Wulf Ratbane
Adventurer
The Quadratic Problem—Speculations on 4e
At its heart, D&D is a game of tactical combat, and the core mechanics are designed to serve this function. The problem with 3e was not that the power curve scaled exponentially, but that the curve was too steep. As a consequence, the period of optimum play (usually described as the “sweet spot”) was too short.
In fact, Power must always scale exponentially—in fact, it scales quadratically. Let’s look at why.
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?
When we analyze how powerful a character is in combat, it is a product of offense and defense: How much damage can he do and how long can he do it?
Let us say that each PC has a “combat rating” CR. His CR is a function of his “expected damage output per action” (k, for kill power), multiplied by the number of actions he can be expected to perform before he is neutralized, or, his staying power (T, a time variable).
CR= (k)(T)
Now, look at what happens when a character advances in level. His effective offense increases linearly, true enough; and so does his staying power (whether by increases in hit points, AC, defenses, etc.)
His kill power advances linearly: k = mx+b; where m is the rate of offensive advancement, x is the number of levels he has gained, and b is his starting point.
His staying power advances linearly: T = dx+a, where d is the rate of defensive advancement, x is (again) the number of levels he has gained, and a is his starting point.
It is also very important to note that m and d, the rates of advancement, are relative to b and a— these increments are relative to the starting values at 1st level.
Thus,
CR=(k)(T)
And substituting for our linear advancement rates we see
CR= (mx+b)(dx+a)
As you can see, the product of two (non-zero) linear advancement rates must be quadratic:
CR=(mdx2)+(mxa)+(dxb)+ab
Specific Analyses
Let’s begin at 1st level, where x is 0—the character has not yet advanced.
Most of the equation drops out, leaving us with
CR = ab
Remember, a and b are our baseline, starting 1st level defense and offense levels. We could arbitrarily set them both to “1” just as a shorthand for “about the amount of offense and defense we can expect from a 1st level character.”
In which case, at 1st level, CR=1.
We know for sure that a 1st level fighter has more staying power (in both hit points and AC) than a 1st level wizard; but we also know that a properly prepared wizard can inflict many more kills than the fighter. Sleep, color spray, and burning hands, even at 1st level, is a lot of “kill power” for the wizard.
We can all probably agree that the design of the wizard is to trade offensive smack-down for his relative fragility.
If the classes are balanced, however, we can also assume that the product ab is probably about the same, even if a and b are reversed and/or adjusted from class to class.
Notice also that the staying power of the wizard is also directly related to his spells per day. When he’s out of applicable spells, he’s effectively neutralized with respect to the fighter. You don’t have to kill a wizard to “kill” him in terms of his combat effectiveness.
Levelling Up
Let’s look at what happens when these two level up. In 3e, their hit points double—doubling their staying power. And, in the case of the wizard in particular, his offense just about doubles—his burning hands goes from 1d4 to 2d4. The fighter, of course, doesn’t enjoy this doubling effect—his BAB goes up, of course, and he may be able to make an extra attack through the use of feats, but he certainly doesn’t get to double the base damage output of his longsword. By 2nd level, he’s already falling behind.
Flattening the Curve
The power curve will never be linear—you don't hear many references to the power line.
But we can change its shape and try to smooth it out by changing the underlying linear rates of advancement of both offense and defense (m and d, respectively) and we can change our starting baselines (a and b) to shift the curve into a more suitable range of play.
The attached chart shows several options for changing the power curve.
Series 1 shows the current power curve of 3e. The starting values a and b are both set to 1 and the linear advancement rate of both offense and defense m and d are also set to 1. All variables are set to 1, so power scales 1, 4, 9, 16, 25, etc.
Series 2 shows a proposed power curve. In this example, we start 1st level character with twice as much staying power as before; but the linear rates of advancement for offense and defense are dialed down to 2/3 and ½, respectively.
Series 3 is identical to Series 2, but offense and defense both increment at a ½ rate.
Series 4 is the “rat bastard” power curve—1st level characters do not receive any additional HD, and both the offense and defense advancements are dialed down to ½. Notice that the shape of the blue Series 4 curve is the same shape as the green Series 3 curve, but there is a baseline shift.
