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

[Wulf Ratbane]

The problem is that Lanchester model cannot be applied to the real combat, especially ground combat.

Incorrect, the Lanchester model is applied to real combat in the real world by military planners, with varying degrees of complexity and success.

It also cannot be applied to D&D combat.

Of course it can.

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 fact, it does not disregard frontage at all. In any engagement where the attacking force is unable to bring its entire force to bear, you would treat it as multiple engagements with a divided force, or an engagement between asymmetrical forces: adjust the model and continue.

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

This is a classic example of asymmetric warfare: The paper that you cite spends most of its time explaining the vast technological advantage enjoyed by the conquistadors despite their vast numerical disadvantage.

It is foolish to assume that 30,000 Incas could bring to bear the entire force at once, even with missile weapons, against 200 Spaniards. You are forced into a model whereby 200 Spaniards (with a significant technological advantage in offense and defense) engage a much smaller force of Incas who are replaced at regular intervals by significant reserves. Not only is it feasible for the smaller (and asymmetrically higher quality) force to win such an engagement, they quite obviously did.

That Spanish steel weapons proved more deadly to the Incan padded armor, and particularly that Spanish steel armor proved all but invulnerable to primitive Incan weapons, raises the quality of Spanish troops to previously unseen levels with respect to their foes. At the moment I can think of no other conflict in history where two conflicting peoples were separated by so vast a technological advantage (certain Civ4 games I have played notwithstanding).

CONCLUSION

First, the Spanish advantage in the technology of war was, in fact, a vital factor in their stunning military victories. This was so, however, not just because of the steepness of the technological gradient against which the Inca armies had to struggle. That would no doubt have been the case in the long run; as we have indicated above, the Spanish technological advantage was all but overwhelming, in both weaponry and the manner in which the weapons were employed in battle. Moreover, we are not concerned purely with differences in military capability on a one-for-one basis; rather, our focus has been on the way combatant groups maintained their cohesion under stress. Cultural and religious factors are crucial components of cohesion, and these, acting in conjunction with superior technology, proved an important Spanish advantage. But while Inca defeat was probably inevitable sooner or later, its reality was, in the event, heavily shaped by the fact that Pizarro and his men understood the nature and degree of their advantage and took steps to maximize it to a degree not commonly appreciated.

. . .

Third, the shock of military contact across so vast a cultural and technological gap was so great that significant technology transfer proved impossible before the Andeans’ ultimate defeat. On occasion, individual Inca captains used captured Spanish swords and helmets to considerable effect, notably in the siege of Cusco, but that was about that.74 Some twenty years, or two generations, were needed for the indigenous populations of the Americas to absorb effectively the military technologies that might have enabled them to survive on their own cultural terms. In 1564 Spanish authorities in Peru discovered large stores of weapons secretly manufactured and stockpiled in preparation for revolt.75 Significantly, the stockpiled weapons included large numbers of pikes, the one weapon that might have enabled the Andeans to prevail against the Spanish horse. At about the same time, the Chilean Araucanians, arguably the most successful indigenous resistors of Spanish penetration, were learning the use of pikes, as well as how to breed horses and ride them in combat.

Before dismissing Lanchester out of hand I would recommend that you actually read both the paper that I linked, as well as the one you linked. With all due respect, you do not seem to have read either.

 

[Wulf Ratbane]

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.

Although I do tend to agree with you that we can drill down and try to define what makes up quality, for the purposes of applying Lanchester's above, I did not do so.

But whether we use your assumptions (which admittedly, farther upthread, are also my assumptions), or use the more abstract model (that I am using directly above), we can clearly agree that a brute is 4x the quality of a mook.

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.

It strikes me that this might create a problem. Lanchester's assumes that the effectiveness of a unit is based on its kill rate. Dealing only 1 hp of damage per round, you have given your creatures a fractional kill rate.

Alternately, we can change the base time unit to mean "As much time as it takes to kill one opposing creature."

Again, in either case, the brute has 4x the quality of the mook.

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.

(Just to make sure we are on the same page, the brutes kill him in 2.5 rounds, and receive 2.5 damage in return, which is 1/64 of their 160 hit points.)

I believe we're looking right at Lanchester's square model there. (Although, again, I think you're dangerously substituting "hit points" for "units" here. DPS <> kill rate.)

Lanchester's square model is the unit's quality, or effectiveness, times the square of their numbers.

The mook has a quality of 1. Now please note, this number "1" is not drawn from their DPS-- I am not using 1 because they do 1 damage/round. Their quality is 1 because they are the baseline. In comparison, the brute has a quality of 4.

So by Lanchester's, mooks perform at a rate = (1)(8^2), and brutes perform at a rate (4)(4^2). In both cases, 64.

