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

[Nonlethal Force]

So basically, we want to ensure that S remains a constant, as P increases. I am afraid my math is a bit lacking at this moment, so I can't tell you if a logarithmic advancement can help here. But since we are using a quadratic function if we k = a*l and T=b*l, wouldn't it be sufficient if we used an advancement based on the square root, so k = a * SQRT(l) and T = b * SQRT(l)?

In a way, yes. Your math is good, by the way - FWIW.

{I'm changing the variable l to x so that it is easier to read in the equation below. Not that there is anything wrong with using l}

is k = (a)(SQRT[x]) and T = (b)(SQRT[x]) then kT = (a)(b)(x).

{For those non-math geeks who are interested: The mathematical rule is that if we are multiplying expressions with the same bases then we can add the corresponding exponents. A square root is technically an exponent of 1/2, thus (SQRT[x])(SQRT[x]) = x because 1/2 + 1/2 = 1.}

Since a and b are merely constants that relate to the amount of power increase, you can see that the only variable is x, and thus this is linear.

However, to use Wulf's original logic, we do not only have an increase each level. We have a constant base starting point. Let's call them c and d, respectively. So the equations are actually:

k = (a)(SQRT[x]) + c
T = (b)(SQRT[x]) + d

Then,

kT = (aSQRT[x] + c)(bSQRT[x] +d)

We need to FOIL that baby, and when we do it isn't really very pretty:

kT = abx + adSQRT[x] +bcSQRT[x] + cd

If we want, we can somewhat simplify the equation into this:

kT = abx +(ad+bc)SQRT[x] + cd

As you can tell by the existance of the pesky SQRT term, this is not a linear progression either. However, it will behave much more like a linear equation over the long haul than a flattened exponential or a flattened quadratic. [This is because the dominant power of the equation is 1]




The real problem with the SQRT approach is not in the mathematical progression of the equations. The real problem is that the numbers are so darn unweildy. Game designers would nee to figure out fractional (decimal, really) values. It isn't too far from the way that saves are done in UA, except that those are linear in and of themselves rather than the blending of two SQRT functions.

I suspect that if a SQRT approach was taken, you would simply see a step-function approach to game play. You would see no improvement from levels 1-4, then a small jump. Then no more improvement until level 9, and then another small jump. It would be the only way to effectively use a SQRT approach to power curve and lasting power. The reason is because people like whole numbers, not slow increasing decimals.

 

[Mustrum_Ridcully]

In a way, yes. Your math is good, by the way - FWIW.

{I'm changing the variable l to x so that it is easier to read in the equation below. Not that there is anything wrong with using l}

is k = (a)(SQRT[x]) and T = (b)(SQRT[x]) then kT = (a)(b)(x).

{For those non-math geeks who are interested: The mathematical rule is that if we are multiplying expressions with the same bases then we can add the corresponding exponents. A square root is technically an exponent of 1/2, thus (SQRT[x])(SQRT[x]) = x because 1/2 + 1/2 = 1.}

Since a and b are merely constants that relate to the amount of power increase, you can see that the only variable is x, and thus this is linear.

However, to use Wulf's original logic, we do not only have an increase each level. We have a constant base starting point. Let's call them c and d, respectively. So the equations are actually:

k = (a)(SQRT[x]) + c
T = (b)(SQRT[x]) + d

Then,

kT = (aSQRT[x] + c)(bSQRT[x] +d)

We need to FOIL that baby, and when we do it isn't really very pretty:

kT = abx + adSQRT[x] +bcSQRT[x] + cd

If we want, we can somewhat simplify the equation into this:

kT = abx +(ad+bc)SQRT[x] + cd

As you can tell by the existance of the pesky SQRT term, this is not a linear progression either. However, it will behave much more like a linear equation over the long haul than a flattened exponential or a flattened quadratic. [This is because the dominant power of the equation is 1]




The real problem with the SQRT approach is not in the mathematical progression of the equations. The real problem is that the numbers are so darn unweildy. Game designers would nee to figure out fractional (decimal, really) values. It isn't too far from the way that saves are done in UA, except that those are linear in and of themselves rather than the blending of two SQRT functions.

