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

[DarkKestral]

One possibility to keep the power-growth higher than linear but less than quadratic would be linear + logarithmic. Since it is the nearest of the infinite-growth curves to being a bounded function, a logarithm helps when you want to create a 'maximum' power level that continuously grows, but you need it to eventually stop growing very much. In this way, characters improvement speeds improve over time, but the speed of that improvement of improvement slows.

From what I can tell, WoW uses something of a similar system for it's stats, and is noted for having a very long 'sweet spot', even though the difference between a character of level 1 vs. a character of equivalent race and class who is level 2 is much less than those same characters when the first is level 69 (the penultimate level) and the other at 70 (the current maximum level).

That way, you can tell the obvious differences, because a character of level 30's 'baseline' chance of success vs. a level 30 target compared to a level 1's can be higher, but not so overpoweringly so that we see the 'autosuccess on everything but 1' that can happen in level 20 games of the current day, as compared to the 'autofail on everything but 1' that you can see in level 1. It would be closer to 60% success baseline at 30, but 40% success baseline at 1. A signficant and notable difference, but one less prone to being 'unfun'.

 

[Nikosandros]

Sorry Wulf, but "exponential" does not simply mean "involving an exponent". If Bruce Cordell is using it that way, he doesn't know what he is talking about.

I liked your formulas and the discussion, but fair is fair - your use of "exponential" is simply wrong.

Exactly... here there is a confusion between exponential (a^x) and power law (x^b). Those are two very different behaviors: a quadratic growth is most definitely not an exponential growth!

 

[Nonlethal Force]

I was just about to post that, Nikosandros, but thanks for beating me to it. For those of you non-math geeks (of which I am amath-geek, so I don't mean it in a snark voice!):

there is a big difference in the behavior between y = 2^x and y = x^2.

I give a specific pair of examples because some people may not be familiar enough with math lingo to realize that a and b are often letters used to represents "unknown constants" where as x and y are often used to represent variables. To those of you that know this already, ignore this paragraph!

Wulf, I find your ideas intriguing. As a total aside ... you could make it linear by making the staying power or damage potential a constant. If Hit Dice never go up, for example, the staying power never changes and the equation is now linear. This is a bad example, though, because if hit dice don't go up but damage potential does, it makes for very lethal combats.

If hit points go up but damage potential doesn't, then combats become far less lethal (and longer). But the power curve is now linear. My assumption is that a linear curve is much easier to make a continuos "sweet spot" of game play than is a quadratic one.

Not sure the above is helpful other than to point out that it is the human desire to need increases to damage potential and staying power that makes it so difficult. And the sad thing is, it is a viscious cycle. Damage output has to increase because staying power increases. That is because it helps keep combats shorter. So with each increase to HD, there is a perceived need for an increase to damage output. But that perceived need is where the problem lies.

Anyway, not sure there is anything useful in this, just sayin' ....

 

[Nikosandros]

I was just about to post that, Nikosandros, but thanks for beating me to it. For those of you non-math geeks (of which I am amath-geek, so I don't mean it in a snark voice!):

there is a big difference in the behavior between y = 2^x and y = x^2.

I give a specific pair of examples because some people may not be familiar enough with math lingo to realize that a and b are often letters used to represents "unknown constants" where as x and y are often used to represent variables. To those of you that know this already, ignore this paragraph!

Thanks for pointing that out. It's a conventional notation that I'm so used to that I take it for granted, but it was a very good idea to explain it...

 

[Lanefan]

Not that I can make head or tails of all the formulas above, but there *is* one factor missing: magic.

Magic makes a character's numbers go up even faster than they would otherwise.

An example: a fighter at 1st level fights at baseline x ability, when all the inherent variables are thrown together. The same fighter at 10th level fights at x + 10^y where y is whatever all the formulas decide the "curve" factor to be. However, a 10th-level fighter has almost always (and particularly so in 3e) acquired magic items that will make her:
- hit more often (+n magic weapon, +s strength device, boots of speed, feats)
- do more damage per hit (+n weapon, +s strength again)
- be harder to hit (+a armour/shield, boots of speed, ring of protection, feats)

Now, when comparing similar fighters against each other this is a wash, as both would in theory gain about equal overall benefit from equipment. But comparing said fighter to the average monster that doesn't have access to The Adventurers' Magic Supply, the monster is in even more trouble. Much the same goes for all other classes; fighters are just the easiest to quantify. End result: the curve goes up even more steeply than you thought. (in 1e, it's magic that topples the curve in the end, not increasing levels; level benefits largely flatten out after about the 9-12 range except for wizards)

A simple yet dull way to flatten the curve a bit is to stop giving out magic items...but that's no fun for anyone.