But a mixed group of 2 brutes and 4 mooks loses only 1/96 of their hit points when they take down a mook.

Two brutes and four mooks have 120 hit points-- 40+40+(10x4).

This puts the rate (using your method) at 1/72, which is confirmed by Lanchester's square law that I performed above.

I just eyeballed the figures and deduced that average quality was the only way for the equation to work out-- I wasn't certain when I did it that it was the proper way to calculate mixed groups. However, as it turns out, the paper I linked above confirms this (3.3 Mixed Forces, page 4).

EDIT: Just a placeholder edit-- remind me to revisit "Drilling Down on Quality."
EDIT2: Lancaster? Fixed. I blame my buddy Lasater who's been emailing me all day.

 

[mmadsen]

That Spanish steel weapons proved more deadly to the Incan padded armor, and particularly that Spanish steel armor proved all but invulnerable to primitive Incan weapons, raises the quality of Spanish troops to previously unseen levels with respect to their foes. At the moment I can think of no other conflict in history where two conflicting peoples were separated by so vast a technological advantage (certain Civ4 games I have played notwithstanding).

Later colonial battles, like the Battle of Omdurman, involved artillery and machine guns against spears.

 

[Wulf Ratbane]

Later colonial battles, like the Battle of Omdurman, involved artillery and machine guns against spears.

Ah, of course, and surely some other British/Zulu conflicts-- but even there, and in the article you linked, the opposing forces by this time were able to adopt more modern technology (rifles) and so close the gap.

Still, I have to imagine that the Spanish armor counted more than Spanish muskets, and might overall count for more than British artillery, machine guns, and other forms of punishing offense. Western armies had largely given up entirely on armor by 1898, and so any "primitive" force able to exploit the weaknesses of artillery and rifle volleys might do significant damage in close combat.

Really I think the Spanish conquest is almost unique for its particular snapshot in military time.

 

[Cheiromancer]

Two brutes and four mooks have 120 hit points-- 40+40+(10x4).

This puts the rate (using your method) at 1/72, which is confirmed by Lanchester's square law that I performed above.

Yeah, that's a pretty bad math error. I blame the cold medicine I'm taking. Lucky that I'm giving tests today- I shudder to think what incoherent ramblings would have slipped out if I were lecturing.

Anyway, it seems that (modulo some arithmetical errors) our methods are in agreement. I'll wait until my head clears before saying too much more, though.

 

[Wulf Ratbane]

More than ever I am convinced that the solution is to rebuild both characters and monsters from the ground up. See "the challenge" several posts back. We're coming back to it-- and I am dramatically expanding the challenge.

I believe it is to our advantage to keep the process abstracted as much as possible. Applying "precision" does not give us any added security in "accuracy." The more moving parts our model has, the more places it will break down. We can all agree that D&D is infinitely more complex than we can imagine or that any model can predict. But that does not invalidate the exercise-- it should simply keep us focused on the big picture.

Assumptions/Processes:

1) We want encounters to remain "moderately difficult."

1a) I believe this means that encounters must be somewhere between "equal power" (the proverbial coin-flip) and 1/4 the power of the party.

1b) Because we are shifting to a per encounter format, with the assumption that the party tackles each successive encounter relatively "fresh," I believe that the proper ratio of party quality to encounter quality should be about 2x (although it looks like Saga is more along the lines of 3x, if starting HD are the primary indicator). Indeed, in Unearthed Arcana, p159, Andy suggests with respect to "Recharge Magic" that PCs can probably handle EL+2 encounters. Ultimately, we want the PCs to be challenged, but we want them to win. I don't think 4x is sufficiently challenging, certainly not once "per encounter" is taken into account.

2) Therefore, a Level 1 brute-- which term I am using quite incorrectly to refer to "a creature designed to meet the PCs in equal numbers," but bear with me-- should be designed with about half the quality of a single PC.

2a) We'll further assume that overall quality in fact may be derived from the product of offense and defense.

2b) We'll assume that the basic unit of offensive quality is measured in either dice of damage or actual points of damage; and the basic unit of defensive quality is measured in either Hit Dice or hit points.

2c) However, for simplicity sake, to keep things easy at 1st level, we'll measure both in dice, not points.

2d) Therefore our Level 1 brute should have 1 HD and do 1 die of damage; by extension the 1st level PC will have 2 HD and do 1 die of damage.

2e) If you prefer using points to dice, which granted is much more granular, assume that 1 die = 4 points. Therefore the PC has 8 hps and averages 4 damage, for a total quality of "32" (abstracted); the brute has 4 hps and averages 4 damage for a total quality of "16" (abstracted).

