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

[Malhost Zormaeril]

I'm afraid I don't follow. BAB increases linearly with level: +1 BAB/level.

Probability of hitting depends on the target AC, which is a moving target. If we only compare nearby levels -- e.g. Ftr1 vs. Ftr2 -- then to-hit probability increases by a factor of approximately 1.1 -- e.g. from 50 percent to 55 percent.

If we assume our target is always an Orc War1 with AC 13, then to-hit probabilities will go from, say, 40 percent, to 45, 50, 55, and so on, up to 95 percent. That's obviously linear with level, to a point. As with hit points, the ratio of one level's to-hit probability to the previous level's drops from level to level: 1.12, 1.11, 1.10, 1.09, and so on.

I apologise; that's what I meant to say. The probability to hit increases by a factor of (1 + 1/[Lvl]), which is what I meant to say. It is what the quote I posted on my original post alluded to, anyway.

Well, I guess I don't have anything else interesting to add. I'll mull this over a bit more.

 

[Piratecat]

This is the problem with us nerds, we can never get past our own intellect to have a discussion about anything other than "I'm smarter than you, na-na-na-boo-boo."


A quadratic IS an exponential function ( f(x)=a^x + b ) where x = 2.

Hmm. Okay, we'll make it explicit: please don't sidetrack this thread with a discussion of what makes up an exponential functions. The actual thread subject is both interesting and meaty (at least to me), and we don't want it to slide off on a non-D&D tangent.

Thank you.

 

[Wulf Ratbane]

I could go back and edit the first post.

 

[mmadsen]

If we look at the four "brute" qualities -- to-hit probability, damage, avoid-hit probability, and hit points -- we should note that to-hit probability increases linearly (if we assume the same target across all levels), damage does not increase at all, avoid-hit probability does not increase at all, and hit points increase linearly.

Thus, a Fighter's power does increase quadratically with level.

This is all going purely from the class though, assuming no improvements in equipment. With equipment, the Fighter increases to-hit even faster and increases damage and AC too.

My understanding is that 4E characters will increase their avoid-hit probability linearly with level, but I don't believe that they will necessarily increase their damage. That would imply that a 4E Fighter's power would increase as a cubic function of level, or a third-degree polynomial.

This all raises the issue of how characters, or martial characters and "brute" monsters, should advance. Should they increase all four "brute" qualities linearly? It is odd that historically Fighters improved to-hit very, very slowly, damage not at all, AC not at all, and hit points rapidly, then very, very slowly.

 

[Nonlethal Force]

I think the current system, where CR-N is as powerful as Nth-level, makes far more sense than creating a second scale, so that five monsters of Nth-monster level are just strong enough to pose a moderate challenge to five PCs of Nth-level.

FWIW, I totally agree with balancing a single versus a party. Personally - and this may just be a personal preference - I always think it is easier and better game design to start at the most basic component and design a method for scaling up. I think is is less intuitive to say that group of creatures are a certain CR and perhaps have to go backwards.

[Sblock=Aside]
FWIW, I know that a few posters have slighted my earlier post regarding mathematical terms. As a former math teacher, I was simply trying to explain math for those people unfamiliar with math but trying to follow the conversation. It was meant as an aide for those who struggle with math. I have found that the typical person struggles with math and is put off by it - and most people appreciate a pleasant attempt to explain it. I have never been trying to sidetrack the conversation or sound pompous. I apologize if it came off that way.[/Sblock]

 

[mmadsen]

Thus, a Fighter's power does increase quadratically with level.

I've created a simple Google spreadsheet (nothing to download) demonstrating that a Fighter (or Warrior) with the barest of equipment (spear and shield) increases in combat power quadratically with level -- until he has a 95-percent chance of hitting his opponent, at which point his power progresses linearly, purely from hit points.

Not only does this ignore improvements in equipment, but it also ignores iterative attacks and feats.

 

[Irda Ranger]

Maybe I'm missing something, but why do people keep referring to Lanchester's Square Law when describing Brute combat in D&D? Wasn't his point that ancient warfare (swords and formations) was linear, while only modern artillery made combat non-linear?