At its heart, D&D is a game of tactical combat, and the core mechanics are designed to serve this function. The problem with 3e was not that the power curve scaled exponentially, but that the curve was too steep. As a consequence, the period of optimum play (usually described as the “sweet spot”) was too short.
In fact, Power must always scale exponentially—in fact, it scales quadratically. Let’s look at why.
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?
When we analyze how powerful a character is in combat, it is a product of offense and defense: How much damage can he do and how long can he do it?
Let us say that each PC has a “combat rating” CR. His CR is a function of his “expected damage output per action” (k, for kill power), multiplied by the number of actions he can be expected to perform before he is neutralized, or, his staying power (T, a time variable).
CR= (k)(T)
Now, look at what happens when a character advances in level. His effective offense increases linearly, true enough; and so does his staying power (whether by increases in hit points, AC, defenses, etc.)
His kill power advances linearly: k = mx+b; where m is the rate of offensive advancement, x is the number of levels he has gained, and b is his starting point.
His staying power advances linearly: T = dx+a, where d is the rate of defensive advancement, x is (again) the number of levels he has gained, and a is his starting point.
It is also very important to note that m and d, the rates of advancement, are relative to b and a— these increments are relative to the starting values at 1st level.
Thus,
CR=(k)(T)
And substituting for our linear advancement rates we see
CR= (mx+b)(dx+a)
As you can see, the product of two (non-zero) linear advancement rates must be quadratic:
CR=(mdx2)+(mxa)+(dxb)+ab
Specific Analyses
Let’s begin at 1st level, where x is 0—the character has not yet advanced.
Most of the equation drops out, leaving us with
CR = ab
Remember, a and b are our baseline, starting 1st level defense and offense levels. We could arbitrarily set them both to “1” just as a shorthand for “about the amount of offense and defense we can expect from a 1st level character.”
In which case, at 1st level, CR=1.
We know for sure that a 1st level fighter has more staying power (in both hit points and AC) than a 1st level wizard; but we also know that a properly prepared wizard can inflict many more kills than the fighter. Sleep, color spray, and burning hands, even at 1st level, is a lot of “kill power” for the wizard.
We can all probably agree that the design of the wizard is to trade offensive smack-down for his relative fragility.
If the classes are balanced, however, we can also assume that the product ab is probably about the same, even if a and b are reversed and/or adjusted from class to class.
Notice also that the staying power of the wizard is also directly related to his spells per day. When he’s out of applicable spells, he’s effectively neutralized with respect to the fighter. You don’t have to kill a wizard to “kill” him in terms of his combat effectiveness.
Levelling Up
Let’s look at what happens when these two level up. In 3e, their hit points double—doubling their staying power. And, in the case of the wizard in particular, his offense just about doubles—his burning hands goes from 1d4 to 2d4. The fighter, of course, doesn’t enjoy this doubling effect—his BAB goes up, of course, and he may be able to make an extra attack through the use of feats, but he certainly doesn’t get to double the base damage output of his longsword. By 2nd level, he’s already falling behind.
Flattening the Curve
The power curve will never be linear—you don't hear many references to the power line.
But we can change its shape and try to smooth it out by changing the underlying linear rates of advancement of both offense and defense (m and d, respectively) and we can change our starting baselines (a and b) to shift the curve into a more suitable range of play.
The attached chart shows several options for changing the power curve.
Series 1 shows the current power curve of 3e. The starting values a and b are both set to 1 and the linear advancement rate of both offense and defense m and d are also set to 1. All variables are set to 1, so power scales 1, 4, 9, 16, 25, etc.
Series 2 shows a proposed power curve. In this example, we start 1st level character with twice as much staying power as before; but the linear rates of advancement for offense and defense are dialed down to 2/3 and ½, respectively.
Series 3 is identical to Series 2, but offense and defense both increment at a ½ rate.
Series 4 is the “rat bastard” power curve—1st level characters do not receive any additional HD, and both the offense and defense advancements are dialed down to ½. Notice that the shape of the blue Series 4 curve is the same shape as the green Series 3 curve, but there is a baseline shift.