Lanefan

 

[mmadsen]

Anyone designing a war game, or a roleplaying game with a lot of combat, should be familiar with Lanchester's Laws:

In 1916, during the height of World War I, Frederick Lanchester devised a series of differential equations to demonstrate the power relationships between opposing forces. Among these are what is known as Lanchester's Linear Law (for ancient combat) and Lanchester's Square Law (for modern combat with long-range weapons such as firearms). In ancient combat, between phalanxes of men with spears, say, one man could only ever fight exactly one other man at a time. If each man kills, and is killed by, exactly one other, then the number of men remaining at the end of the battle is simply the difference between the larger army and the smaller, as you might expect (assuming identical weapons).

In modern combat, however, with artillery pieces firing at each other from a distance, the guns can attack multiple targets and can receive fire from multiple directions. Lanchester determined that the power of such an army is proportional not to the number of units it has, but to the square of the number of units. This is known as Lanchester's Square Law. It relies on the fact that when either side has more units it means that they have also more surface-area that opponent can hit (thus diluting the amount of fire hitting each unit, reducing their rate of attrition) and also more firepower (thus increasing the enemy's rate of attrition).

Note that Lanchester's Square Law does not apply to technological force, only numerical force; so it takes an N-squared-fold increase in quality to make up for an N-fold increase in quantity.​

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.

 

[RedShirtNo5]

Linear Power Curve

Make both kill power and staying power logarithmic.

I guess that should be advance as square root ito be more accurate.

 

[BryonD]

I was just about to post that, Nikosandros, but thanks for beating me to it. For those of you non-math geeks (of which I am amath-geek, so I don't mean it in a snark voice!):

there is a big difference in the behavior between y = 2^x and y = x^2.

Well, I'm glad we cleared that up.
Heaven forbid a good conversation and common use of terms stand in the way of thread destruction and niche conventions.

 

[Badkarmaboy]

Umm......Kittens are cute.....

(I opened this thread up and felt like I was back in HS Math class again..neat info though)

 

[Mustrum_Ridcully]

I was just about to post that, Nikosandros, but thanks for beating me to it. For those of you non-math geeks (of which I am amath-geek, so I don't mean it in a snark voice!):

there is a big difference in the behavior between y = 2^x and y = x^2.

I give a specific pair of examples because some people may not be familiar enough with math lingo to realize that a and b are often letters used to represents "unknown constants" where as x and y are often used to represent variables. To those of you that know this already, ignore this paragraph!

Wulf, I find your ideas intriguing. As a total aside ... you could make it linear by making the staying power or damage potential a constant. If Hit Dice never go up, for example, the staying power never changes and the equation is now linear. This is a bad example, though, because if hit dice don't go up but damage potential does, it makes for very lethal combats.

If hit points go up but damage potential doesn't, then combats become far less lethal (and longer). But the power curve is now linear. My assumption is that a linear curve is much easier to make a continuos "sweet spot" of game play than is a quadratic one.

Not sure the above is helpful other than to point out that it is the human desire to need increases to damage potential and staying power that makes it so difficult. And the sad thing is, it is a viscious cycle. Damage output has to increase because staying power increases. That is because it helps keep combats shorter. So with each increase to HD, there is a perceived need for an increase to damage output. But that perceived need is where the problem lies.

Anyway, not sure there is anything useful in this, just sayin' ....

But you're also pointing out a weakness in your stated linear curve: Either the game becomes to deadly, or it becomes a lot less deadly at high levels. Neither of which is really what people want.

Basically, for a good game, there is a second equation you have to take into account.
Let's call it "Survivability" (S):
S = k/T
P = k*T
S(L) = k(l)/T(l)
P = k(l) * T(l)
(S = Survivability,P = Power, k = Attack Power, T = Staying Power. l= Level)
k,T are based on level only, not on a individual character (since it's not important to know how much attack power a character can direct against himself, but against other opponents of his level or challenge rating)

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)?