Wouldn't that mean that Brute monsters and Artillery monsters (two "roles" that Mearls has described as being used in the 4e MM) would have to have different rates of advancement for different qualities? This is perhaps the driving reason for breaking up the MM into roles in the first place. Mearls has explicitly stated there will be a "brute" advancement and a "artillery" advancement (among other roles), and I expect those rates will be informed by this thinking.

Further, certain classes (Fighters and Paladins, namely) would obey the Linear law while Wizards and Sorcerers (and possibly ranges Strikers, such as the Ranger) would have their level advancement influenced by the Square law. Leaders, OTOH, probably answer to a strange mix of the two, since they fight in a Linear manner but produce non-linear effects that depend on the number of allies they can provide bonuses to. Put another way, we have to ask: how far does a PC group's aggregate (k) resulting from a Leader's presence scale with party size?

Further, Lanchester would have pointed out that surface area is different from number of combatants. 9 wizards, no two of whom are within 15' of each other, may be far more Survivable than 9 wizards squeezed into a 15'x15' room, depending on whether their enemy uses precision weapons (Finger of Death) or area weapons (DB Fireball).

Lastly, I'm surprised I haven't see more discussion on how the limitations of using a single 20-sided die effects outcome resolution. I can barely follow the algebra, so probability theory is right beyond me, but it seems to me that the lack of granularity of outcomes and the lack of any kind of probability curve have a huge impact on D&D's ability to scale with level. The game would probably handle some things a lot better with a "25d4" game engine, even if it was unplayable in other (obvious) respects. This is one advantage that WoW will always have.

 

[Irda Ranger]

I've created a simple Google spreadsheet (nothing to download) demonstrating that a Fighter (or Warrior) with the barest of equipment (spear and shield) increases in combat power quadratically with level -- until he has a 95-percent chance of hitting his opponent, at which point his power progresses linearly, purely from hit points.

Not only does this ignore improvements in equipment, but it also ignores iterative attacks and feats.

Using SWSE as an educated springboard for 4e discussion, you can probably give him a +1 dmg per level.

 

[Cheiromancer]

Or a +9 bonus to AC might mean that a monster that was getting hit half the time (50 percent) now gets hit one-tenth as often (5 percent), but, if it was getting hit just a bit more often, 60 percent of the time, a +9 bonus means it's now getting hit one-fourth as often -- nowhere near as big an improvement as we'd planned.

That's exactly the problem with the AC and attack numbers; they are nowhere near as easy to manage as the hit points and damage.

... a monster that does 25 times as much damage should be 25 times as powerful, but most of that extra damage will go to waste against one-hit-die orcs.

And even hit points and damage are not as simple as one would like. These calculations are all rough calculations, giving the general scheme of things.

Under the old system, a single CR-N monster was supposed to be a moderate challenge for four Nth-level PCs, but I think they forgot Lanchester's Square Law there, because four PCs should be 16 times as powerful as one.

Under the new system, a MonN should be one-sixteenth as powerful as a PCN, if we want to maintain the old, easy standard for a "moderate" challenge. Given the exponential nature of levels, that would set MonN+4 = PCN. So a single Mon5 would be a moderate challenge for a single PC1.

I start with a different set of assumptions. I'm assuming that when they find a monster that is a moderate challenge (25% resources) against four Nth-level PCs, they say it is a MonN. (Thanks for the improved notation, btw).

Now given Lanchester's Square Law, that means that a MonN is 4 times as powerful as a single PCN. Which means that it is the same power as a PC(N+2) - assuming the exponential law follows for PCs as well as monsters. Which is the assumption in 3.5.

An alternative way of saying the same thing is that the CR of PCN is N-2. Which means that a single Mon5 would be a cointoss against a PC7, and would use up 25% of the resources of a PC9. On average, that is: not all characters are equally capable against all challenges.

 

[mmadsen]

Maybe I'm missing something, but why do people keep referring to Lanchester's Square Law when describing Brute combat in D&D? Wasn't his point that ancient warfare (swords and formations) was linear, while only modern artillery made combat non-linear?

Ancient warfare is only linear to the extent that there's a (short) line of men fighting one on one. What makes modern warfare follow Lanchester's Square Law is that every unit is in the fight -- offensively and defensively.