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

[Wulf Ratbane]

FORK: Quadratic XP Advancement

WARNING: Potential thread-fork. And it's also an idea in infancy, looking for "thought fertilizer."

3rd Edition
XP Awards are not fixed, but rather, relative to the quadratic power ratio (chi/rho)
XP Required to Level = 1000 * current level

Proposal
XP Awards are fixed at (CR)^2 * 300
XP Required to Level = 1000 * current level ^ 2

Aside: Using 300 as the base XP award is an interesting choice. 300 is divisible by 2, 3, 4, 5, and 6, for a wide range of "less than 1st Level" options. I went with 300 because it is a number familiar to 3e, but if the numbers are too large for your tastes, consider using 60.

Level, XP Award of Level-Equivalent Encounter, XP Table
Someone else may be able to format this into a table. My vbcode-fu is weak.

These numbers are based on a 4-person party, and 13 1/3 encounters of your level to advance.

1/300/0
2/1200/1000
3/2700/5000
4/4800/14k
5/7500/30k
6/10800/55k
. . .
20/120000/2470000

Using the Numbers
As we know, 4e will allow you to "build" encounters based on a total XP value, and by "purchasing" monsters towards this total.

For example, in 4e, a "moderate" encounter for four 1st-level PCs would be four Orcs. Therefore, four orcs must be worth 300 XP. The XP value of a single orc is 75xp.

We also know that a "moderate" encounter could be twenty goblins; therefore a goblin is worth 15 XP.

We could then construct a "moderate" encounter from 10 goblins (10x15=150) and two orcs (2x75=150).

After 40/3 of these encounters, the party has earned (in total) = (40/3)(300) = 4000 xp, or 1000 xp per PC. (Ding! Level up.)

Caveat
These numbers will only hold if the advancement curve, the XP award curve, and the power curve all match.

(This system would assume that the PCs could handle 20 goblins at 1st level, and 80 goblins at 2nd level. We know for certain that power curve is being dramatically lowered.)

 

[mmadsen]

Using the Numbers
As we know, 4e will allow you to "build" encounters based on a total XP value, and by "purchasing" monsters towards this total.

For example, in 4e, a "moderate" encounter for four 1st-level PCs would be four Orcs. Therefore, four orcs must be worth 300 XP. The XP value of a single orc is 75xp.

We also know that a "moderate" encounter could be twenty goblins; therefore a goblin is worth 15 XP.

We could then construct a "moderate" encounter from 10 goblins (10x15=150) and two orcs (2x75=150).

After 40/3 of these encounters, the party has earned (in total) = (40/3)(300) = 4000 xp, or 1000 xp per PC. (Ding! Level up.)

This system treats experience-per-monster as a constant, so defeating four orcs at once is worth no more than defeating four orcs, one at a time. Experience is linear with respect to monsters, when it should increase with the square of the number of monsters, just like combat-power.

 

[Wulf Ratbane]

This system treats experience-per-monster as a constant, so defeating four orcs at once is worth no more than defeating four orcs, one at a time. Experience is linear with respect to monsters, when it should increase with the square of the number of monsters, just like combat-power.

I'll be very surprised if 4e accounts for that. The description of fixed XP awards and encounter building so far doesn't give any indication that it will.

Am I missing something?

EDIT: I just can't imagine that the fixed XP awards / encounter building advice will, at any point, recommend to the DM to start performing quadratic equations based on the number of monsters he used.

 

[Wulf Ratbane]

This thread has not been particularly active, but my brain has been. And it's starting to hurt. Time for a braindump.

I'm still unable to completely (ok, let's be frank, remotely) incorporate Lanchester's Laws into the "Encounter Building" model.

Let us say that for any given party of four PCs, we can challenge them equally with either four equally-matched brutes, or eight lesser mooks. Lanchester says that an N-fold increase in numbers must be countered with an N-squared increase in quality. From this we can simply deduce that a single PC, or a single brute, must have four times the quality of a single mook.

We also know that the combat/encounter value of four brutes is the same as eight mooks-- both of these encounters are equal in difficulty. If the Encounter Design system says that you may buy either four brutes or eight mooks from the same pool of 1000 xp, for example, we can deduce that a brute costs 250 xp and a mook costs 125 xp.

So it would seem elementary that you could use your 1000 xp to build an encounter with TWO brutes (250x2) and FOUR mooks (125x4), right?

Not exactly. According to Lanchester's Laws, the combat value of this heterogenous group is NOT equivalent to either 4 brutes or 8 mooks-- it's actually a little bit stronger than either of the homogenous groups.

If a mook has a "combat quality" rating of 1, then 8 mooks have a "combat quality" of 8^2 (unit size) x 1 (their average quality) = 64. Similarly, four brutes, each with an individual "combat quality" of 4, have a total combat quality of 4^2 (unit size) x 4 (their average quality) = 64.

However, two brutes and four mooks (6 units) would have a total combat quality of 6^2 x [(4+4+1+1+1+1)/6] = 72, and increase in "difficulty" of 12.5%.

The main takeaway from Lanchester's Laws is that numbers count for more than quality. To say that five PCs can handle twenty goblins (a fourfold increase in numbers) is to say, in effect, that PCs are sixteen times as effective in combat as goblins. And if orcs are the brutes to our goblin mooks, what must an orc statblock look like?

Again, I have a hard time envisioning any kind of statblock for goblins, orcs, or the PCs that has any resemblance to the D&D that I know.

And that is before I have even begun to consider that a single "boss" monster designed to be on an even footing with five PCs must have twenty-five times their power.

To say that I am eager to see how WoTC solves this problem would be an understatement.