I suspect that if a SQRT approach was taken, you would simply see a step-function approach to game play. You would see no improvement from levels 1-4, then a small jump. Then no more improvement until level 9, and then another small jump. It would be the only way to effectively use a SQRT approach to power curve and lasting power. The reason is because people like whole numbers, not slow increasing decimals.

Essentially, it might work best in a computer game (yeah! Video Gamey!), where the numbers can be abstracted away from the player.
Or they can be a lot higher, then rounding the fractions. Instead of a Base Attack Bonus of +1, you get one of +100. At second level, you get one at +141 (rounded SQRT(2), and at 3rd a bonus of 173, at 4th a bonus of +200, at 5th a bonus of 223, at 6th a bonus of 244...
Doesn't really look to nice, I am afraid, the curve is to flat...

I am not sure that the logarithmic curve will fare much better..

 

[Nonlethal Force]

Essentially, it might work best in a computer game (yeah! Video Gamey!), where the numbers can be abstracted away from the player.
Or they can be a lot higher, then rounding the fractions. Instead of a Base Attack Bonus of +1, you get one of +100. At second level, you get one at +141 (rounded SQRT(2), and at 3rd a bonus of 173, at 4th a bonus of +200, at 5th a bonus of 223, at 6th a bonus of 244...
Doesn't really look to nice, I am afraid, the curve is to flat...

I am not sure that the logarithmic curve will fare much better..

Actually, to be honest I think the square root function behaves much more in line with how I enjoy gaming. The fasted changes would be from levels 1 to four, and then the power curve would actually begin to taper off. It would imply that the power differential between an ECL (to use a 3.5 term) 3 and an ECL 4 would be greater than an ECL 12 to an ECL 13. The way I think about it, I don't think that's a bad thing.

 

[mmadsen]

Linear Advancement
We all remember f(x)= mx + b from our algebra days. This is the equation for a line. To apply this equation to a character’s power advancement, m is the slope, or rate of advancement; x is the number of levels advanced (or, character level-1), and b is the baseline level of power (its actual value is not important but can be substituted as “what we can expect from a 1st level character").

But Power Is a Quadratic Function
The problem is that power does not, and can not, scale linearly; it is a quadratic function. Why?

I think we need to ask ourselves whether we should be comparing power to level, when the real issue is power as a function of time -- or, more accurately, encounters defeated.

Under the current system, the experience point requirements for each level increase faster than linearly, but the experience point rewards also increase faster than linearly.

 

[Remathilis]

The problem is CR= (mx+b)(dx+a) means NOTHING in terms of the game.

What is m stand for? How do we determine it? Is it based on BAB? If so, a cleric and a bard have the same damage output ratio. Same with a raging half-orc barbarian and a rapid-shot halfling ranger. If we had some way of proving what mx+b actually was in game (rather than an abstract concept of attack power) then we could see the curve in action.

 

[KarinsDad]

It's that last paragraph that's the most important to our current discussion. If we measure troop quality with a single variable -- let's say that ogres kill orcs twice as fast as orcs kill ogres -- then two ogres might seem like they'd defeat four orcs easily, but really they'd be overpowered, because multiplying the number of troops multiplies its offense and its defense. More orcs have more attacks, and there are more orcs to kill.

What confuses this is that D&D level includes multiple measures of offensive stength (to-hit and damage) and defensive strength (AC and hit points); it's not a single linear measure. For instance, in going from first to second level, an NPC fighter might multiply his to-hit chance by 1.1, his damage by 1.0, his avoid-a-hit chance by 1.0, and his hit points by 2.0, for a total quality factor of something like 2.2. As you can see, at lower levels, without better equipment, it's almost entirely about improved defense through extra hit points. As characters accumulate magic weapons, armor, etc., they can improve across all four of those dimension, and a 10-percent improvement in everything isn't a 10-percent improvement in fighting ability; a 10-percent improvement across four factors is a 46-percent improvement. Now compound that over multiple levels.

