D&D 4E The Quadratic Problem—Speculations on 4e

[gizmo33]

First of all, in the OP "kill power" and "staying power" are multiplied. I don't know why.

Let's say every time a fighter goes up a level, he gains an apple, and orange, and a peach. A first level fighter has one apple, one orange, and one peach. A 3rd level fighter has three times as much wealth (3 of each fruit). That's linear. If someone with 20 hitpoints and +2 to hit is a match for someone with 30 hitpoints and +1 to hit, then AFAICT things work like they do for fruit.

Now I don't think that fighting power isn't exponential in 3E. I'm just saying that I don't see how the OP proves it one way or another. I think the OP makes some sensible predictions, but AFAICT it assumes the thing that it attempts to prove.

And also, people say that quadratics increase faster than linear, and that's not exactly true (or completely stated) and can cause you to reach incorrect conclusions. A slope of a parabola increases as x goes to infinity, but it doesn't mean that within a given range* (say levels 1 to 30) that a parabola would ever outstrip a linear. For example y=9200x is greater over the given range than y=1/4x^2 even though the first is a linear function.

Thus, I think there are too many variables here that are undefined and too many assumptions that are being made by people trying to use these formulas. With some tweaking, I think you could get a polynomial function that approximated a linear function over the numbers of interest (1 to 30). Basically, I don't think that the math is doing much more than putting a gloss over things people already believe to be true.

*(edit: er, domain - you get the idea, don't get all exponential on me)

 

[Wulf Ratbane]

Actually, I think that's about right.

Extend it to 30th level and then see if it still looks right.

Imagine five Ogre30's being a moderate encounter for 30th level PCs.

And Upper_Krust fielded a very good question-- if five Ogre30's are a good encounter for five 30th level PCs, what does a good single monster encounter look like?

I am just having a hard time buying those numbers.

(It's not helped by Cheiro staying half-in, half-out of 3e mindset. I think if you're going to project numbers, you have to be ready for the 1-on-1-- rather, 5-on-5-- shift in encounter design.)

 

[mmadsen]

First of all, in the OP "kill power" and "staying power" are multiplied. I don't know why.

Perhaps this Google Spreadsheet can help.

A combat unit's power is effectively its kills-per-round times the number of rounds it can expect to last before being killed.

 

[Wulf Ratbane]

First of all, in the OP "kill power" and "staying power" are multiplied. I don't know why.

Because "combat power" is the product of "how much damage can he do" and "how long can he do it."

Let's say every time a fighter goes up a level, he gains an apple, and orange, and a peach.

Let's not... Back up until you understand the above concept. Don't fruit the thread.

A slope of a parabola increases as x goes to infinity, but it doesn't mean that within a given range*

*(edit: er, domain - you get the idea, don't get all exponential on me)

Big points for that!

 

[Wulf Ratbane]

I really had intended this to be a separate Quadratic Problem discussion, but I will go ahead and throw it out there now:

Bloodied.

It changes things. A lot.

It toys with Lanchester's laws.

What happens when a creature who drops below 50% of its hit points suddenly doubles its offensive capability? (The dragon in the combat example.)

What happens when a creature who drops below 50% of its hit points suddenly doubles the offensive capability of its opponents? (Unusual rogue abilities, power words, etc.)

It doesn't have to be an exact "doubling"-- it could be any increase. I use "doubling" because it's a convenient shorthand to work with our force multiplier rules.

(It's one more clue that WoTC's designers are thinking about this stuff. IMO there is some genius up there in WoTC design.)

 

[AdmundfortGeographer]

What happens when a creature who drops below 50% of its hit points suddenly doubles its offensive capability? (The dragon in the combat example.)

What happens when a creature who drops below 50% of its hit points suddenly doubles the offensive capability of its opponents? (Unusual rogue abilities, power words, etc.)

It doesn't have to be an exact "doubling"-- it could be any increase. I use "doubling" because it's a convenient shorthand to work with our force multiplier rules.

