increase their overall effectiveness without having to increase the "statblock" as much.

 

[Wisdom Penalty]

Wulf -

All the finer points and mathematical posturing aside, which system - to you, based upon what you know - seems to better measure the challenge of combat, thereby allowing DMs to better plan encounters for their group: 3E or 4E.

4E? I agree.

So time is better spent looking at what it does well and (if you don't plan on using the system), porting those aspects over to 3.XX inasmuch as possible. Can you "fix" 3E by doing so? Probably not. But you can certainly improve upon it, much as you did with GT.

All the other stuff is just gobildigook.

Wis

 

[Mustrum_Ridcully]

It depends on how "strict" you want to be about the 3E rules.

For example, you could try change the CR/EL and HD guidelines to get some aspects of the 4E into 3E.

A rough idea: Try to extract a level equivalent from monsters, that fits the "average" or typical AC/HP/Attack ratings for a monster. (One could try to assign monster roles like in 4E).
It is very hard to do so, since the variance in 3E is very high. You really have to use a lot of averages, and you can use neither CR or HD as your sole basis, and you don't really know beforehand if a monster might be more like a Minion, Elite or Solo (in as much as its average damage or hit points diverge from the base). Assymetric abilities (Save or Die effects especially) make it hard to do so.

If you ever found such a benchmark, you could assign a (new) CR to it to use for XP rewards using the old tables, but add multiplies based on "weight" (Minion/Elite/Solo status)

Some work is - even if hidden away - probably done. Iron Heroes removes the magical items from character classes. You could use appropriately statted IH characters for each level as your benchmark for monsters. If the average attack and AC/Save is close to a IH character of Level X, the monster might be of that level. Now compare damage x hit points x #attacks to figure out its "weight".

 

[Victim]

I've read an account of 4 first level PCs losing (pretty handily, IIRC) to a solo succubus, so I'm not sure that numbers prove so overwhelming. On the other hand, both of the devil's main tricks are assymetric. The Angel of Valor has Resistances and high defenses (especially when unbloodied), so the cleric is pretty useless offensively and the paladin isn't much better. It's At-Will power gives it two attacks. It does have an area attack, but only as an encounter power. I don't really expect it to go down easily against first level characters either, despite fitting into their encounter XP budget.

Area attacks also tend to diminish the impact of numbers, as was mentioned. Half of the pregen characters do have multitarget powers: Cleave and Passing Strike for the Fighter, the Ranger's daily can hit 2 targets, the wizard has lots of AoE). The cleric also has an area power, but only turn undead. The paladin usually gets extra defense when outnumbered.

So it seems like Lanchester's law is being taken account in the design of characters and monsters.

 

[Wulf Ratbane]

Wulf -

All the finer points and mathematical posturing aside, which system - to you, based upon what you know - seems to better measure the challenge of combat, thereby allowing DMs to better plan encounters for their group: 3E or 4E.

4E? I agree.

So time is better spent looking at what it does well and (if you don't plan on using the system), porting those aspects over to 3.XX inasmuch as possible. Can you "fix" 3E by doing so? Probably not. But you can certainly improve upon it, much as you did with GT.

All the other stuff is just gobildigook.

I like what I see of 4e better than what I know of 3e. So far that's indisputably true. But I don't know that I will like what I know of 4e (when we all finally know it) better than what I know of 3e. Remember, CR/EL was very well received when 3e debuted, and it wasn't until well into the process of "mastery" on the part of the playerbase that its flaws (both minor and deep) were revealed.

4e has a huge leg up on 3e because we know that 3e is both flawed, and difficult to use. 4e looks VERY easy to use. But if 4e's ease of use comes at the price of... well, the math not actually working out... that's not a plus.

"Mathematical posturing and gobilidigook." Look, the easiest thing in the world is for a designer to just throw up his hands and say, " it, good enough." And in all honesty, I may yet do that. (I would like to think that I am at least doing due diligence AND setting the "good enough" bar pretty high.) Certainly it is true that all the precision in the world isn't necessarily going to gain us any accuracy.

But it's not "posturing" to want to understand "the math behind the system."

It depends on how "strict" you want to be about the 3E rules.

For example, you could try change the CR/EL and HD guidelines to get some aspects of the 4E into 3E.

A rough idea: Try to extract a level equivalent from monsters, that fits the "average" or typical AC/HP/Attack ratings for a monster. (One could try to assign monster roles like in 4E).
It is very hard to do so, since the variance in 3E is very high. You really have to use a lot of averages, and you can use neither CR or HD as your sole basis, and you don't really know beforehand if a monster might be more like a Minion, Elite or Solo (in as much as its average damage or hit points diverge from the base). Assymetric abilities (Save or Die effects especially) make it hard to do so.

Honestly, having studied this for a while, I don't think CR is that far off. Yes, there's a lot of variance inside a single measurement of CR. A CR8 creature such as the mind flayer feels very different in combat than a CR8 stone giant. But I think it still has considerable value in the system as a benchmark.

If you ever found such a benchmark, you could assign a (new) CR to it to use for XP rewards using the old tables, but add multiplies based on "weight" (Minion/Elite/Solo status)

That's good thinking, and exactly what I am looking at right now. It's very interesting to look back at the 1e XP tables. 1e did a much better job of breaking out "normal" creatures from creatures with loads of extraordinary abilities.

Take a creature of a given CR = X. Now give it X "choices" from the Defense column, and X "choices" from the Offense column. The Defense column includes Hit Dice, high AC, DR, SR. The Offense column includes size increases, multiple attacks, spellcasting and other extraordinary asymmetric abilities. You can also just give a creature 2X choices from one big list of Defense and Offense.

If I were a designer inclined towards the TLAR method, one disinclined towards mathematical posturing and all that gobildigook, that's how I'd determine CR.

 

[Wisdom Penalty]

Good reply/rebuttal Wulf. I really have nothing to counter that. I hope what you see of 4E matches what you know of 4E, for all of our sake.

If not, I'm heading to GURPS.

Wis