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

[mmadsen]

There are two ways to flatten the power curve. The first is to heavily stack the first level (or base character).

Ah, but that doesn't play out the way you'd think if you're working with an exponential curve. If power doubles with every level, then "stacking" first level just feels like starting off at third level, and the jump to fourth still feels like a doubling in power. (Only now you have no way to "really" start at first level.)

The second method would seemingly be the 'drop off'. This would be a sort of tapering of the 1st/2nd Edition design (though not as obviously abrupt) whereby the actual power of a single level 'unit' is reduced over time.

The current step-up in power is supposed to be exponential, with a doubling at every level, and the current EL system is predicated on that.

We don't need a discontinuous "drop off" at some point to flatten the curve. First, we could keep the exponential curve but reduce the "interest rate" to where combat-power doubles every other level.

Or we could flatten the curve to a polynomial of fourth, third (cubic), or even second (quadratic) degree. In fact, if you look at the current curve for "brute" Fighters -- see the Google Spreadsheet -- they don't advance anywhere near exponentially; their curve is already quadratic, at least without increasing equipment bonuses over time.

 

[Mustrum_Ridcully]

It seems that monsters will have TWO numbers describing their power: Level AND fixed XP. The level of the monster is the middle of the range of PC levels he can meaningfully engage. XP is the overall danger posed by the monster.

In effect, there will be monsters of level 5 with high XP, designed to fight 5 level party by itself, and level 5 monsters with low XP, designed to fight level 5 party in groups of 10. But both will have BAB high enough to be able to hit the fighter, at least from time to time, and AC high enough to avoid being hit, at least from time to time. Level 5 minion dies after 1 hit, level 5 boss can survive 10 hits, but both are designed to fight level 5 PCs.

That looks like an interesting solution, too. I already got the impression that some monsters (like Dragons) would have more abilities to use per round than usual - these are probably boss monsters, that have to emulate the ability of 4-5 minions of their level. (Increasing Hit Points is only one way of getting them to be a suitable foe for 5 PCs. And the Dragon certainly seemed complex)

 

[Upper_Krust]

Hi mmadsen!

Ah, but that doesn't play out the way you'd think if you're working with an exponential curve. If power doubles with every level, then "stacking" first level just feels like starting off at third level, and the jump to fourth still feels like a doubling in power. (Only now you have no way to "really" start at first level.)

Just like an Ogre, unless we 'Savage Species' him.

The current step-up in power is supposed to be exponential, with a doubling at every level, and the current EL system is predicated on that.

As for the doubling of power, that won't happen at every level where each level gains roughly the same total benefits.

We don't need a discontinuous "drop off" at some point to flatten the curve. First, we could keep the exponential curve but reduce the "interest rate" to where combat-power doubles every other level.

Indeed, in fact this increasingly looks like how 4E will handle it. Though I fail to see the benefit.

Or we could flatten the curve to a polynomial of fourth, third (cubic), or even second (quadratic) degree. In fact, if you look at the current curve for "brute" Fighters -- see the Google Spreadsheet -- they don't advance anywhere near exponentially; their curve is already quadratic, at least without increasing equipment bonuses over time.

Is there a variant of that table which takes average equipment bonuses into consideration?

 

[mmadsen]

As for the doubling of power, that won't happen at every level where each level gains roughly the same total benefits.

Ah, but it will, UK, or it will come close, because the math is sneaky that way.

If a Fighter gained nothing at all per level except his one extra hit die, then his combat power would increase linearly. If he improves (linearly with level) just one of the other combat-power variables -- to-hit, damage, or armor class -- then his overall combat power increases quadratically (x²). If he improves three of the four variables, his combat-power increases cubically (x³). If he improves all four variables, his combat-power increases even faster (x⁴).

Then we need to combine that with the sneaky math around armor class, where a linear increase in armor class yields a much faster increase in combat-power, if we assume low-level opponents.

Is there a variant of that table which takes average equipment bonuses into consideration?

I just added another sheet with 3E NPC Fighter stats -- and with iterative attacks calculated, all assuming a 1st-level NPC Fighter as the baseline (and the default opponent).

Addendum: A 20th-level NPC Fighter has more than 4,000 times the combat-power of a 1st-level NPC Fighter.

 

[Cheiromancer]

Addendum: A 20th-level NPC Fighter has more than 4,000 times the combat-power of a 1st-level NPC Fighter.

But if power doubled every level, it should be about 500,000 times, right? 4000 times the combat power of a 1st-level fighter should occur at 13th level.

 

[mmadsen]

But if power doubled every level, it should be about 500,000 times, right? 4000 times the combat power of a 1st-level fighter should occur at 13th level.

As you point out, the NPC Fighter's combat-power progression is not exponential -- or at least it doesn't double every level. It is more than quadratic though; it's almost cubic.

Addendum: A little curve-fitting in Excel shows that the combat-power curve can be nicely approximated by the following ugly third-degree polynomial:

y = 0.4641x³ + 3.5233x² - 47.917x + 77.565

Less accurate is this power curve:

y = 0.2454x^3.1642

 

[Wulf Ratbane]

Less accurate is this power curve:

y = 0.2454x^3.1642

I think your numbers are just a little bit off. Obviously what they're shooting for is the basic grid unit times X to power of pi. Only they converted the grid from inches to decimeters, of course.

It makes perfect sense.

 

[Sepulchrave II]

Just to say that I find this thread fascinating, and I hope it doesn't die.

 

[Cheiromancer]

As you point out, the NPC Fighter's combat-power progression is not exponential -- or at least it doesn't double every level. It is more than quadratic though; it's almost cubic.

Addendum: A little curve-fitting in Excel shows that the combat-power curve can be nicely approximated by the following ugly third-degree polynomial:

y = 0.4641x³ + 3.5233x² - 47.917x + 77.565

Less accurate is this power curve:

y = 0.2454x^3.1642

Well then, if PCs don't have an exponential power curve, then neither should monsters. And if the power curve is too steep at high levels, the higher order factors should be scaled back; i.e. the cubic term makes the curve rise too high.

I think that a quadratic power curve would be ideal.

 

[mmadsen]

In my earlier analysis, I compared Fighters of levels 1 through 20 to a static foe of level 1. This was straightforward, and it made it easy to figure out just how many minions a hero could cut through, but it didn't capture the full power of higher-level Fighters in a higher-level context.

More recently -- see the updated Google Spreadsheet -- I compared Fighters of levels 1 through 20 to a dynamic foe of one level lower. That is, I compared a 2nd-level Fighter to a 1st-level Fighter and computed a combat-power ratio. Then I compared a 3rd-level Fighter to a 2nd-level Fighter and computed another combat-power ratio.

This gave vastly different results, as combat-power typically more than doubled with level. In fact, the curve was exponential -- e^x.

Intriguing.