(It's one more clue that WoTC's designers are thinking about this stuff. IMO there is some genius up there in WoTC design.)

Sounds quite a bit like the berserker strength alternate class ability in the PHB2 for barbarians.

After a certain amount of damage was taken by the barbarian level (based on an equation that takes into account the number of barbarian class levels), then the barbarian's berserker strength kicked in . . . and stayed active until the barbarian was healed passed the trigger level of hit points.

 

[Irda Ranger]

His point was that ancient combatants could only ever fight one-on-one-- clearly that's not the case in D&D

It is the case for 4e Fighters though. Especially with Full Attack removed, and expected 1:1 ratios in monster encounters, the only way a Fighter approaches one:many is with Cleave or a Grenade weapon (or Whirlwind Attack, if that still exists). However, you still don't get true many:many, in the way that wizards-with-wands do.

Interestingly, also recall that the Encounter examples Mearls has given there have been "mixes" of Monsters - ogres (brutes) backed up by gnoll archers (artillery). Like any good PC party, I think the designers recognize that a monster encounter needs to have its tactical bases covered, or the XP award will be "off" relative to the difficulty in overcoming it. A monster attack consisting solely of hostile centaurs charging with lances will be particularly susceptible to some elven archers and a well placed Earth to Mud spell* ...

Bloodied.

It changes things. A lot.

It toys with Lanchester's laws.

What happens when a creature who drops below 50% of its hit points suddenly doubles its offensive capability? (The dragon in the combat example.)

Every fight is a knock-down, drag-out one. Getting the jump on someone is not decisive. Cornered rats lose that "special something" that make them unique. You won't like them when they're angry.

Having run out of cliches, it means to me that once anyone in a fight with this quality reaches <50% HP, the fight ends quickly, but without certainty. If you have 100 HP, and the Dragon goes from doing 35 HP damage to 55 HP damage with a successful hit, the variability of outcomes has spiked. Suddenly each 20% of being hit becomes much more crucial, because you've left the "If I'm hit, I'm hurt" territory to enter "If I'm hit, I'm a '-37 HP' kind of dead" land.

Or did the Dragon just get a bonus to hit, not a bonus to damage? I didn't read the article you are referring to. If it's just a bonus to hit, that means the certainty of outcome remains the same, but the rate of resource usage (in HP and Cure spells) accelerates near the end of the fight.

It doesn't really change your tactics though. Due to Lanchester's Square Law, the quickest way to reduce the other side's combat effectiveness is to concentrate all your firepower on one guy at a time, killing him ASAP, so that the opponent's offensive power is reduced from x^2 to (x-1)^2. Spreading your damage around makes no sense, since this allows him to keep his (k) at x^2 for far longer. In a 'Bloodied' world, spreading damage around makes even less sense - but since the quantum outcome (choice of tactics) remains the same, it really doesn't change anything.

What happens when a creature who drops below 50% of its hit points suddenly doubles the offensive capability of its opponents? (Unusual rogue abilities, power words, etc.)

Death spiral?


* quite possibly the least poorly disguised military history reference ever.

 

[Cheiromancer]

Here's another sequence of critters. Call them Orcs, though I haven't used the Orc statistics. It's the Ogre stats, but with a higher starting AC. Each row is four times as powerful as the row above it, but the end result is weaker than an Ogre20, even though it starts off stronger.

Orc4:  Attack +8,  damage 16,  AC 21, hp 32
Orc6:  Attack +8,  damage 16,  AC 27, hp 32   
Orc8:  Attack +14, damage 16,  AC 27, hp 32    
Orc10: Attack +14, damage 16,  AC 27, hp 128   
Orc12: Attack +14, damage 32,  AC 27, hp 256   
Orc14: Attack +14, damage 32,  AC 33, hp 256   
Orc16: Attack +20, damage 32,  AC 33, hp 256   
Orc18: Attack +20, damage 64,  AC 33, hp 512   
Orc20: Attack +20, damage 64,  AC 39, hp 512

edit:

(It's not helped by Cheiro staying half-in, half-out of 3e mindset. I think if you're going to project numbers, you have to be ready for the 1-on-1-- rather, 5-on-5-- shift in encounter design.)