There's a very good paper describing Lanchester's Laws at the link below:

http://arxiv.org/PS_cache/math/pdf/0606/0606300v1.pdf

Be warned that it is sometimes heavy reading (especially if you are already intimidated by this thread, which at times I have been myself). However, equations aside, it does have some very readable and very enjoyable explanations including the terms "a constant bloody melee" and "Never split the party."

 

[Baduin]

The problem is that Lanchester model cannot be applied to the real combat, especially ground combat. It also cannot be applied to D&D combat.
It simply disregards the question of frontage. In hand to hand combat there is a limited number of opponents which can engage you, especially if you are a part of formation. In Peru, 200 Spanish soldiers could defeat an Inka army of any size - unless they grew to tired. Elite armies of 30 thousand Indians were simply destroyed by them.

http://www.angelfire.com/ga4/guilmartin.com/Edge.html

As for D&D - the difference is that Lanchester model assumes that if you are 2 times as good as your opponent, in the same unit of time you can kill 2 of them, and they can kill 1 of you. It is as if all fighters were exactly equal and the only difference were the hit points.

 

[Cheiromancer]

Wulf- I'm not quite sure that the average quality of a mixed group is just the mean of the individual qualities. Suppose that each creature differed only in hit points, and did an average of, say, 1 hp/round.

A mook has 10 hp say. If 2 mooks are an even match for one brute, then the brute has 40 hp (as you calculated).

8 mooks have 80 hp and do 8 hp per round. That's 8 x 8 = 64 times as much as a single mook.

4 brutes have 160 hp and do 4 hp per round. That's 16 x 4 = 64 times as much as a single mook.

However, 2 brutes and 4 mooks do 6 hp/round and have 160 hp. That's 96 as much as a mook. 50% the power of a more homogenous group!

 

[Wulf Ratbane]

Wulf- I'm not quite sure that the average quality of a mixed group is just the mean of the individual qualities. Suppose that each creature differed only in hit points, and did an average of, say, 1 hp/round.

A mook has 10 hp say. If 2 mooks are an even match for one brute, then the brute has 40 hp (as you calculated).

8 mooks have 80 hp and do 8 hp per round. That's 8 x 8 = 64 times as much as a single mook.

4 brutes have 160 hp and do 4 hp per round. That's 16 x 4 = 64 times as much as a single mook.

However, 2 brutes and 4 mooks do 6 hp/round and have 160 hp. That's 96 as much as a mook. 50% the power of a more homogenous group!

I'd like to engage you on this but I can't follow your line of reasoning.

And something's off with your conclusion there regardless. 50% more power than... or 150% the power of... But not 50% the power of...

If you edit it in with some common terminology...? What are you considering a unit? (1 creature or 1 hit point?) Which term is your power (aka quality)?

 

[mmadsen]

I'm still unable to completely (ok, let's be frank, remotely) incorporate Lanchester's Laws into the "Encounter Building" model.

As I said before, if the new encounter-building system uses a constant experience-cost per monster, then it's inherently linear with respect to the quantity of monsters -- whereas Lanchester's model is naturally quadratic with respect to the quantity of combat units.

I don't see any way to reconcile such an encounter-building system with Lanchester's Square Law. Either the encounter-building system has to move away from costs per monster, or the combat system has to move away from directed attacks.

 

[Wulf Ratbane]

I don't see any way to reconcile such an encounter-building system with Lanchester's Square Law. Either the encounter-building system has to move away from costs per monster, or the combat system has to move away from directed attacks.

There's a third, more likely, option: Hand wave it.

Designate a Nth-level-apprpriate "band" of creatures, with power ranging from 1/4N to 4N, allow the DM to "buy" his encounter warband within that range-- which will naturally limit the number of units within a reasonable distance of 1:1 with the PCs-- and just assume that anything within +/- 25% of the appropriate power is "close enough."

 

[Cheiromancer]

I'd like to engage you on this but I can't follow your line of reasoning.

And something's off with your conclusion there regardless. 50% more power than...

Oops. Yeah, 50% more power.

I'm thinking that quality would be the product of damage dealing capacity and staying power. In this abstract example (where each creature's AC, attack bonus and damage/attack is set so the damage dealt is 1 hp/round) the differences between different creatures can be set solely in terms of hit points (since multiplying by 1 doesn't change things). I guess that's the quality of each individual creature.

Now I'm thinking that a band of creatures can be assigned a number of hit points just by summing the individual contributions- that band of creatures will also do a certain number of hit points damage per round; in this toy example, at the rate of 1 hp/round/creature.

The unit I'm using is the mook. I figure that if 8 mooks gang up on 1 mook, they will lose 1/64 of their total hit points. They kill him in 1.25 rounds, and he does 1.25 damage in that interval. 1.25/80 = 1/64. Similarly for 4 brutes against a mook. But a mixed group of 2 brutes and 4 mooks loses only 1/96 of their hit points when they take down a mook.

Does that make more sense?

The power of a band diminishes over time, as its members drop. This is the death spiral familiar to those who have experienced a TPK. But I'm not considering that at this point.

As I said before, if the new encounter-building system uses a constant experience-cost per monster, then it's inherently linear with respect to the quantity of monsters -- whereas Lanchester's model is naturally quadratic with respect to the quantity of combat units.

I don't see any way to reconcile such an encounter-building system with Lanchester's Square Law. Either the encounter-building system has to move away from costs per monster, or the combat system has to move away from directed attacks.

If the number of creatures in an encounter is always equal to the number of characters, then the distinction becomes less important. Or if extra monsters wait in line to engage the hero, or something. But if you want to fiddle with the number of combatants the math won't work out properly.