CR, EL, and Lanchester's Law

[mmadsen]

Well, as mmadsen said,

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.

Unfortunately, it seems that Lanchester never got round to naming this principle. Perhaps we could call it Lanchester's Fireball Law?

Actually, Lanchester's Linear Law is his Fireball Law. From Lanchester Equations and Scoring Systems:

It is usually said that the square law applies to "aimed fire" (e.g., tank versus tank) and the linear law to "unaimed fire" (e.g., artillery barraging an area without precise knowledge of target locations).

With aimed fire, under the Square Law, the defender's rate of attrition is proportional to the number of attackers (and their quality). With unaimed fire, under the Linear Law, the defender's rate of attrition is proportional to the number of attackers and the number of defenders (and their quality).

 

[FireLance]

Right - I'd forgotten the bit about similar combatants. Instead of a wizard against a gang of goblins, it would be a wizard against a cabal of equal wizards, all of whom are capable of casting Fireball. I can see how the Linear Law would apply in this case.

 

[mmadsen]

Quantity vs Quality

In the real world, Lanchester's Square Law has been used to argue for quantity rather than quality. If twice as many tanks are four times as effective, and three times as many tanks are nine times as effective, why bother paying for quality? A tank that's twice as good is only twice as effective. A tank that's three times as good is only three times as effective.

What this neglects is that quality is often multi-dimensional. A tank gun that's twice as accurate (that hits twice as often) is twice as effective; doubling its accuracy doubles its kill-rate. A tank gun that's twice as powerful (that kills in half as many hits) is also twice as effective. Thus, a tank armed with a gun that's twice as accurate and twice as powerful is four times as effective.

We see this in D&D monsters and characters all the time. A CR 2 creature has twice the hit points of a similar CR 1 creature, making it twice as effective -- but it doesn't stop there. The CR 2 creature is also harder to hit from its higher AC. (Depending on the attacker's to-hit bonus, it may be twice as hard to hit, but probably not.) The CR 2 creature also hits more often and does more damage when it does hit.

Doubling all four of those attributes (difficulty to hit, difficulty to hurt, ability to hit, ability to hurt) multiplies a monster's combat effectiveness by a factor of 16 -- which is more-or-less what we saw between the Goblin and the Ogre.

 

[mmadsen]

Right - I'd forgotten the bit about similar combatants. Instead of a wizard against a gang of goblins, it would be a wizard against a cabal of equal wizards, all of whom are capable of casting Fireball. I can see how the Linear Law would apply in this case.

I guess you could set up a simulation (i.e., a spreadsheet) where one side is "linear" (e.g., the Goblins, taking casualties proportional to their own numbers and the Wizards') and the other is "square" (e.g., the Wizards, taking casualties proportional purely to the Goblins' numbers).