Well, I don't understand otherwise how CR is to be assigned to monsters. The double number = +2 EL I can understand (if supplemented by the notion that a single monster of CR N is an EL N encounter). But for this to generate a sequence of brutes (OgreN, OrcN, whatever) you need to be anticipating 2 on 1 combats. Although by Lanchester's Square Laws this implies that on a 1 on 1 combat the bigger loses 25% of its hit points while squashing the smaller. As long as their CRs differ by 2.

If one MonN is a challenge to a group of 4, then 4 Mon(N-2) is the same challenge to that group. If the exponential law holds, that is. But I'm using the 3.5 convention that a monster is CR N if it is a moderate challenge to a group of four level N PCs.

The difference between the OgreN and the OrcN sequences makes me wonder whether it is helpful to use a sequence of "Brutes" to investigate power curves. An Orc20 is nowhere near as powerful as an Ogre20. In fact, an Ogre20 is 20 times as powerful!

Unless I've really screwed up the math.


edit2:

And Upper_Krust fielded a very good question-- if five Ogre30's are a good encounter for five 30th level PCs, what does a good single monster encounter look like?

The thread in question is A 30th-level Party needs 50th-level Monsters.... Upper_Krust doesn't buy the exponential power increase, at least at epic levels. Try extending the Ogre and Orc tables and you'll see why! He is now using a cubic power curve, which means that a single 50th level monster has the power of a five member party of 30th level PCs.

 

[Wulf Ratbane]

It is the case for 4e Fighters though. Especially with Full Attack removed, and expected 1:1 ratios in monster encounters, the only way a Fighter approaches one:many is with Cleave or a Grenade weapon (or Whirlwind Attack, if that still exists). However, you still don't get true many:many, in the way that wizards-with-wands do.

Unless I am really misreading him, though, what I think he was saying was that in a 2 Good Guys fight vs. 1 Bad Guy fight, the one Bad Guy got to line up and fight one Good Guy at a time, instead having to fight both at once.

And all things being equal, trading blows back and forth, one Good Guy kills one Bad Guy at about the same time that the Bad Guy killed him, so you could predict the outcome of the combat by just subtracting the size of the smaller force from the large.

Obviously that's not the case. Two good guys team up on the one bad guy, they do twice as much damage in the same amount of time, the bad guy dies twice as fast, so he only inflicts half as much damage as he would otherwise. The disparity is 4:1. (Quadratic.)

So while it is true that the Fighter in D&D is largely going to be focusing his offense on one opponent, he can, of course, still pitch in against a common foe with a teammate and so benefit from quadratic force projection.

It doesn't really change your tactics though. Due to Lanchester's Square Law, the quickest way to reduce the other side's combat effectiveness is to concentrate all your firepower on one guy at a time, killing him ASAP, so that the opponent's offensive power is reduced from x^2 to (x-1)^2.

That was basically the gist of my intended post. Spreading damage around makes even less sense.

It makes opening with a fireball against a mob of Bloodied-Bonus creatures a bad idea. The wizard can really sour things for the party.

It also makes the Knight-like power (calling out a foe to man-to-man combat) particularly useful.

 

[Cheiromancer]

It makes opening with a fireball against a mob of Bloodied-Bonus creatures a bad idea. The wizard can really sour things for the party.

I don't know. Suppose that there are 5 fighter/wizards facing a group of 5 bloody-bonusers. Suppose it takes 4 rounds for each PC to take down their opponent; two round they are fighting a non-bloodied monster, and two rounds they are fighting a bloodied monster.

If one of the wizard lobs a fireball and reduces the five BBs to half hit points, what changes? Well, each PC has to spend 2 rounds fighting a bloodied monster before killing it.

So the fireball helped, or at least didn't hurt. It just didn't help as much since, presumably, the BBs do most of their damage while bloodied.