3) Now we can also design the Level 1 mook, who is designed to be used in a 2:1 ratio against the party. This creature, therefore, will have 1/4 the overall quality of a 1st level PC-- he will do 1 die of damage and have 1/2 HD.

3a) If you are using the more granular points, we know that the PCs have a total quality of "32" (abstracted) so we want the mooks to have a total quality of "8" (abstracted). We can convert this to 8 hit points, averaging 1 hp/round damage; 4 hit points averaging 2 hp/round in damage; we can even fudge it a little and call it 3 hit points, averaging 2-3 hp/round damage.

4) Now we go back and increment the PC to 2nd level in a way that makes sense, and hopefully bears some resemblance to the game we know. He gains 1 HD in defense (because everyone likes rolling a new hit die when they level up!), and he gains some (likely smaller) improvement in offense-- anywhere from +1 die (in the case of a spell based on full caster level, for example) or some portion less than one full die, all the way down to as little as a 5% increase from BAB and nothing else.

4a) Derive the new total quality of the 2nd level PC as a product of offense and defense.

4b) Derive the new total quality of a Level 2 brute-- half the quality of the 2nd level PCs-- and determine stats accordingly.

4c) Derive the new total quality of a Level 2 mook-- 1/4 the quality of the 2nd level PCs-- and determine stats accordingly.

5) Repeat this process as a complete redesign of all PC levels, and all monsters, across all levels from 1-30.

ENworld's rycanada has already performed an analysis of all 3e monsters across all CRs, giving us a "fudge table" of expected, typical, or "appropriate" values for all monsters.

Our final work should look very much the same-- the barest mechanical skeleton on which we can hang all the creative, fluffy design we want. The brute-1s we call "orcs," our mook-1s are "goblins," our brute-2 are "troglodytes" and so forth. We may even have "mook's mooks" at 1st level and beyond-- Level-1 "kobolds" designed to outnumber 1st level PCs four to one (at 1/16 the quality).

We can push the numbers around in all sorts of ways and eventually we flesh out the entire MM.

6) Now we build our new XP charts and new Encounter Design system. First, graph the power curve of the PC from 1st-30th level.

6a) Assign a fixed XP value for "A 1st level encounter" such that ~13.333 encounters, with all XP totalled and divided by 5 PCs, accumulates enough XP to make 2nd level. (There's nothing special about 13.333 encounters that I know of, other than it is probably the "average" number of encounters that "average" game groups like to tackle before levelling up-- in other words, it probably has its basis in real-world game time.)

6b) Using 300 xp as particularly utile base value, we can derive from this that 5 brutes are worth 300 xp; 1 brute is worth 60 xp. An equal encounter, 10 mooks, is also worth 300 xp; thus 1 mook is worth 30 xp. (And our kobolds, worth 15 xp.)

6c) Multiply out the baseline XP awards across the same power curve of PC power, and derive encounter XP values for 2nd-30th level. (Yes, there will be some considerable "smoothing out" of the actual values.)

6d) Derive a new XP advancement chart using these new fixed XP awards.

7) As pointed out above, using mixed groups cannot account for Lanchester's square law. However, by giving the DM some guidelines ("Use anywhere between 1/2x and 2x the size of the party, and/or monsters within 2 levels of the average PC level.") we should be able to get any encounter into a reasonable framework, and handwave any differential predicted by Lanchester. Give or take 25% seems reasonable to handwave, especially as the system has already been pre-designed with a bias in the PCs favor.

Comments?

 

[jgbrowning]

1a) I believe this means that encounters must be somewhere between "equal power" (the proverbial coin-flip) and 1/4 the power of the party.

Great thread.

I'd think that between 1/2 and 1/4 the power would be better for moderate difficulty than a coin flip. To me a coin flip would be more like "very difficult"

joe b.

 

[Wulf Ratbane]

Great thread.

I'd think that between 1/2 and 1/4 the power would be better for moderate difficulty than a coin flip. To me a coin flip would be more like "very difficult"

joe b.

Oh, I absolutely agree. I was just setting two margins, admittedly very wide. Neither of these margins is appropriate.

I have run some 50/50 encounters before, but only when they are thematically appropriate-- at the end of a campaign.

A 50/50 encounter is actually more dangerous than an encounter that is obviously more dangerous. In the latter case, it is obvious to the party that they are outmatched, and so they have strong clues that they are meant to run, not fight.

The 50/50 encounter "feels doable" and so it is particularly alluring to players.

 

[Wulf Ratbane]

Looks like I'll use 10 encounters per level instead of 40/3.

http://www.enworld.org/showthread.php?t=208172

 

[Wulf Ratbane]

No takers on the new challenge?

EDIT: Can someone explain to me how to use vbcode to insert formatted tables?