CR, EL, and Lanchester's Law

In D&D 101: A lesson in fun, Faerl'Elghinn makes a number of assertions about numbers in combat:

Faerl'Elghinn said:

First of all, the whole "EL" concept to me seems ridiculous. How can it not be exactly twice as difficult to face twice as many monsters? The answer is that it is objectively twice as difficult to face 2 monsters as it is to face one. This can be demonstrated through the staggeringly complex equation: 1+1=2. The math doesn't lie.

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First, of course, Faerl'Elghinn ignores the fact that EL is on a log scale; it's not linear.

Second, though, is the fact that doubling the number of troops in a force doesn't double its strength; it quadruples it. This is known as Lanchester's Law:

John Allen Paulos said:

Although usually couched in terms of differential equations (the context in which I first came across it), Lanchester's Law can be paraphrased as follows: "The strength of a military unit — planes, artillery, tanks, or just soldiers with rifles — is proportional not to the size of the unit, but to the square of its size."

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Amusingly, if EL is on a log scale, and combat strength is exponential, then doubling the number of troops/monsters should double the EL.

It's been a while since I've worked with Logs so forgive me if I'm off a bit. I think the military might scale might increase exponentially and be charted on an x-y axis in a shape similar to a capital "J". The CR/EL function, however, I think can be more effectively pictured as the upper half of the letter "C". It rises quickly in the beginning, but incurs diminishing returns over time.

Also, keep in mind that the CR of a creature is not tied so much to overall strength as it is directly tied to the amount of experience awarded. The need to plot the EL of an encounter vs. the average level of the party muddys things up a bit.

Leaving the CR/EL system out of it for a moment, Faerl'Elghinn's logic is still unsound. He forgets the concept of synergy.

To wit - say we have one ogre. An encounter with two ogres that you face one after the other without a break will be twice as hard - you will have to suffer twice as many attacks, take twice as much damage, and deal out twice as much damage before you are through.

But when you face the two ogres together, you get synergy - the ogres can now flank you, for example. With two, one of them can take a special manuver like trip and the other can take advantage of it. One can grapple you and then hold you for the other to punch, and so on. The whole is equal to more than the simple sum of it's parts, because with more creatures available, they have more options available to them.

Umbran said:

Leaving the CR/EL system out of it for a moment, Faerl'Elghinn's logic is still unsound. He forgets the concept of synergy.

To wit - say we have one ogre. An encounter with two ogres that you face one after the other without a break will be twice as hard - you will have to suffer twice as many attacks, take twice as much damage, and deal out twice as much damage before you are through.

But when you face the two ogres together, you get synergy - the ogres can now flank you, for example. With two, one of them can take a special manuver like trip and the other can take advantage of it. One can grapple you and then hold you for the other to punch, and so on. The whole is equal to more than the simple sum of it's parts, because with more creatures available, they have more options available to them.

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A very good point. Another thing to keep in mind about encounters with multiple creatures is that it occasionally makes for greater efficiency in xp awards for the players. In the example given with the two ogres, a wizard pc can possibly fireball both of them getting more bang for his spell use. If the ogres were encountered separately then the wizard would have to expend another spell (theoretically) to gain the same amount of xp. I have found that numerous encounters with small numbers of creatures instead of large battles will exhaust more party resources and is therefore more potentially challenging than a big fight on many occasions. I still think the EL system needs work though. There are too many encounter circumstances that cannot be figured into a monster's CR to have a real reliable measure of true EL. This is why I do not really use the EL stuff. I do what I have done since the old Basic D&D days- know my party's capabilities and design accordingly.

I take Lanchester's Law with a large handful of salt when applied to D&D's Challenge Rating/Encounter Level system.

Armour Class, Damage Reduction and other aspects of the game really muck things up.

10,000 kobolds still wouldn't be able to take out a Tarrasque, for instance.

Cheers!

MerricB said:

I take Lanchester's Law with a large handful of salt when applied to D&D's Challenge Rating/Encounter Level system.

Armour Class, Damage Reduction and other aspects of the game really muck things up.

10,000 kobolds still wouldn't be able to take out a Tarrasque, for instance.

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I don't consider making the system model something that it wasn't ever intended to represent as a major failure. This is about on par with arguments about how all elves will be 50th level. The experience system in made for PCs. The encounter level system is made for PCs.

As was pointed out in the other post, another factor that messes with the equation is area attacks and other effects that affect multiple enemies at the same time. Four 1st-level goblin warriors are not sixteen times as difficult to fight as one for a wizard with fireball or even sleep. It is more difficult, certainly - more goblins means that it is more likely that one will win initiative and grapple the wizard, more will succeed at saving throws, the wizard may not be able to target all of then with a single spell, etc - but perhaps not sixteen times as difficult.

Psion said:

I don't consider making the system model something that it wasn't ever intended to represent as a major failure. This is about on par with arguments about how all elves will be 50th level. The experience system in made for PCs. The encounter level system is made for PCs.

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Bad example... imagine if the PCs took on 1,000 kobolds compared to 1 Tarrasque. It's not hard for the PCs to be virtually invulnerable against the kobolds whilst the Tarrasque stomps the PCs every time.

There are many low-level creatures that are ignorable by high-level PCs, regardless of what "Encounter Level" might say. This doesn't mean that the EL system doesn't work; but Lanchester's Law doesn't take into account the wide disparity in power that can apply in D&D.

Cheers!

mmadsen said:

Second, though, is the fact that doubling the number of troops in a force doesn't double its strength; it quadruples it. This is known as Lanchester's Law.

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Actually, the law I cited is know as Lanchester's Square Law; there's also a Lanchester's Linear Law.

Lanchester's Square Law models modern aimed-fire combat -- not modern barrage fire or ancient hand-to-hand combat -- where the rate of enemy attrition is proportional to the number of friendly combatants, and the rate of friendly attrition is proportional to the number of enemy combatants; e.g. each combatant has a 10% chance of killing an enemy per round of combat.

If a force of 100 combatants faces another force of 100 combatants, and each combatant disables 0.1 enemies per turn, then each force will have 100 combatants, then 90, then 81, and so on, until both have 0 (more or less). If force A has 200 combatants, and force B has 100, then the losses aren't symmetrical: 200 and 100, then 190 and 80, then 182 and 61, then 176 and 43, then 172 and 25, then 169 and 8, then 168 and 0.

A naive linear model would have predicted that 200 vs. 100 would have ended up 100 vs. 0, not 168 vs. 0. The smaller force loses offensive capabilities faster than the larger force, and this spirals dramatically.

Again, this is only for combat where the rate of attrition is proportional to the number of attackers. Obviously, in melee combat, it's not always the case that every attacker can target a defender. If Horatio holds the bridge, he fights a series of one-on-one fights. This follows Lanchester's Linear Law -- 200 vs. 100 does turn into 100 vs. 0 if the battle is just a series of one-on-one combats.

In barrage combat, the rate of attrition isn't proportional just to the number of attackers but to the number of defenders too. If you cram enough defenders into an area, one fireball can kill all of them.

MerricB said:

Bad example... imagine if the PCs took on 1,000 kobolds compared to 1 Tarrasque. It's not hard for the PCs to be virtually invulnerable against the kobolds whilst the Tarrasque stomps the PCs every time.

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This should not come as any surprise to you. The CR/EL system explicitly states that more than 12 creatures are not well modeled by the system, and if it takes more than 12 of a creature to challenge the party, they should no longer be considered a challenge. A group of PCs that has a fair chance against the Tarrasque should expect kobolds to be a non-issue and the DMG says as much.

IOW, you are pointing out limitations to the system that the DMG has already pointed out for you.