Lanchester’s Laws

From- http://games.groups.yahoo.com/group/sfconsim-l/message/45945

Lanchester’s laws are concerned with balancing military
strength of various units.

If one is dealing with ancient combat (i.e., pre-gunpowder)
Lanchester Linear Law applies, the common sense
“relative strength is proportional to number of combatants”.
(e.g., if the unit Alfa has twice as many men as unit Bravo,
unit Alfa is twice as strong)
It applies because one man is engaging in combat with
only one hostile man. If each pair of combatants kills
each other, the number of men remaining after the battle
is the larger army minus the smaller army.

But post gunpowder, Lanchester Square Law applies.
The relative strength is proportional to the *square*
of the number of combatants.
If unit Alfa has three times as many men, it is 3^2 = 9
times as powerful.

This is because with gunpowder combatants can engage
more than one hostile and come under attack from
more than one hostile.

Unit Alfa is concentrating three times as much firepower
on unit Bravo compared to Bravo’s firepower. And as
important, unit Bravo’s firepower is being diluted over
three times as many targets.

The number of units remaining would be
R = sqrt( a^2 – b^2)

If Alfa’s rifles are twice as efficient as Bravo’s,
if the two units are of equal size, Alfa will win.
But the Square law makes it easy to overcome the
efficiency with mere numbers. If Bravo has three
times as many units, they will win even with
Alfa’s advantage in weapons.

Specifically, if Bravo is three times bigger, it
has a strength nine times that of Alfa. Alfa’s
weapons reduces that strength to “only” 4.5,
so Bravo still destroys Alfa.

As Dr. Paulos put it, it takes an N-squared-fold increase in
quality to make up for an N-fold increase in quantity.
That’s a tall order.

Lanchester Laws do not take into account many other
important factors, but they can come in handy when setting
up the cost and production rate of different unit types.

One thought on “Lanchester’s Laws

  1. ….what a weird thing to discuss, but; okay:

    http://robynpaterson.com/?p=1532

    >If each pair of combatants kills each other, the number of men remaining after the battle is the larger army minus the smaller army

    ….which assumes individual troops of equal ability. If Alfa is an army of Zulus, and Bravo are pygmies, this formula doesn’t (and didn’t) hold true. The superior combatant can eliminate a disproportionate number of enemy, which changes the formula. If Alfa is a batallion of Zulus, and Bravo a batalion of extraterrestrial killing machines with stinger tails and acid for blood, well…. balancing “point cost” for any sort of game once more becomes difficult.

    Or for another example, picture yourself vs agroup fo 5 year olds. Sure, they’re gonna get you in the end, but you could probably take two, maybe three of ’em first.

    >with gunpowder combatants can engage more than one hostile and come under attack from more than one hostile

    Eventually. Remember the first rifles were good for one shot, and at the effective range they had; after that shot they become right nice clubs for the duration of the engagement. (There’s a reason pointy bits were stuck on the end.) Provided they didn’t kill YOU when you fired the damned thing.

    ….and looking at the comments to the original article, the REAL advantage of gunpowder wasn’t cost effectiveness: early gunpowder weapons were expensive; OR amount of training. (It’s a LOT easier to use a bow or crossbow than an old school black powder weapon.) The REAL advantage was that a gun could put a bullet through armour. Which is why you notice a correlation between mass produced rifles, and a decline in production of battlefield armour. (Although officers would occasionally wear breastplates; both as ornamentation, and ‘cos they’d stop a pistol round.)

    >If Alfa’s rifles are twice as efficient as Bravo’s, if the two units are of equal size, Alfa will win. But the Square law makes it easy to overcome the efficiency with mere numbers. If Bravo has three times as many units, they will win even with Alfa’s advantage in weapons.

    NOW we’re cookin’…. although there are a LOT of variables that fall under the “more efficient” category, and each brings with it a separate bundle of circumstance. Are you referring to rate of fire? Accuracy? Range? ROF was the idea behind the assault rifle: you can make a person deadly with a minimal amount of training…. but that was offset in actual combat: during ‘Nam it was estimated the US troops made one successful hit for every 250 rounds fired! So the faster rate wasn’t as big a benefit as you’d expect. (Today, US rifles can only fire 3 round bursts, not full auto.)

    >As Dr. Paulos put it, it takes an N-squared-fold increase in quality to make up for an N-fold increase in quantity. That’s a tall order.

    ….unless you bring grendaes. Those sorta even things up in a hurry. That’s the other problem with trying to simplify things like this; you miss out on a lot of weird bits…. like grenades, and shock, and surprise, and home field advantage. It’s generally accepted that an attack on a defended position favours the defenders 10 to 1. That is, for every one defender dead it costs the attacker 10 troops.

    Don C.

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