Well said.

 

[Mustrum_Ridcully]

The problem is CR= (mx+b)(dx+a) means NOTHING in terms of the game.

What is m stand for? How do we determine it? Is it based on BAB? If so, a cleric and a bard have the same damage output ratio. Same with a raging half-orc barbarian and a rapid-shot halfling ranger. If we had some way of proving what mx+b actually was in game (rather than an abstract concept of attack power) then we could see the curve in action.

That's yet another level of complexity.

Average Damage Per Round is a good approximation for mx+b, but if we now begin to look closer, we get to see that it's even more complicated.
Attack Bonus clearly mostly affects Attack Power, but in D&D (thanks to trip, sunder and disarm for example) it can also affect your Staying Power (because you reduce the ability of an enemy to deal damage to you, just like a high AC does). But you give up Attack Power.

Spells are also strange in this manner: As was pointed out in the first post, you can take a wizard out of a fight even if you never deal damage to him - the amount of spells cast per day limits is part of his staying power. So we have abilities that might help our staying power (like a defensive spell), but also reduce it. An we have offensive abilities that reduce the staying power, too.

But that's not really _that_ important, unless you really want to make a perfectly balanced game system. The important part is that you have to consider the rate of advancement of the base numbers that are used. Which means, attack bonus and caster level on the Attack Power side, and hit points, AC, saves and spells per day.

I think we need to ask ourselves whether we should be comparing power to level, when the real issue is power as a function of time -- or, more accurately, encounters defeated.

Under the current system, the experience point requirements for each level increase faster than linearly, but the experience point rewards also increase faster than linearly.

That is a good point, too. But the current D&D style doesn't fit a slower advancement at higher levels (to counter the quadratic improvement, you would need to use a quadratic rate of XP cost, meaning level 2 might cost 1.000 XP, Level 3 4.000 XP, Level 4 9.000 XP and so on).
Part of the fun of the game is that the character improves regularly. Unfortunately, in D&D, these improvements come in big chunks (levels), which means if you delay getting another "chunk of improvement", you take out some of the fun. (Though that's not bad for everybody, because many might have enough fun playing their characters as he is and enjoying the stories and challenges.) So, If you (and your group) likes this, this might be a very good house rule for your Advancement XP chart.

Non-level based games have it a bit easier here. You can purchase each advancement individually, so you can always improve a little bit (even if it is nothing compared to D&D, it feels like advancement and thus like a reward.)
I have mostly experience with Shadowrun as a non-level based game, and it takes basically forever to change your character notably. If you're really lucky, you get 10 Karma points at the end of a longer session or adventure, and you might be able to advance two skills or maybe a weak attribute with that.
But it's not that bad, because you also got a lot of money you can throw at new toys (read: better weapons, cyberware and maybe some magical equipment. Or just improve your lifestyle!). It doesn't really change your character _that_ much (not compared to magical items), but all these improvements still feel like something, even if they pale compared to the options and abilities you gain in D&D for a level. But it doesn't matter, because there is a reward, and you get it now, not after 5 adventures.

 

[Remathilis]

The true measure of power in D&D has always been just f'ing metal your character is.

Do Warforged win D&D then?

 

[Squire James]

Not necessarily, because warforged are assumed to have organic parts (or even be mostly organic) except for metal shells akin to armor. They would win an Awsomeness Contest in a greater variety of situations (such as conflicts involving Fighting Naked), but a stout warrior in heavy armor might out-Awesome a warforged in the long run. Especially if you're running Rules Encyclopedia D&D, where Warforged are noticeably absent and are therefore hard-coded to 0 Awesomeness (because 0 x any amount of metal is still 0).

 

[Remathilis]

Warforged are elemental metal, like sporks or toasters; This is f'ing metal, like a Gwar concert or an Iron Maiden album cover.

Now warforged *can* be both elemental and f'ing metal. When they are, you end up with something like a viking version of Darth Vader.

That is a mental image worthy of signatures. Bravo! Dio would be proud...